ON TRANSFER OPERATORS FOR CONTINUED FRACTIONS WITH RESTRICTED DIGITS

O. JENKINSON, L.F. GONZALEZ, AND M. URBAŃSKI

Abstract. For I ⊂ ℕ, let Λ_{I} denote those numbers in the unit interval whose continued fraction digits all lie in I. Define the cor-

responding transfer operator L_{I,β}f(z) = ∑_{n∈I} (\frac{1}{n+z} )^{2β} f (\frac{1}{n+z} ) for Re(β) > max(0, θ_{I}), where Re(β) = θ_{I} is the abscissa of convergence of the series ∑_{n∈I} n^{−2β}.

When acting on a certain Hilbert space H_{I,β}, we show that the operator L_{I,β} is conjugate to an integral operator K_{I,β}. If furthermore β is real, then K_{I,β} is selfadjoint, so that L_{I,β} : H_{I,β} → H_{I,β} has purely real spectrum. It is proved that L_{I,β} also has purely real spectrum when acting on various Hilbert or Banach spaces of holomorphic functions, on the nuclear space C^{ω}[0, 1], and on the Fréchet space C^{∞}[0, 1].

The analytic properties of the map β ↦ L_{I,β} are investigated. For certain alphabets I of an arithmetic nature (eg. I = {primes}, I = {squares}, I an arithmetic progression, I the set of sums of two squares) it is shown that β ↦ L_{I,β} admits an analytic continuation beyond the half-plane Re(β) > θ_{I}.

1. Introduction

For a non-empty subset I ⊂ ℕ, let Λ_{I} denote the set of real numbers x ∈ [0, 1] for which the infinite continued fraction expansion

x = \frac{1}{a_{1} + \frac{1}{a_{2}+···}}

has each partial quotient a_{i} belonging to the alphabet I. Let T : [0, 1] → [0, 1] be the continued fraction (or Gauss) map, given by T (x) = \frac{1}{x} (mod 1). The set Λ_{I} is invariant under T.

Date: April 2002. The first author acknowledges the warm hospitality of the University of North

Texas, where some of this research took place. The second author was supported by the CONACYT grant no. 110864/110990. The third author was supported in part by the NSF Grant no. DMS 0100078. This paper has greatly benefited from various enlightening discussions with Oscar Bandtlow on some fine points of functional analysis. We are also grateful to Richard Sharp for some helpful comments on Theorem 5(vii).

When the alphabet I is finite, the irrationals in Λ_{I} are all badly approximable by rationals (cf. [45]). The arithmetic properties of such Λ_{I} have therefore been widely studied, with particular emphasis on the set E_{2}, corresponding to the choice I = {1, 2} (cf. [17], [8], [22]). The Hausdorff dimension of sets such as E_{2} yields insight into various problems in Diophantine approximation, particularly in connection with the Markoff and Lagrange spectra (cf. [8], [13], [25]). Further geometric measure-theoretic properties have been studied by Mauldin & Urbański [32], who extended this investigation to the case of infinite alphabets I, as part of a wider analysis of infinite conformal iterated function systems [31].

A powerful approach to these problems is provided by the thermodynamic formalism of Ruelle [40]. To a continuous function g : [0, 1] → ℝ, we associate the Ruelle transfer operator L_{I,g}, defined by the formula

L_{I,g}f(x) = ∑_{n∈I} exp [g (\frac{1}{x + n})] f (\frac{1}{x + n}) ,  (1)

whenever this sum is convergent.

The choice g(x) = g_{β}(x) = −β log |T′(x)| = log x^{2β}, for a real parameter β, is of particular importance, leading to the formula

L_{I,β}f(z) = L_{I,g_{β}}f(z) = ∑_{n∈I} (\frac{1}{z + n})^{2β} f (\frac{1}{z + n}) ,  (2)

where again if I is infinite then care is needed in the choice of β and f to ensure convergence of the sum. Specifically, we suppose Re(β) > θ_{I}, where Re(β) = θ_{I} is the abscissa of convergence of the series ∑_{n∈I} n^{−2β}, and that f ∈ C^{0}[0, 1]. In what follows, we will consider this operator acting on functions f that are at least C^{∞}, and this will additionally ensure that it has countable spectrum (see §3).

One of the reasons for the importance of L_{I,β} is its relation to Hausdorff dimension. Under appropriate hypotheses on I (see §2) there is a unique real value of β for which L_{I,β} has leading eigenvalue equal to 1, and this value is precisely the Hausdorff dimension of Λ_{I} (see [7], [31]). This characterisation has been used to approximate the Hausdorff dimension of Λ_{{1,2}} (see [8], [22], [25]). The eigenfunction of this leading eigenvalue is, suitably normalised, the density function for the natural geometric T -invariant probability measure supported on Λ_{I}. For I = ℕ this density is \frac{1}{log 2} \frac{1}{1+x}, corresponding to the well-known Gauss measure. The second eigenvalue of L_{I,β} determines the rate of correlation decay for the dynamical system T : Λ_{I} → Λ_{I} with respect to the geometric invariant measure; for I = ℕ this corresponds to Wirsing’s constant (cf. [14], [35], [52]). It can also be related to the efficiency of numerical algorithms for approximating the Hausdorff dimension of Λ_{I} (cf. [24]). The lower-lying eigenvalues (the so-called Ruelle resonances, cf. [41], [42]) determine finer mixing properties of T : Λ_{I} → Λ_{I}.

It is conjectured that the full C^{∞} spectrum of various Ruelle transfer operators associated to expanding dynamical systems (including, for example, the linearised Feigenbaum renormalisation operator [11]) is real. Empirical evidence supports these conjectures, although a satisfactory explanation for the phenomenon is lacking. If the underlying expanding map is piecewise linear, then it is easily proved that the spectrum is real. For nonlinear expanding maps where one branch dominates the other in a certain sense, Rugh [44] has shown that the spectrum of certain associated transfer operators is indeed real. However, reality of the spectrum is not ubiquitous, as demonstrated by Levin [30].

A remarkable analysis of Babenko [3] established the reality of the spectrum of L_{ℕ,1}, the Perron-Frobenius operator for the Gauss map on [0, 1]. His method is very different from the perturbative approach of Rugh, hinging instead on an algebraization of the problem by means of an integral transform. This analysis of the Perron-Frobenius operator was further elaborated by Mayer & Roepstorff [35, 36], while a significant generalisation of Babenko’s theorem to the family L_{ℕ,β}, β > 1/2, was given by Mayer [33]. The main purpose of our article is to further develop this algebraization program. It will be shown that the transfer operators L_{I,β} associated to general invariant sets Λ_{I} may also be transformed in such a way that the reality of their spectra is readily apparent. These operators will act primarily on canonical spaces such as C^{∞}[0, 1] and C^{ω}[0, 1]. For example we prove:

Theorem. Suppose I is a non-empty subset of ℕ, and β > max(0, θ_{I}). Then the operator L_{I,β} : C^{∞}[0, 1] → C^{∞}[0, 1] has real spectrum. This spectrum is a sequence of real numbers union its accumulation point at zero.

This result will follow from Theorems 2 and 3, in sections 5 and 6 respectively, where identical spectral properties are proved for L_{I,β} acting on the nuclear space C^{ω}[0, 1] and on various spaces of holomorphic functions. The scheme of proof is as follows. To each alphabet I we will associate a theta function ϑ_{I}(s) = ∑_{n∈I} e^{−ns}, a Stieltjes measure dm_{I}(s) = ϑ_{I}(s) ds, and an integral transform given by T_{I,β}φ(z) = ∫_{0}^{∞} s^{β−1/2}e^{−sz}φ(s) dm_{I}(s). The transfer operator will be shown to preserve a Hilbert space H_{I,β} = T_{I,β}(L^{2}(m_{I})) of holomorphic functions, on which its action is similar to a selfadjoint integral operator on L^{2}(m_{I}). It follows that L_{I,β} : H_{I,β} → H_{I,β} has real spectrum. Since it can be shown (Lemma 6) that H_{I,β} is densely and continuously embedded in C^{∞}[0, 1], a recent intertwining theorem of Bandtlow [4] implies that the spectrum of L_{I,β} on these two spaces is the same, thus establishing the above theorem. In the same way it will be shown that the non-essential spectrum of L_{I,β} acting on Hölder function spaces, or on spaces of C^{k} functions, is also real.

In the second part of this article we consider the analytic dependence of L_{I,β} on the parameter β. If β ∈ ℂ, with Re(β) > θ_{I}, then L_{I,β} is clearly well-defined. For such β it can again be shown (Theorem 1) that L_{I,β} is similar to an (in general nonselfadjoint) integral operator. (If I is finite, so that (2) is convergent for all β ∈ ℂ, this result requires the assumption Re(β) > 0).

We consider L_{I,β} as an element of the complex Banach space of bounded linear operators on a suitable Hilbert space. This ensures a canonical notion of analyticity for the map β ↦ L_{I,β}, equivalent to complex differentiability. The analyticity of this map in the half-plane Re(β) > θ_{I} suggests the possibility of an analytic continuation to a wider domain in the β-plane. The original result in this direction is due to Mayer [33], who analysed the case I = ℕ. Using a somewhat weaker notion of analyticity, he proved that β ↦ L_{ℕ,β} has a meromorphic continuation to the whole β-plane, the only poles being simple ones at β = (1 − k)/2, k ∈ ℤ_{≥0}.

The situation for general alphabets I is rather more recondite. The simplest situation is when I is finite; in this case θ_{I} = −∞, and the analytic continuation, in this case to the entire β-plane, is immediate. When I is infinite then β = θ_{I} is a non-removable singularity, and in general there is no analytic continuation past this point. However for certain infinite alphabets I a meromorphic continuation does exist, as described in Theorem 5. For example if I is an arithmetic progression {a + bn}, a monomial sequence {n^{b}}, or a geometric progression {b^{n}}, then there is a meromorphic continuation to the whole complex β-plane. If I is either the set of all prime numbers, or the set of numbers which are the sum of two squares, then θ_{I} = 1/2. In both these cases an analytic continuation past β = θ_{I} is possible, but there is no meromorphic extension to the whole plane. If I is a sequence of superexponential growth, then θ_{I} = 0, and the line Re(β) = 0 is a natural boundary, so that no analytic continuation is possible.

The article is organised as follows. In §2 we introduce the fundamental objects necessary for our analysis. The action of L_{I,β} on various function spaces is discussed in §3. In §4 we develop the basic properties of the integral operator and the integral transform. The conjugacy of L_{I,β} with the integral operator is established in §5, as is the reality of the spectrum for β real. In §6 this spectral property is extended to various other function spaces. In §7 we consider the analytic properties of the map β ↦ L_{I,β}. This is related (Theorem 4) to the analytic properties of a certain Hurwitz zeta function, which then yields precise results (Theorem 5) for certain alphabets I of an arithmetic nature.

2. The fundamental objects

A nonempty subset I of ℕ will be called an alphabet. We associate to I the following objects.

• The limit set Λ_{I}, defined as the set of real numbers in [0, 1] whose continued fraction partial quotients all belong to I.

• The Hausdorff dimension δ_{I} of the limit set Λ_{I}.

• The finiteness parameter θ_{I}, defined as the infimum of those real numbers θ for which

∑_{n∈I} \frac{1}{n^{2β}}  (3)

converges for all β ∈ ℂ with Re(β) > θ. If I = ℕ then Λ_{I} is the set of irrationals in [0, 1], and in general when

I is infinite Λ_{I} is homeomorphic to the irrationals (in particular it is not compact). If I is finite with cardinality greater than one, then Λ_{I} is a Cantor subset of [0, 1]. Moreover, we have 0 < δ_{I} < 1 unless I = ℕ (when the dimension is 1) or I is a singleton (when the dimension is zero).

If I is finite then clearly θ_{I} = −∞. If I is infinite then θ_{I} ∈ [0, 1/2], and the series (3) has a singularity at β = θ_{I} since it is a Dirichlet series all of whose coefficients are non-negative (cf. [2], Thm. 11.13).

To the alphabet I we further associate the following objects: • N_{I} := min I.

• The zeta function ζ_{I}(s) = ∑_{n∈I} n^{−s}, defined initially for Re(s) > 2θ_{I}, but in some cases admitting an analytic continuation to a larger domain in the s-plane.

• The Hurwitz zeta function ζ_{I}(s, z) = ∑_{n∈I}(n + z)^{−s}, defined initially for z ∈ ℂ \ (−∞, −N_{I}], Re(s) > 2θ_{I}, but in some cases admitting an analytic continuation to a larger domain in the s-plane.

• The theta function ϑ_{I}(t) = ∑_{n∈I} e^{−nt}, defined for t ∈ ℝ^{+}.

• The theta measure m_{I} on ℝ^{+}, defined by dm_{I}(t) = ϑ_{I}(t) dt.

• The right half-plane R_{I} = {z ∈ ℂ : Re(z) > −N_{I}/2}. The nomenclature is motivated by that used in classical analytic

number theory, in spectral geometry (eg. [9], [10], [50]), and in recent generalisations of Jorgenson & Lang (eg. see [26, 27]). For example if I = ℕ then ζ_{I} is Riemann’s zeta function^{1}, and the theta function is simply θ_{I}(t) = \frac{1}{e^{t}−1}.

For any β ∈ ℂ, the kernel function G_{β} : ℝ^{+} × ℝ^{+} → ℂ is defined by

G_{β}(s, t) = ∑_{k=0}^{∞} \frac{(−1)^{k}(st)^{k+β−1/2}}{k! Γ(k + 2β)} .

In fact this kernel function is related to a Bessel function of the first kind (cf. [51], p. 40) by G_{β}(s, t) = J_{2β−1}(2\sqrt{ts}). The most important property of G_{β} is that if β ∈ ℝ then G_{β}(s, t) = \overline{G_{β}(t, s)}, so that any associated integral operator is selfadjoint.

For Re(β) > θ_{I} we associate to the pair (I, β) the following objects. • The integral transform T_{I,β} defined by

T_{I,β}φ(z) = ∫_{0}^{∞} s^{β−1/2}e^{−sz}φ(s) dm_{I}(s).  (4)

• The space of integral transforms H_{I,β} = T_{I,β}(L^{2}(m_{I})). This is a Hilbert space, its inner product inherited from L^{2}(m_{I}) via the map T_{I,β}, which is easily seen to be injective and which we declare to be a Hilbert space isomorphism. It will be shown that each f ∈ H_{I,β} is a function analytic in the right half-plane R_{I}.

• The integral operator K_{I,β} defined by

K_{I,β}φ(t) = ∫_{0}^{∞} G_{β}(s, t)φ(s) dm_{I}(s).

• The transfer operator L_{I,β} defined by

L_{I,β}f(z) = ∑_{n∈I} (\frac{1}{z + n})^{2β} f (\frac{1}{z + n}) .  (5)

The main focus of our attention is the transfer operator L_{I,β}. As mentioned in §1, one of the reasons for this interest is its relation with Hausdorff dimension δ_{I}. If ζ_{I}(2θ_{I}) = ∞ (in which case we say that I is regular, as in [31, 32], where examples of irregular sets can also be found) then there is a unique real value of β for which L_{I,β} has spectral radius 1, and this value is precisely δ_{I} (see [7], [31]). The transfer operator will be studied by means of the various auxiliary objects we have introduced. The integral transform T_{I,β} resembles a Laplace-Mellin transform (cf. [26], p. 9), although the exponent β −1/2 is carefully chosen for our purposes (cf. Remark 3). The reason for the choice of kernel function G_{β} will become apparent in §5.

1Note, however, that the classical Hurwitz zeta function is defined by a summation over all non-negative (rather than positive) integers (cf. [6], p. 502). In this way our convention differs from the classical one.

3. Trace class transfer operators

As is well known, the spectral properties of the transfer operator L_{I,β} depend strongly upon the space on which it acts. In fact it will be convenient for L_{I,β} to act on several different spaces, which we now specify.

For 0 < α ≤ 1, let C^{α}[0, 1] denote the space of complex-valued α-Hölder functions on [0, 1], equipped with its Banach norm. For real β > max(0, θ_{I}), the operator L_{I,β} : C^{α}[0, 1] → C^{α}[0, 1] enjoys various Perron-Frobenius type properties. In particular it has an essential spectral radius strictly smaller than its spectral radius (see [43]).

Since z ↦ (z + n)^{−2β} extends as a C^{ω} function to the unit interval, the operator L_{I,β} in fact preserves various subspaces of C^{α}[0, 1], notably the Banach spaces C^{k}[0, 1] for k ∈ ℕ, the Fréchet space C^{∞}[0, 1], and the nuclear space C^{ω}[0, 1]. When acting on C^{k}[0, 1], L_{I,β} is again quasicompact, although its essential spectral radius shrinks to zero as k → ∞ (see [43]). In Proposition 2 it will be shown that the spectrum of L_{I,β} on both C^{∞}[0, 1] and C^{ω}[0, 1] is a countable set accumulating at zero.

L_{I,β} also preserves various spaces of holomorphic functions, and there is considerable freedom in selecting such spaces. For example, suppose D is a complex domain satisfying

(i) D does not intersect −I, (ii) \overline{D} ⊂ int (∩_{n∈I}S_{n}(D)), where S_{n}(z) := \frac{1}{z} − n, (iii) D is bounded and simply connected.

There is an abundance of such domains D. For example D = {z ∈ ℂ : |z − 1| < \frac{3}{2}\} (cf. [33]) satisfies (i)–(iii) for any I ⊂ ℕ. In certain applications it may be advantageous to choose larger domains, and the main restriction on this is the value N_{I} = min I. If I were the set of all prime numbers larger than one hundred, say, then the disc D = {z ∈ ℂ : |z| < 100} also satisfies (i)–(iii).

In general, if f is holomorphic on a complex domain D satisfying (i)–(iii), then L_{I,β}f is holomorphic on some open neighbourhood of D. A suitably chosen space of functions holomorphic on D, and satisfying some prescribed condition on the boundary of D, will then be invariant under L_{I,β}. Ruelle [39] worked with the disc algebra A_{∞}(D) of holomorphic functions which extend continuously to the boundary of D, equipped with the supremum norm. This space has since been the most common choice, although Bergman, Dirichlet, or Hardy spaces (see [12]) are also invariant under L_{I,β}, as are various more exotic (in general non-normable) spaces (cf. [38], Ch. VIII, §2,3). However, for many reasons it is preferable to work with a Hilbert space. If D is a disc of radius r, centred at c, and in addition satisfying (i),(ii) above, then a convenient choice is the Hardy space H^{2}(D) of those functions holomorphic on D and such that sup_{ϱ<r} \frac{1}{2π} ∫_{0}^{2π} |f(c + ϱe^{it})|^{2} dt < ∞. Note that any function holomorphic on an open neighbourhood of D is automatically in H^{2}(D). The inner product on H^{2}(D) is defined by (f, g) = \frac{1}{2π} ∫_{0}^{2π} f(c + re^{it})\overline{g(c + re^{it})} dt, which is well-defined since any element of H^{2}(D) extends as an L^{2} function of ∂D.

The very freedom in choosing these holomorphic function spaces means that none of them are canonical. By contrast C^{∞}[0, 1] and C^{ω}[0, 1] are both canonical, although neither of these is normable. Therefore the holomorphic function spaces may usefully be considered as intermediate spaces, on which certain properties of the transfer operator L_{I,β} are more readily apparent. The intermediate nature of these spaces means we do not strive for the utmost generality in our choice of the domain D. For simplicity we will only consider the Hilbert space H^{2}(D) and the Banach space A_{∞}(D), where D is a disc satisfying conditions (i) and (ii). When acting on either of these spaces, it will be seen that the operator L_{I,β} is trace class (and hence compact), a concept we now briefly review.

Given a compact operator L : H → H on a Hilbert space H, define the n^{th} approximation number (or n^{th} singular value) to be s_{n}(L) = min{||L − F || : F is a linear operator of rank ≤ n − 1}. Then L has a norm convergent expansion L = ∑_{n} s_{n}(L)(φ_{n}, ·)ψ_{n} for orthonormal sets {φ_{n}}, {ψ_{n}} (cf. [46], Thm. 1.4). For p > 0, the Schatten-von Neumann ideal S_{p}(H) is the collection of those operators for which ∑_{n} s_{n}(L)^{p} < ∞. The case p = 1 is of particular importance; L is said to be trace class if ∑_{n} s_{n}(L) < ∞, in which case its trace tr(L) = ∑_{n} s_{n}(L)(φ_{n}, ψ_{n}) is well-defined. Lidskii’s theorem then asserts that tr(L) = ∑_{n} λ_{n}(L), where λ_{n}(L) are the non-zero eigenvalues of L, counted with multiplicity. The Fredholm determinant for L is an entire function of z defined as exp [tr (log(I − zL))] for small z, and then by analytic continuation. Its zeros are precisely the reciprocals of the non-zero eigenvalues of L, the order of any zero being the algebraic multiplicity of the corresponding eigenvalue.

Grothendieck [19, 20] generalised the notion of trace class operators to Banach spaces B. If L : B → B can be written as L(f) = ∑_{n} l_{n}(f)f_{n}, for some l_{n} ∈ B′, f_{n} ∈ B satisfying ∑_{n} ||l_{n}||_{B′}||f_{n}||_{B} < ∞, then L is said to be nuclear. If ∑_{n}(||l_{n}||_{B′}||f_{n}||_{B})^{p} < ∞, then L is called nuclear of order p. If L is nuclear of order p for every p > 0 then it is said to be nuclear of order 0. The class of nuclear operators of order 2/3 (the Grothendieck trace class) is of particular importance, for in that case the trace tr(L) = ∑_{n} l_{n}(f_{n}) is well-defined, and independent of the sequences l_{n}, f_{n}. Moreover, for such operators Grothendieck [19] proved that the analogue of Lidskii’s theorem holds, namely tr(L) = ∑_{n} λ_{n}(L). The Fredholm determinant can also be defined for such operators L, in the same way as for trace class operators on Hilbert space, and its zeros are again related to the eigenvalues of L as above.

If ||l_{n}||_{B′}||f_{n}||_{B} = O(γ^{n}) for some 0 < γ < 1, we will say that L is exponentially nuclear (such operators are clearly nuclear of order zero). Moreover we have the tail estimate ∑_{n=N}^{∞} ||l_{n}||_{B′}||f_{n}||_{B} = O(γ^{N}), so that s_{N}(L) = O(γ^{N}), and hence ∑_{n} s_{n}(L) < ∞. Therefore an exponentially nuclear operator on a Hilbert space is of trace class.

Proposition 1. Suppose I ⊂ ℕ is non-empty, and Re(β) > θ_{I}. Let D ⊂ ℂ be a disc satisfying conditions (i) and (ii).

(a) L_{I,β} : H^{2}(D) → H^{2}(D) belongs to every Schatten-von Neumann ideal S_{p}(H^{2}(D)), p > 0, and in particular is trace-class.

(b) L_{I,β} : A_{∞}(D) → A_{∞}(D) is nuclear of order 0, and in particular is a Grothendieck trace class operator.

Proof. The assumptions on D guarantee that if f is holomorphic on D then L_{I,β}f is holomorphic on an open neighbourhood of D. In particular we may choose an open disc D′, concentric with D, and with strictly larger radius, such that L_{I,β}f is holomorphic on D′. Let r (resp. r′) denote the radius of D (resp. D′), and let c denote their common centre. We have the Taylor series

L_{I,β}f(z) = ∑_{k=0}^{∞} ψ_{k}(f)g_{k}(z),

where

ψ_{k}(f) = \frac{(L_{I,β}f)^{(k)}(c)}{k!} , g_{k}(z) = (z − c)^{k}.

Now ψ_{k}(f) = O((r′)^{−k}), since L_{I,β}f is holomorphic on D′.

In particular, if E denotes either H^{2}(D) or A_{∞}(D) then L_{I,β}f ∈ E, and moreover ||ψ_{k}||_{E′} = O((r′)^{−k}). Clearly also g_{k} ∈ E, and ||g_{k}||_{E} = r^{k}. Therefore ||g_{k}||_{E}||ψ_{k}||_{E′} = O ((r/r′)^{k}), so that L_{I,β} is exponentially nuclear. The discussion preceding this proposition then implies that L_{I,β} is (Grothendieck) trace class. □

We now consider the action of L_{I,β} on the spaces C^{∞}[0, 1] and C^{ω}[0, 1]. These are not Banach spaces, so care is needed in the definition of its spectrum, which we will always understand in the sense of Allan [1] (see also §A.1 of [4]). The spectrum of a linear operator L : E → E on a topological vector space E will be denoted by σ(L|E). Proposition 2. Suppose I ⊂ ℕ is non-empty, and Re(β) > θ_{I}. Both L_{I,β} : C^{∞}[0, 1] → C^{∞}[0, 1] and L_{I,β} : C^{ω}[0, 1] → C^{ω}[0, 1] have countable spectrum, the only accumulation point being at zero. Proof. First we consider the operator L_{I,β} acting on the space C^{ω}[0, 1]. Let Δ_{j} be a decreasing sequence of open rectangles with ∩_{j}Δ_{j} = [0, 1], such that \overline{Δ_{j}} ⊂ int (∩_{n∈I}S_{n}(Δ)) and −I ∩ Δ_{j} = ∅. For definiteness we could choose Δ_{j} to be centred at a_{j} = \frac{1}{2} + \frac{1}{j}, of height 1/j and length 1 + \frac{1}{2} \sqrt{(2a_{j} + 3)(2a_{j} − 1)}.

By Riemann’s mapping theorem there is a holomorphic diffeomorphism Φ_{j} : int (∩_{n∈I}S_{n}(Δ_{j})) → 𝔻(0, 1) sending the simply connected Jordan domain int (∩_{n∈I}S_{n}(Δ_{j})) onto the open unit disc 𝔻(0, 1). There is some 0 < ϱ_{j} < 1 such that Φ(Δ_{j}) ⊂ 𝔻(0, ϱ), the open disc of radius ϱ_{j} centred at 0. Let C_{j} denote the composition operator C_{j}f = f ∘ Φ_{j}. The Riemann mapping Φ_{j} extends to a boundary homeomorphism ∂ (∩_{n∈I}S_{n}(Δ_{j})) → S^{1} (see [23], Thm. 17.5.3). This ensures that C_{j} is a continuous operator A_{∞} (𝔻(0, 1)) → A_{∞} (int (∩_{n∈I}S_{n}(Δ_{j}))), with unit norm. The operator C_{j}^{−1} is also continuous, for the same reason.

By an argument analogous to that of Proposition 1 we see that C_{j} ∘ L_{I,β} ∘ C_{j}^{−1} is exponentially nuclear when acting on the space A_{∞}(Φ(Δ_{j})). From the continuity of C_{j} and C_{j}^{−1} we deduce that L_{I,β} : A_{∞}(Δ_{j}) → A_{∞}(Δ_{j}) is also exponentially nuclear. It follows that the non-zero eigenvalues of L_{I,β} : A_{∞}(Δ_{j}) → A_{∞}(Δ_{j}), counted with algebraic multiplicity, are precisely the reciprocals of the zeros, again counted with multiplicity, of the Fredholm determinant D_{I,β}(z) of L_{I,β}. But by Ruelle [39], the Fredholm determinant for a transfer operator can be expressed in terms of the periodic points of the underlying dynamical system as

D_{I,β}(z) = exp − ∑_{n=1}^{∞} \frac{z^{n}}{n} ∑_{{T^{n}x=x \atop x∈Λ_{I}}} \frac{∏_{i=0}^{n−1}(T^{i}x)^{2β}}{1 − (−1)^{n} ∏_{i=0}^{n−1}(T^{i}x)^{2}} ,

an expression which is manifestly independent of the particular domain Δ_{j}. Thus the spectrum of L_{I,β} : A_{∞}(Δ_{j}) → A_{∞}(Δ_{j}) is independent of Δ_{j}. Let {λ_{i}}_{i∈ℕ} ∪ {0} denote this common spectrum.

Now C^{ω}[0, 1] is topologised as the inductive limit of the injective inductive system of Banach spaces E_{j} := A_{∞}(Δ_{j}). Each L_{I,β} : E_{j} → E_{j} is nuclear, hence compact, and these operators all have identical spectrum {λ_{i}}_{i∈ℕ} ∪ {0}. A result of Bandtlow ([4], Cor. A.7.11) then allows us to deduce that the spectrum of L_{I,β} : C^{ω}[0, 1] → C^{ω}[0, 1] is also {λ_{i}}_{i∈ℕ} ∪ {0}.

To treat the operator L_{I,β} : C^{∞}[0, 1] → C^{∞}[0, 1], we first note that C^{∞}[0, 1] is the projective limit of the countable injective projective system of Banach spaces C^{k}[0, 1]. This means, by Prop. A.7.20 of [4], that the spectrum σ (L_{I,β}|C^{∞}[0, 1]) is the intersection of the spectra σ (L_{I,β}|C^{k}[0, 1]), k ∈ ℕ.

We claim that L_{I,β} : C^{k}[0, 1] → C^{k}[0, 1] is quasicompact, and that its essential spectral radius shrinks to zero as k → ∞. This will essentially follow from Ruelle [43]. Since Re(β) > θ_{I} then condition (i) on p. 176 of [43] is satisfied for the operator L_{I,β}, and also for L_{I,β}^{2}. Now L_{I,β}^{2} is a transfer operator for the infinite iterated function system determined by the collection of strict contractions T_{m,n}(z) = \frac{1}{m+ \frac{1}{z+n}},

(m, n) ∈ I^{2}. Since |T′_{m,n}(z)| ≤ 1/4 for all (m, n) ∈ I^{2}, z ∈ [0, 1], the results of [43] can be applied to L_{I,β}^{2} (the subtlety here is that if 1 ∈ I then Ruelle’s results would not apply verbatim to L_{I,β}, since T_{1}(z) = \frac{1}{z+1} is not a strict contraction). By Theorem 1.3 of [43] we see that L_{I,β}^{2} : C^{k}[0, 1] → C^{k}[0, 1] is quasicompact, with essential spectral radius bounded by c (\frac{1}{4})^{k}, where c > 0 is independent of k. If we consider both L_{I,β} and L_{I,β}^{2} as acting on the Calkin algebra (i.e. bounded operators modulo compact ones), then an application of the spectral mapping theorem shows that L_{I,β} : C^{k}[0, 1] → C^{k}[0, 1] is also quasicompact, with essential spectral radius bounded by \sqrt{c} (\frac{1}{2})^{k}.

In particular, the essential spectral radius of L_{I,β} does tend to zero as k → ∞, so that σ (L_{I,β}|C^{∞}[0, 1]) = ∩_{k=1}^{∞}σ (L_{I,β}|C^{k}[0, 1]) is countable, its only accumulation point being at zero. □

Remark 1. In the course of the above proof it was shown that the spectrum of L_{I,β} on C^{ω}[0, 1] coincides with that on A_{∞}(D), for suitable domains D. In fact it can be proved that the spectrum of L_{I,β} on each of the spaces C^{∞}[0, 1], C^{ω}[0, 1], H^{2}(D), A_{∞}(D) is identical. The proof of this is postponed until Theorem 3, when the Hilbert space H_{I,β} will also intervene.

4. The integral operator and integral transform

The following lemma is elementary, and its proof is left as an exercise.

Lemma 1. (Properties of the theta function ϑ_{I})

Suppose I ⊂ ℕ is non-empty. Let ϑ_{I}(t) = ∑_{n∈I} e^{−nt} be the corresponding theta function. Then

(a) ϑ_{I} : ℝ^{+} → ℝ^{+} is monotone decreasing, with lim_{t→∞} ϑ_{I}(t) = 0,

(b) lim_{t↘0} ϑ_{I}(t) = ∞ if and only if I is infinite,

(c) e^{−N_{I}t} ≤ ϑ_{I}(t) ≤ (1 − e^{−1})^{−1} e^{−N_{I}t} for all t ≥ 1.

The following lemma is a standard tool in Mellin transform theory, and will be used several times. Its proof follows from the simple change of variable u = at.

Lemma 2. (Mellin transform trick)

∫_{0}^{∞} t^{s−1}g(at) dt = \frac{1}{a^{s}} ∫_{0}^{∞} u^{s−1}g(u) du

for Re(s) > 0, a > 0, and any locally integrable function g for which either side of the equality is finite.

An immediate application is to the following.

Lemma 3. Suppose I ⊂ ℕ is non-empty. If Re(s) > max(0, 2θ_{I}) then

ζ_{I}(s, z) = \frac{1}{Γ(s)} ∫_{0}^{∞} t^{s−1}e^{−zt}dm_{I}(t).

Proof. Since Re(s) > 0 we have

ζ_{I}(s, z)Γ(s) = ∑_{n∈I} \frac{1}{(z + n)^{s}} ∫_{0}^{∞} u^{s−1}e^{−u}du.

Applying Lemma 2, then interchanging integration and summation (which is justified since both are absolutely convergent) gives

ζ_{I}(s, z)Γ(s) = ∑_{n∈I} ∫_{0}^{∞} t^{s−1}e^{−t(z+n)}dt

= ∫_{0}^{∞} t^{s−1}e^{−zt} ∑_{n∈I} e^{−tn}dt

= ∫_{0}^{∞} t^{s−1}e^{−zt}dm_{I}(t).

□

Remark 2. The case z = 0 in Lemma 3 yields the following integral expression for the zeta function ζ_{I}(s):

ζ_{I}(s) = \frac{1}{Γ(s)} ∫_{0}^{∞} t^{s−1}dm_{I}(t),

valid for Re(s) > max(0, 2θ_{I}).

Lemma 4. Suppose I ⊂ ℕ is non-empty, and Re(β) > max(0, θ_{I}). If φ ∈ L^{2}(m_{I}) then T_{I,β}(φ) is holomorphic in the right half plane R_{I} = {z ∈ ℂ : Re(z) > −N_{I}/2}.

Proof. Suppose φ ∈ L^{2}(m_{I}). By Lemma 1.1 of [26] it suffices to show that T_{I,β}φ(z) = ∫_{0}^{∞} s^{β−1/2}e^{−sz}φ(s) dm_{I}(s) = ∫_{0}^{∞} s^{β−1/2}e^{−sz}φ(s)ϑ_{I}(s) ds is uniformly convergent in z, on any compact subset of R_{I}. To do this we will split the range of integration, considering separately the integrals ∫_{1}^{∞} and ∫_{0}^{1}.

We will first show that ∫_{1}^{∞} |s^{β−1/2}e^{−sz}φ(s)|ϑ_{I}(s) ds < ∞, uniformly on any compact subset of R_{I}.

Since φ ∈ L^{2}(m_{I}) then ∫_{0}^{∞} |φ(s)|^{2}ϑ_{I}(s) ds < ∞, and so |φ(s)|^{2}ϑ_{I}(s) → 0 as s → ∞. By Lemma 1 (c) we deduce that |φ(s)| ≤ e^{sN_{I}/2} for all sufficiently large s. Consequently there exists K > 0 such that

∫_{1}^{∞} |s^{β−1/2}e^{−sz}φ(s)| ϑ_{I}(s) ds ≤ K ∫_{1}^{∞} s^{β−1/2}e^{−s(Re(z)+N_{I}/2)} ds,

and this integral is convergent for Re(z) > −N_{I}/2. The convergence is uniform on any compact subset of R_{I} (indeed on any half-plane Re(z) ≥ −N_{I}/2 + δ, for δ > 0).

We now consider the integral ∫_{0}^{1} |s^{β−1/2}e^{−sz}φ(s)|dm_{I}(s). We first claim that s ↦ φ(s)s^{β−1/2} is in L^{1}(m_{I}). To prove this it suffices to show, by the Cauchy-Schwarz inequality, that s^{β−1/2} ∈ L^{2}(m_{I}). An application of Remark 2 gives

∫_{0}^{∞} |s^{β−1/2}|^{2} dm_{I}(s) = ∫_{0}^{∞} s^{2 Re(β)−1}dm_{I}(s)
= ζ_{I}(2 Re(β))Γ(2 Re(β)) < ∞

for Re(β) > max(0, θ_{I}), as required.

For Re(z) > −N_{I}/2 we have

∫_{0}^{1} |s^{β−1/2}e^{−sz}φ(s)| dm_{I}(s) ≤ e^{N_{I}/2} ∫_{0}^{1} s^{Re(β)−1/2}|φ(s)| dm_{I}(s)
≤ e^{N_{I}/2} ∫_{0}^{∞} s^{Re(β)−1/2}|φ(s)| dm_{I}(s)
< ∞

since φ(s)s^{β−1/2} ∈ L^{1}(m_{I}). Indeed the convergence of this integral is uniform on all of R_{I}.

We have checked that the integral ∫_{0}^{∞} s^{β−1/2}e^{−sz}φ(s) dm_{I}(s) is absolutely convergent, uniformly on any compact subset of R_{I}, so we are done. □

The integral operator K_{I,β} will always act on the space L^{2}(m_{I}), which we now show it leaves invariant.

Lemma 5. Suppose I ⊂ ℕ is non-empty, and Re(β) > max(0, θ_{I}). The integral operator K_{I,β}

(a) preserves the space L^{2}(m_{I}),

(b) is trace class when acting on L^{2}(m_{I}),

(c) is selfadjoint in the case where β is real.

Proof. Part (a) will follow from part (b), which we now prove. The straightforward expansion

K_{I,β}φ(t) = ∑_{k=0}^{∞} \frac{(−1)^{k}t^{k+β−1/2}}{k! Γ(k + 2β)} ∫_{0}^{∞} s^{k+β−1/2}φ(s) dm_{I}(s)

is in fact not a nuclear representation. However, the Bessel function can be expanded (cf. [48], p. 8) in terms of Laguerre polynomials L_{n}^{(2β−1)} (which form an orthogonal basis for L^{2}(ℝ^{+}, ds)) as

G_{β}(s, t) = J_{2β−1}(2\sqrt{st}) = (st)^{β−1/2}e^{−t} ∑_{n=0}^{∞} \frac{L_{n}^{(2β−1)}(s) t^{n}}{Γ(n + 2β)} .

This gives the representation

K_{I,β}φ = ∑_{n=0}^{∞} l_{n}(φ)f_{n},

where

f_{n}(s) = s^{β−1/2}L_{n}^{(2β−1)}(s) , l_{n}(φ) = ∫_{0}^{∞} \frac{t^{n+β−1/2}e^{−t}}{Γ(n + 2β)} φ(t) dm_{I}(t).

A calculation (cf. [35], p. 155), again using Lemma 2, reveals this to be an exponentially nuclear representation, with the exponential decay of the ||l_{n}|| dominating the exponential growth of the ||f_{n}||. The discussion preceding Proposition 1 then implies that L_{I,β} is trace class.

Part (c) is clear since G_{β}(s, t) = \overline{G_{β}(t, s)} for β ∈ ℝ. □

5. The transfer operator on the space of integral transforms.

Theorem 1. Suppose I ⊂ ℕ is non-empty, and Re(β) > max(0, θ_{I}). Then

(a) the transfer operator L_{I,β} preserves the space H_{I,β} = T_{I,β}(L^{2}(m_{I})),

(b) if K_{I,β} : L^{2}(m_{I}) → L^{2}(m_{I}) is the integral operator with kernel function G_{β}, then

L^{2}(m_{I}) ⟶^{K_{I,β}} L^{2}(m_{I})
T_{I,β} ↓  ↓^{T_{I,β}}
H_{I,β} ⟶_{L_{I,β}} H_{I,β}

is a commutative diagram.

Proof. At several points during this proof we swap order of summation, or interchange summation and integration. On each occasion this is justified by absolute convergence, stemming from the hypothesis Re(β) > max(0, θ_{I}). Now

L_{I,β} ∘ T_{I,β}φ(z) = ∑_{n∈I} (\frac{1}{z + n})^{2β}T_{I,β}φ (\frac{1}{z + n})

= ∑_{n∈I} (\frac{1}{z + n})^{2β} ∫_{0}^{∞} s^{β− \frac{1}{2}} e^{−s \frac{1}{z+n}} φ(s) dm_{I}(s) (6)

= ∫_{0}^{∞} s^{β− \frac{1}{2}} φ(s) ∑_{n∈I} (\frac{1}{z + n})^{2β} e^{− \frac{s}{z+n}} dm_{I}(s).

Manipulating part of the integrand we have

∑_{n∈I} (\frac{1}{z + n})^{2β}e^{− \frac{s}{z+n}} = ∑_{n∈I} (\frac{1}{z + n})^{2β} ∑_{k=0}^{∞} (\frac{−s}{z + n})^{k} \frac{1}{k!}

= ∑_{k=0}^{∞} ∑_{n∈I} (\frac{1}{z + n})^{k+2β} \frac{(−s)^{k}}{k!}

= ∑_{k=0}^{∞} \frac{(−s)^{k}}{k!} ζ_{I}(k + 2β, z)

= ∑_{k=0}^{∞} \frac{(−s)^{k}}{k! Γ(k + 2β)} ∫_{0}^{∞} t^{k+2β−1}e^{−zt} dm_{I}(t),

where Lemma 3 was used for the final step.

Substituting into (6) gives us

L_{I,β} ∘ T_{I,β}φ(z) = ∫_{s=0}^{∞} ∫_{t=0}^{∞} s^{β− \frac{1}{2}} φ(s) ∑_{k=0}^{∞} \frac{(−s)^{k}}{k! Γ(k + 2β)}t^{k+2β−1}e^{−zt}dm_{I}(t)dm_{I}(s)

= ∫_{s=0}^{∞} ∫_{t=0}^{∞} φ(s) G_{β}(s, t) t^{β−1/2}e^{−zt} dm_{I}(t)dm_{I}(s)

= ∫_{t=0}^{∞} t^{β−1/2}e^{−zt} [∫_{s=0}^{∞} G_{β}(s, t)φ(s) dm_{I}(s)] dm_{I}(t)

= T_{I,β} ∘ K_{I,β} φ(z).

So indeed the operators L_{I,β} and K_{I,β} are conjugated by the integral transform T_{I,β}, and in particular L_{I,β} preserves the space H_{I,β}. □

Remark 3. It is now clear why the exponent β − 1/2 was chosen in the integral transform T_{I,β}φ(z) = ∫_{0}^{∞} s^{β−1/2}e^{−sz}φ(s) dm_{I}(s). It ensures L_{I,β} is conjugate to an integral operator with symmetric kernel.

Theorem 2. Let I be a non-empty subset of ℕ. Suppose β ∈ ℝ with β > max(0, θ_{I}). Then the operator L_{I,β} : H_{I,β} → H_{I,β} is conjugate to a selfadjoint operator, and in particular has real spectrum.

This spectrum is a sequence of real numbers union its accumulation point at zero.

Proof. The continuous conjugacy of Theorem 1 between the integral operator K_{I,β} and the transfer operator L_{I,β} : H_{I,β} → H_{I,β} implies that their spectra coincide.

K_{I,β} is trace class, by Lemma 5 (b), so in particular it is compact. Therefore the only non-discrete point in the spectrum of K_{I,β} is the accumulation point at zero. K_{I,β} is selfadjoint, by Lemma 5 (c), so its spectrum is real. □

6. The full C^{∞} spectrum of the transfer operator

In this section it will be shown that the space H_{I,β} is large enough to capture the full spectrum of L_{I,β} acting on C^{∞}[0, 1] (and on various spaces of analytic functions), as well as the discrete part of the spectrum of L_{I,β} acting on either C^{k}[0, 1] or C^{α}[0, 1]. This will follow from the fact that H_{I,β} is densely and continuously embedded in these various spaces, a result we now prove.

Lemma 6. Let I ⊂ ℕ be non-empty, and Re(β) > max(0, θ_{I}). Suppose the open disc D satisfies [0, 1] ⊂ D and \overline{D} ⊂ R_{I}. Let E denote any of the spaces A_{∞}(D), H^{2}(D), C^{ω}[0, 1], C^{∞}[0, 1], C^{k}[0, 1] (for k ∈ ℕ), or C^{α}[0, 1] (for 0 < α ≤ 1). The natural inclusion ι_{E} : H_{I,β} → E is continuous, and ι_{E}(H_{I,β}) is a dense subspace of E.

Proof. First we show that the inclusions ι_{E} are continuous. For any k ∈ ℕ, α ∈ (0, 1], each inclusion in the chain

A_{∞}(D) ↪ H^{2}(D) ↪ C^{ω}[0, 1] ↪ C^{∞}[0, 1] ↪ C^{k}[0, 1] ↪ C^{α}[0, 1]

is continuous, as is readily verified, so it suffices to check the continuity of ι_{A_{∞}(D)} : H_{I,β} → A_{∞}(D). In the definition of H_{I,β} we declared T_{I,β} to be a Hilbert space isomorphism, so ||T_{I,β}φ||_{I,β} = ||φ||_{L^{2}(m_{I})} for every φ ∈ L^{2}(m_{I}), where || · ||_{I,β} denotes the norm in H_{I,β}.

Now for any z ∈ \overline{D} and Re(β) > max(0, θ_{I}), the function r_{z,β} : s ↦ s^{β−1/2}e^{−sz} belongs to L^{2}(m_{I}), since

∫_{0}^{∞} |s^{β−1/2}e^{−sz}|^{2} dm_{I}(s) = ∫_{0}^{∞} s^{2 Re(β)−1}e^{−2sz} ∑_{n∈I} e^{−ns} ds
= ζ_{I}(2 Re(β), 2z) Γ(2 Re(β)),

which is finite because Re(2z) > −N_{I}, and Re(β) > max(0, θ_{I}). Indeed there is a constant C_{D,β} > 0 such that ||r_{z,β}||_{L^{2}(m_{I})} ≤ C_{D,β} for all z ∈ \overline{D}, since D is compactly contained in R_{I}.

For all φ ∈ L^{2}(m_{I}), z ∈ \overline{D}, the Cauchy-Schwarz inequality then gives

|T_{I,β}φ(z)| = | ∫_{0}^{∞} s^{β−1/2}e^{−sz}φ(s) dm_{I}(s)|
≤ ||r_{z,β}||_{L^{2}(m_{I})} ||φ||_{L^{2}(m_{I})}
≤ C_{D,β} ||T_{I,β}φ||_{I,β}.

That is, ||T_{I,β}φ||_{A_{∞}(D)} ≤ C_{D,β}||T_{I,β}φ||_{I,β}, so the inclusion H_{I,β} ↪ A_{∞}(D) is continuous, as required.

It now remains to prove that each ι_{E}(H_{I,β}) is dense in E. Define f_{j}(z) = \frac{1}{z+j}, for j ∈ ℕ with j > N_{I}. Each f_{j} ∈ E, and we claim that the linear span of {f_{j}}_{j>N_{I}} is a dense subspace of E. To see this it suffices to show that any monomial z^{m} can be approximated by linear combinations of the f_{j}, since the space of all polynomials is certainly dense in E. If

g_{p,m}(z) = \frac{(p + N_{I})(p + 1 + N_{I}) … (p + m + N_{I})}{(z + p + N_{I})(z + p + 1 + N_{I}) … (z + p + m + N_{I})} z^{m}

then clearly g_{p,m} approaches z^{m} in E as p → ∞. On the other hand, there is a partial fractions decomposition of g_{p,m} into a linear combination of elements of {f_{j}}, thus establishing the claim.

Now f_{j} is the Laplace transform of e^{−js}, so if

φ_{β,j}(s) = \frac{e^{−js}}{s^{β−1/2}ϑ_{I}(s)}

then T_{I,β}φ_{β,j} = f_{j}. However in general (i.e. for β ≥ 1), φ_{β,j} does not belong to L^{2}(m_{I}), so that f_{j} is not in H_{I,β}.

Nevertheless, if φ_{β,j,n} := φ_{β,j}χ_{[1/n,∞)} then it is readily verified that φ_{β,j,n} ∈ L^{2}(m_{I}) for all β > max(0, θ_{I}). Therefore f_{β,j,n} := T_{I,β}φ_{β,j,n} ∈ H_{I,β}. Now f_{β,j,n}(z) = e^{−(z+j)/n}/(z + j), so that f_{β,j,n} ∈ E (and in fact does not depend on β). Clearly f_{β,j,n} converges to f_{j} in E as n → ∞. So the linear span of {f_{β,j,n}}_{j>N_{I},n∈ℕ} is E-dense in the linear span of {f_{j}}_{j>N_{I}}, which itself is E-dense in E, and we are done. □

Theorem 3. Let I ⊂ ℕ be nonempty, and Re(β) > max(0, θ_{I}). Suppose the open disc D satisfies [0, 1] ⊂ D, \overline{D} ⊂ R_{I}, and \overline{D} ⊂ int (∩_{n∈I}S_{n}(D)), where S_{n}(z) = \frac{1}{z} − n.

(a) If E denotes any of the spaces A_{∞}(D), H^{2}(D), C^{ω}[0, 1], or C^{∞}[0, 1], then σ(L_{I,β}|E) = σ(L_{I,β}|H_{I,β}). If furthermore β is real, then σ(L_{I,β}|E) is real.

(b) If E denotes either of the spaces C^{k}[0, 1] (for k ∈ ℕ) or C^{α}[0, 1] (for 0 < α ≤ 1), then the non-essential spectrum of L_{I,β} : E → E is a subset of σ(L_{I,β}|H_{I,β}). If furthermore β is real, then the non-essential spectrum of L_{I,β} : E → E is real.

Proof. Grabiner’s intertwining theorem [18], which was initially formulated in the context of Banach spaces, has been extended by Bandtlow [4], Thm. A.6.2, to more general topological vector spaces. This extension asserts the following. Suppose E is a sequentially complete and barrelled topological vector space, with a dense subspace F. Suppose F carries a topology stronger than the one induced by E, and that F is itself sequentially complete and barrelled with respect to this topology. If both E and F are invariant under a continuous linear operator L, then the discrete part of σ(L|E) is a subset of the discrete part of σ(L|F). Moreover, if both L : E → E and L : F → F have countable spectrum, then these spectra coincide.

We choose L = L_{I,β} and F = H_{I,β}. If E is any of the spaces in the statement of the theorem then F is continuously and densely embedded in E by Lemma 6. Part (b) therefore follows from Bandtlow’s intertwining theorem, and from Theorem 2 when β is real. If now E is any of the spaces in the statement of part (a), then we recall from §3 that σ(L_{I,β}|E) is countable. Therefore (a) follows from the intertwining theorem as well, and from Theorem 2 when β is real. □

In fact it is possible to say a little more about any C^{∞} eigenfunction of L_{I,β} corresponding to a non-zero eigenvalue, as described in the following corollary.

Corollary 1. Let I ⊂ ℕ be nonempty, and Re(β) > max(0, θ_{I}). Every C^{∞} eigenfunction corresponding to a non-zero eigenvalue of L_{I,β} is holomorphic in the cut plane ℂ \ (−∞, −1].

Proof. Let A denote the restriction of L_{I,β} to H_{I,β}, and let B denote the restriction of L_{I,β} to C^{∞}[0, 1]. By Theorem 3 we know that every eigenvalue λ of B is also an eigenvalue of A. If λ ≠ 0 then it is an isolated element of both spectra.

Let us suppose that the geometric multiplicity of λ as an eigenvalue of A coincides with its geometric multiplicity as an eigenvalue of B. Then any C^{∞} eigenfunction f corresponding to λ is actually an element of H_{I,β}. Since λ ≠ 0, the eigenequation can be written λ^{−1}L_{I,β}f = f, and iterated to give λ^{−n}L_{I,β}^{n}f = f for all n ≥ 0. But f is holomorphic in the half plane R_{I}, from which we successively deduce that λ^{−n}L_{I,β}^{n}f = f is holomorphic in some region Ω_{n}, where Ω_{0} ⊂ Ω_{1} ⊂ … and ∪_{n=0}^{∞}Ω_{n} = ℂ \ (−∞, −1]. Since λ was an arbitrary non-zero eigenvalue, and f an arbitrary eigenfunction, it follows that every C^{∞} eigenfunction for a non-zero eigenvalue is holomorphic in the cut plane ℂ \ (−∞, −1], as required.

It remains to show that indeed the geometric multiplicity of λ as an eigenvalue of A coincides with its geometric multiplicity as an eigenvalue of B. For the following argument we are grateful to O. F. Bandtlow, who pointed out its omission in our original proof (see also [5] for a thorough investigation of related questions).

If ι : H_{I,β} → C^{∞}[0, 1] denotes the inclusion map, then the relation B ∘ ι = ι ∘ A gives

(z − B)^{−1} ∘ ι = ι ∘ (z − A)^{−1}  (7)

for all z belonging to the resolvent set of both A and B.

Allan’s functional calculus [1] applies to continuous linear operators L on a certain class of locally convex topological vector spaces which includes C^{∞}[0, 1]. This generalises Taylor’s functional calculus for closed operators on Banach spaces, and asserts that, under suitable hypotheses (see [1], Thm. 5.3), F (L) = \frac{1}{2πi} ∫_{∂C} \frac{F (z)}{z−L} dz for functions F holomorphic in a neighbourhood of the spectrum of L, where C is a suitable Cauchy domain (see [1], Defn. 5.2). Combining this with (7) we can derive the relations

ι ∘ Π(λ; A) = Π(λ; B) ∘ ι  (8)

and

ι ∘ A ∘ Π(λ; A) = Π(λ; B) ∘ B ∘ ι,  (9)

where Π(λ; A), Π(λ; B) denote the respective spectral projections at λ. From (8) it easily follows that

ι (R (Π(λ; A))) ⊂ R (Π(λ; B)) ,  (10)

where R (Π(λ; A)), R (Π(λ; B)) are the respective generalised eigenspaces corresponding to λ. Now Prop. A.6.5 of [4] implies that R (Π(λ; A)), R (Π(λ; B)) have the same dimension (i.e. the algebraic multiplicity of λ is the same for both A and B). Now K_{I,β} : L^{2}(m_{I}) → L^{2}(m_{I}) is trace class by Proposition 5, hence so is the operator A, by Theorem 1 (b). In particular A is compact, so that R (Π(λ; A)) is finite dimensional ([15], Thm. 7.4.5). Therefore equation (10), together with the injectivity of ι, implies that ι is an isomorphism between the finite dimensional generalised eigenspaces R (Π(λ; A)) and R (Π(λ; B)).

Using (9), and the fact that B commutes with Π(λ; B), we see that the restrictions B|R (Π(λ; B)) and A|R (Π(λ; A)) are similar, so that their Jordan decompositions are identical. In particular,

nullity [(λ − A)|R (Π(λ; A))] = nullity [(λ − B)|R (Π(λ; B))] .

To complete the proof, we observe that since Π(λ; A) and Π(λ; B) are spectral projections then ker(λ − A) = ker (A|R (Π(λ; A))) and ker(λ − B) = ker (B|R (Π(λ; B))). Hence nullity (λ − A) = nullity (λ − B), as required. □

Remark 4. Suppose I is a singleton I = {n}, and β ∈ ℂ is arbitrary. In this case L_{I,β} is just a weighted composition operator L_{I,β}f(z) = (n + z)^{−2β}f (1/(n + z)), whose spectrum can be determined explicitly (cf. [28], [34]). If

z_{n} = \frac{1}{n + \frac{1}{n+···}}

is the fixed point of 1/(n + z), then

σ (L_{I,β}|C^{∞}[0, 1]) = {(−1)^{k}z_{n}^{2k+2β} : k ∈ ℤ_{≥0}} ∪ {0}.

7. Analytic properties of β ↦ L_{I,β}

In this section we consider the analytic properties of the map β ↦ L_{I,β}, where L_{I,β} is considered as a trace class operator on a suitable Hardy space H^{2}(D). In particular L_{I,β} is an element of the complex Banach space B (H^{2}(D)) of bounded linear operators from H^{2}(D) to itself. The analyticity of a map from some complex domain into B (H^{2}(D)) is then understood (cf. [47], p. 205) to mean the existence of a derivative at every point.

The map β ↦ L_{I,β} is certainly holomorphic in the half-plane Re(β) > θ_{I}, and for general alphabets I ⊂ ℕ there is no reason to expect an analytic continuation to a larger region. However for alphabets of an arithmetic nature, such as those studied in [32], such continuations are often possible, as will be described in this section. The original result in this direction is due to Mayer [33], and asserts that β ↦ L_{ℕ,β}, extends meromorphically to the entire plane, the only poles being simple ones at β = \frac{1−k}{2}, k ∈ ℤ_{≥0}. The notion of analyticity used by Mayer is weaker than the one we consider here; in [33] it means that β ↦ L_{I,β}f(z) is analytic for every z ∈ D, f ∈ H^{2}(D).

Theorem 4. Let I ⊂ ℕ be non-empty. Let D ⊂ ℂ be an open disc with [0, 1] ⊂ D, \overline{D} ⊂ R_{I}, and such that \overline{D} ⊂ int (∩_{n∈I}S_{n}(D)), where S_{n}(z) = \frac{1}{z} − n.

Let ζ_{I}(s, z) = ∑_{n∈I}(z + n)^{−s} denote the Hurwitz zeta function for the alphabet I. If the map β ↦ ζ_{I}(2β, z) has an analytic continuation to a complex domain U, then:

(a) β ↦ L_{I,β} also has an analytic continuation to U.

(b) The singularities of the analytic continuation of β ↦ L_{I,β} are at the points (p − k)/2, k ∈ ℤ_{≥0}, where p is a singularity of s ↦ ζ_{I}(s, z).

(c) If \overline{U} = ℂ, and the analytic continuation of β ↦ ζ_{I}(2β, z) to U is a meromorphic one to \overline{U} = ℂ, then β ↦ L_{I,β} also has a meromorphic continuation to ℂ, with poles of order m at the points (p − k)/2, k ∈ ℤ_{≥0}, where p is a pole of order m of s ↦ ζ_{I}(s, z).

(d) If s ↦ ζ_{I}(s, z) has an algebraic (resp. logarithmic) branch singularity at the point p, then β ↦ L_{I,β} has algebraic (resp. logarithmic) branch singularities at those points (p − k)/2, k ∈ ℤ_{≥0} which lie in \overline{U}.

(e) For each β ∈ U, L_{I,β} : H^{2}(D) → H^{2}(D) is a trace class operator.

Proof. We follow Mayer [33] in expressing β ↦ L_{I,β} as the sum of two maps, the first of which is meromorphic with range the class of rank- (N + 1) operators, the second of which is entire and trace class-valued. We will then let N → ∞.

Define the operators π_{k}, P_{N} : H^{2}(D) → H^{2}(D) by

π_{k}f(z) = \frac{f^{(k)}(0)}{k!}z^{k}  ,  P_{N} = ∑_{k=0}^{N} π_{k}.

Letting Q_{N} = I − P_{N} we write, for any N ∈ ℤ_{≥0},

L_{I,β} = L_{I,β} ∘ P_{N} + L_{I,β} ∘ Q_{N}.  (11)

Each of the rank-one projections L_{I,β} ∘ π_{k} can be expressed as

L_{I,β} π_{k}f(z) = \frac{f^{(k)}(0)}{k!}ζ_{I}(2β + k, z)

for f ∈ H^{2}(D), Re(β) > \frac{1}{2}(θ_{I} − k), and for z ∈ D′, some open disc concentric with D and of strictly larger radius, as in Proposition 1.

Fix f ∈ H^{2}(D) and z ∈ D′. Let P denote the set of singularities of s ↦ ζ_{I}(s, z). Since β ↦ ζ_{I}(2β, z) extends analytically to U, with singularities in \frac{1}{2}P, then β ↦ L_{I,β} π_{k}f(z) extends to the translated region U −k, with singularities translated to lie in the set \frac{1}{2}(P −k)∩\overline{U}. Therefore β ↦ L_{I,β}P_{N}f(z) extends analytically to U, with singularities in the set \{(p − k)/2 : 0 ≤ k ≤ N, p ∈ P \} ∩ \overline{U}.

Now we claim that for arbitrary fixed f ∈ H^{2}(D), z ∈ D′, the map β ↦ L_{I,β}Q_{N}f(z) can be extended holomorphically to the half-plane Re(β) > \frac{θ_{I}−N−1}{2}. If I is finite then this claim is clearly true, so suppose I is infinite, and write I = {n_{k}}_{k=1}^{∞}, where n_{1} < n_{2} < . . ..

If ξ > 0 is sufficiently small, then every f ∈ H^{2}(D) has a power series expansion around 0, uniformly convergent for |z| < ξ. It follows from Cauchy’s theorem that there is a constant C_{f} > 0 such that

|Q_{N}f(z)| ≤ C_{f}|z|^{N+1}  (12)

for all |z| < ξ.

Since I = {n_{k}} is infinite, we may choose m ∈ ℕ large enough so that |\frac{1}{z+n_{k}} | < ξ for any z ∈ D′ and k > m. We now write

L_{I,β}Q_{N}f(z) = ∑_{k=1}^{m} (\frac{1}{z + n_{k}} )^{2β} Q_{N}f (\frac{1}{z + n_{k}} )

+ ∑_{k=m+1}^{∞} (\frac{1}{z + n_{k}} )^{2β} Q_{N}f (\frac{1}{z + n_{k}} ) .

The first summation in this expression is finite, so can be extended to an entire function of β. Using (12) we see that the second summation is absolutely and uniformly bounded on D′ by a constant multiple of ∑_{n∈I} n^{−2Re(β)−N−1}, which in particular is convergent for Re(β) > \frac{θ_{I}−N−1}{2}, and therefore defines a holomorphic function in this half-plane.

Therefore for any fixed f ∈ H^{2}(D), z ∈ D′, the map β ↦ L_{I,β}f(z) = L_{I,β}P_{N}f(z) + L_{I,β}Q_{N}f(z) has an analytic extension to U ∩ {β ∈ ℂ : Re(β) > \frac{θ_{I}−N−1}{2}\}, for any N ∈ ℤ_{≥0}. Letting N → ∞ shows that β ↦ L_{I,β}f(z) has an analytic extension to U. The nature of any singularity of β ↦ L_{I,β}f(z) is clearly the same as the corresponding singularity of s ↦ ζ_{I}(s, z).

We now claim that β ↦ L_{I,β} is weakly analytic on U; that is, given any f ∈ H^{2}(D) and any bounded linear functional l ∈ H^{2}(D)′, the map β ↦ l(L_{I,β}f) has an analytic extension to U. Every functional has a representation l = (·, g), for some g ∈ H^{2}(D), so that

l(L_{I,β}f) = \frac{1}{2π} ∫_{0}^{2π} L_{I,β}f(c + re^{it})\overline{g(c + re^{it})} dt,  (13)

where c denotes the centre of D, and r denotes its radius. We have shown that for every t ∈ [0, 2π), the map β ↦ L_{I,β}f(c + re^{it}) is holomorphic. Moreover, on any compact subset K of U, the integral (13) is uniformly convergent for β ∈ K. The analyticity of β ↦ l(L_{I,β}f) then follows from Lemma 1.1 of [26].

The weak analyticity of β ↦ L_{I,β} in fact implies strong analyticity (see [47], Thm. 4.4-F); that is, for any f ∈ H^{2}(D) the map β ↦ L_{I,β}f is analytic on U (as an H^{2}(D)-valued map). But strong analyticity in turn implies the analyticity of β ↦ L_{I,β} as a B (H^{2}(D))-valued map (see [47], Thm. 4.4-G), which is the result we want. Once again the nature of any singularity of β ↦ L_{I,β} is clearly the same as the corresponding singularity of s ↦ ζ_{I}(s, z).

The operator L_{I,β} ∘ Q_{N} is trace class for each β ∈ U, by the same calculations as in Proposition 1. Adding the finite rank operator L_{I,β} ∘ P_{N} gives the analytically continued operator L_{I,β}, which is therefore itself trace class for each β ∈ U. □

Theorem 5. Let I ⊂ ℕ be non-empty. Let D ⊂ ℂ be an open disc with [0, 1] ⊂ D, \overline{D} ⊂ R_{I}, and such that \overline{D} ⊂ int (∩_{n∈I}S_{n}(D)), where S_{n}(z) = \frac{1}{z} − n.

Define L_{I,β} : H^{2}(D) → H^{2}(D) by (5) on the half-plane Re(β) > θ_{I}. For the following alphabets I, the map β ↦ L_{I,β} has an analytic continuation to the region stated, and in each case L_{I,β} is a trace class operator for every β in the extended region.

(i) I finite (and non-empty). β ↦ L_{I,β} is an entire function of β. (ii) I = aℕ + b for a, b ∈ ℕ. β ↦ L_{I,β} extends meromorphically to ℂ with simple poles at β = (1 − k)/2 for k ∈ ℤ_{≥0}. (iii) I = {n^{a}}_{n=1}^{∞} for a ∈ ℕ. β ↦ L_{I,β} extends meromorphically to the complex plane with simple poles at β = \frac{1}{2}(\frac{1}{a} − k) for k ∈ ℤ_{≥0}. (iv) I = {ψ(n)}_{n=1}^{∞}, ψ ∈ ℕ[z] a polynomial of degree d > 1. Then θ_{I} = \frac{1}{2d}. β ↦ L_{I,β} extends meromorphically to ℂ, the only poles being simple ones among the points β = \frac{1−k}{2d}, k ∈ ℤ_{≥0}.

(v) I = {a^{n}}_{n=1}^{∞} for a ∈ ℤ_{≥2}. β ↦ L_{I,β} extends meromorphically to the complex plane with poles at \{\frac{−k}{2} + \frac{πi}{loga}m : (k, m) ∈ ℤ_{≥0} × ℤ\}. (vi) I = {prime numbers}. Then θ_{I} = 1/2. There is a sequence {β_{r}}_{r=1}^{∞}, with β_{1} = 1/2, whose set of accumulation points is the imaginary axis Re(β) = 0, such that β ↦ L_{I,β} extends analytically to a domain \{β ∈ ℂ : Re(β) > 0\} \ ∪_{r=1}^{∞}L_{r}, where L_{r} = \{\frac{1}{2}β_{r} − t : t ≥ 0\}. The map has a logarithmic singularity at each point β_{r}. (vii) I the set of integers which are the sum of two squares (of natural numbers). Then θ_{I} = 1/2, and L_{I,β} = (β − 1/2)^{−1/2}M_{I,β}, where β ↦ M_{I,β} extends analytically to a domain \{β ∈ ℂ : Re(β) > \frac{1}{2} − \frac{A}{2} [log (2|Im(β)| + 2)]^{−A}\} for some A > 0. (viii) I = {n_{k}}_{k=1}^{∞} is such that there exists γ > 0 with n_{k+1} ≥ n_{k}^{1+γ} for all k. Then θ_{I}=0, and the line Re(β) = 0 is a natural boundary for the map β ↦ L_{I,β}.

Proof. (i) is clear.

By Theorem 4, the analytic properties of β ↦ L_{I,β} are closely related to those of s ↦ ζ_{I}(s, z). In case (ii) we can exploit this directly.

(ii) We can write ζ_{I}(s, z) = ∑_{n=1}^{∞} (\frac{1}{an+b+z})^{s} = a^{−s}ζ_{ℕ} (s, \frac{z+b}{a}) . If ζ(s, z) = ∑_{n=0}^{∞}(z+n)^{−s} denotes the classical Hurwitz zeta function, then we have ζ_{ℕ}(s, \frac{z+b}{a}) = ζ(s, \frac{z+b}{a}) − (\frac{a}{z+b})^{s}. Now ζ (s, \frac{z+b}{a}) has a meromorphic extension to the entire complex plane, with a simple pole at s = 1 ([6], p. 502), so the same is true of ζ_{ℕ} (s, \frac{z+b}{a}), and hence of ζ_{I}(s, z). The result then follows by Theorem 4.

In cases (iii)–(viii) it will be more convenient to relate ζ_{I}(s, z) to ζ_{I}(s). The analytic properties of this latter zeta function are known for the alphabets I considered here.

We claim that if ζ_{I}(s) has an analytic extension to a complex domain U, with set of singularities Q, then for z ∈ D, the map s ↦ ζ_{I}(s, z) has an analytic extension to the same domain U, and that its singularities are contained in \{q − k : q ∈ Q, k ∈ ℤ_{≥0}\} ∩ \overline{U}.

Let us prove this claim. Write I = {n_{k}}_{k=1}^{∞}, where n_{1} < n_{2} < . . .. We can choose m ∈ ℕ sufficiently large to ensure that the closure of D is contained in the disc of radius \frac{n_{m}}{2} centred at 0. Define

g(s, z) = ∑_{k=1}^{m−1} (\frac{1}{z + n_{k}} )^{s} , h(s, z) = ζ_{I}(s, z) − g(s, z),

for z ∈ D and Re(s) > 2θ_{I}. For fixed z ∈ D, the map s ↦ g(s, z) is clearly entire. We will consider the analytic extension of s ↦ h(s, z). Let J = {n_{k}}_{k=m}^{∞} be the tail of the alphabet I. For z ∈ D and Re(s) > 2θ_{I} we have

h(s, z) = ∑_{k=m}^{∞} \frac{1}{n_{k}^{s}} (\frac{1}{1 + \frac{z}{n_{k}}} )^{s}

= ∑_{k=m}^{∞} \frac{1}{n_{k}^{s}} ∑_{l=0}^{∞} \binom{−s}{l} (−\frac{z}{n_{k}} )^{l}

= ∑_{l=0}^{∞} \binom{−s}{l}(−z)^{l} ∑_{k=m}^{∞} \frac{1}{n_{k}^{s+l}}

= ∑_{l=0}^{∞} \binom{−s}{l}(−z)^{l}ζ_{J} (s + l),

where the interchange of order of summation is clearly justified, and where \binom{−s}{l} = \frac{Γ(1−s)}{Γ(l+1)Γ(−s−l+1)} (so in particular if −s ∈ ℕ with −s < l then \binom{−s}{l} = 0).

We will prove that the map s ↦ ∑_{l=N}^{∞} \binom{−s}{l}(−z)^{l}ζ_{J} (s + l) extends analytically to U ∩ {s ∈ ℂ : Re(s) > 2θ_{I} − N}, for all N ∈ ℕ. This will complete the proof of the claim, since it is easily seen that s ↦ ∑_{l=0}^{N−1} \binom{−s}{l}(−z)^{l}ζ_{J} (s + l) extends analytically to U, with singularities in the set \{q − k : q ∈ Q, 0 ≤ k ≤ N − 1\} ∩ \overline{U}.

Fix N ∈ ℕ and set h_{N}(s, w) = ∑_{l=N}^{∞} \binom{−s}{l}(−w)^{l}. By a calculation analogous to the one above we have

∑_{l=N}^{∞} \binom{−s}{l}(−z)^{l}ζ_{J} (s + l) = ∑_{k=m}^{∞} \frac{1}{n_{k}^{s}} h_{N} (s, \frac{z}{n_{k}} ) .

Let K be any compact subset of the half plane Re(s) > 2θ_{I} − N. By the Cauchy formula there is a constant A > 0 such that |h_{N}(s, w)| ≤ A|w|^{N} for s ∈ K and |w| ≤ \frac{1}{2}. Now if z ∈ D then |z/n_{k}| ≤ 1/2 for all k ≥ m. Therefore

| ∑_{k=m}^{∞} \frac{1}{n_{k}^{s}} h_{N} (s, \frac{z}{n_{k}} ) | ≤ A max_{z∈D} |z|^{N} ∑_{k=m}^{∞} (\frac{1}{n_{k}} )^{Re(s)+N} ,

for all z ∈ D, s ∈ K.

The right hand side of this inequality is finite, since Re(s) > 2θ_{I} −N, so indeed the map s ↦ ∑_{l=N}^{∞} \binom{−s}{l}(−z)^{l}ζ_{J} (s+l) = ∑_{k=m}^{∞} \frac{1}{n_{k}^{s}} h_{N} (s, \frac{z}{n_{k}} ) is holomorphic on Re(s) > 2θ_{I} −N, and the claim is proved. The claim will now be used to establish parts (iii)–(viii) of the theorem. (iii) In this case we have ζ_{I}(s) = ζ(as), where ζ is Riemann’s zeta function, which has a meromorphic extension to all of ℂ, with a simple pole at s = 1. Therefore ζ_{I}(s) has a meromorphic extension to the whole plane, with a simple pole at s = \frac{1}{a}. The above claim means that ζ_{I}(s, z) has a meromorphic continuation to ℂ, with simple poles at s = \frac{1}{a} − k, k ∈ ℤ_{≥0}. (In the case p = 1 all these poles except for k = 0 are cancelled by zeros). The result follows by Theorem 4. (iv) ζ_{I}(s) has a meromorphic extension to all of ℂ, with a simple pole at s = 1/d, and possible further simple poles at s = −k/d for k ∈ ℤ_{≥0} [37]. By the above claim, the same is true of ζ_{I}(s, z), and the result then follows by Theorem 4.

(v) Now ζ_{I}(s) = ∑_{m=1}^{∞} a^{−ms} = \frac{1}{1−a^{−s}} ,, which extends meromorphically to ℂ with simple poles at s = \frac{2πi}{loga}m for m ∈ ℤ. The above claim implies that ζ(s, z) has a meromorphic continuation to ℂ, with simple poles at points s = \frac{2πi}{loga}m − k for k ∈ ℤ_{≥0}, m ∈ ℤ. The result then follows by Theorem 4. (vi) For I = {primes}, the Dirichlet series ζ_{I}(s) is holomorphic for Re(s) > 1, has a logarithmic singularity at s = 1, and can be continued into the half-plane Re(s) > 0 with logarithmic singularities at points s_{r} which accumulate on the imaginary axis and do not accumulate elsewhere [29]. Removing suitable half-lines, for example L′_{r} = {s_{r} − t : t ≥ 0}, gives a domain {s ∈ ℂ : Re(s) > 0} \ ∪_{r=1}^{∞}L′_{r} on which ζ_{I}(s) has a holomorphic continuation. The Hurwitz zeta function ζ_{I}(s, z) is also holomorphic on this domain, by the above claim, and the result follows by Theorem 4. (vii) The Dirichlet series ζ_{I}(s) is holomorphic for Re(s) > 1, and can be written as ζ_{I}(s) = (s − 1)^{−1/2}Z(s), where Z(s) is holomorphic in a domain of the form Re(s) > 1 − \frac{A}{[log(|Im(s)|+2)]^{A}} (see p. 61–2 of [21]). An analogous statement is true for ζ_{I}(s, z), by the above claim, and the result then follows by Theorem 4. (viii) Clearly θ_{I} = 0. Write s = it, where t ∈ ℝ. We then have the formal expression

ζ_{I}(it) = ∑_{k=1}^{∞} ξ^{log n_{k}},  (14)

where ξ = e^{−it}. Now n_{k+1} ≥ n_{k}^{1+γ}, so that log n_{k+1} ≥ (1 + γ) log n_{k}. A minor modification of Hadamard’s gap theorem (see §7.43 of [49]) then implies that the circle |ξ| = 1 is a natural boundary for the function of ξ defined by (14). Therefore Re(s) = 0 is a natural boundary for ζ_{I}(s), and hence for ζ_{I}(s, z). The result follows by Theorem 4. □

References

[1] G. R. Allan, A spectral theory for locally convex algebras, Proc. London Math. Soc, 15 (1965), 399–421.

[2] T. Apostol, Introduction to analytic number theory, Springer-Verlag, New York, 1976.

[3] K. Babenko, On a problem of Gauss, Dokl. Akad. Nauk SSSR, 238 (1978), 1021–1024. (English translation in Soviet Math. Dokl. 19 (1978), 136–140).

[4] O. F. Bandtlow, Spectral analysis of dynamical systems, Ph.D. thesis, Cambridge University, 1997, available at http://www2.elec.qmul.ac.uk/∼oscar

[5] O. F. Bandtlow, Fine-structure of the spectra of intertwined operators, manuscript, 2001.

[6] C. Berenstein & R. Gay, Complex variables, an introduction, New York, Springer-Verlag, 1991.

[7] R. Bowen, Hausdorff dimension of quasi-circles, Publ. Math. (IHES), 50 (1979), 11–25.

[8] R. T. Bumby, Hausdorff dimensions of Cantor sets, J. Reine Angew. Math., 331 (1982), 192–206.

[9] U. Bunke & M. Olbrich, Selberg zeta and theta functions, a differential operator approach, Akademie Verlag, Berlin, 1995.

[10] P. Cartier & A. Voros, Une nouvelle interprétation de la formule des traces de Selberg, Progress in Mathematics vol. 87, Birkhäuser, Boston, 1990, pp. 1–68.

[11] F. Christiansen, P. Cvitanović, and H-H. Rugh, The spectrum of the perioddoubling operator in terms of cycles, J. Phys. A, Math. Gen., 23 (1990), L713S-L717S.

[12] C. Cowen & B. Maccluer, Composition operators on spaces of analytic functions, CRC Press, 1995.

[13] T. W. Cusick, M. E. Flahive, The Markoff and Lagrange spectra, AMS Mathematical Surveys and Monographs 30, 1989.

[14] H. Daudé, P. Flajolet, B. Vallée, An average-case analysis of the Gaussian algorithm for lattice reduction, Comb. Probab. Comput., 6 (1997), 397-433.

[15] N. Dunford & J. T. Schwartz, Linear Operators. Part I: General Theory, Interscience, New York, 1957.

[16] I. Gohberg, S. Goldberg, N. Krupnik, Traces and Determinants of Linear Operators, Basel, Birkhäuser Verlag, 2000.

[17] I. J. Good The fractional dimension of continued fractions, Proc. Camb. Phil. Soc., 37 (1941), 199–228.

[18] S. Grabiner, Spectral consequences of the existence of intertwining operators, Ann. Soc. Math. Polonae Ser. 1, 22 (1970), 227–238.

[19] A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc., 16, 1955.

[20] A. Grothendieck, La théorie de Fredholm, Bull. Soc. Math. France, 84 (1956), 319–384.

[21] G. H. Hardy, Ramanujan, Chelsea, New York, 3rd edition, 1978.

[22] D. Hensley The Hausdorff dimensions of some continued fraction Cantor sets, J. Number Theory, 33 (1989), 182–198.

[23] E. Hille, Analytic function theory, vol. II, Ginn & Co., 1962.

[24] S. Heinemann & M. Urbański, Hausdorff dimension estimates for infinite conformal iterated function systems, Preprint, (2001)

[25] O. Jenkinson & M. Pollicott Computing the dimension of dynamically defined sets : E_{2} and bounded continued fractions, Ergod. Th. & Dyn. Sys., to appear.

[26] J. Jorgenson & S. Lang, Basic Analysis of Regularized Series and Products, Lecture Notes in Mathematics 1564, Springer-Verlag, 1993.

[27] J. Jorgenson & S. Lang, Explicit formulas, Lecture Notes in Mathematics 1593, Springer-Verlag, 1994.

[28] H. Kamowitz, The spectra of endomorphisms of the disc algebra, Pacific J. Math., 46 (1973), 433–440.

[29] E. Landau & A. Walfisz, Uber die Nichtfortsetzbarkeit einiger durch dirichletsche Reihen definierter Funktionen, Rendiconti di Circ. Mat. di Palermo, 44 (1919), 82–86.

[30] G. Levin, On Mayer’s conjecture and zeros of entire functions, Ergod. Th. Dyn. Sys., 14 (1994), 565–574.

[31] R. D. Mauldin & M. Urbański, Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc. (3), 73 (1996), 105–154.

[32] R. D. Mauldin & M. Urbański, Conformal iterated function systems with applications to the geometry of continued fractions, Trans. Amer. Math. Soc. 351 (1999), 4995–5025.

[33] D. H. Mayer, On the Thermodynamic Formalism for the Gauss Map, Commun. Math. Phys. 130 (1990), 311–333.

[34] D. H. Mayer, On composition operators on Banach spaces of holomorphic functions, J. Funct. Anal., 35 (1980), 191–206.

[35] D. H. Mayer & G. Roepstorff, On the relaxation time of Gauss’ continuedfraction map. I: The Hilbert space approach (Koopmanism), J. Stat. Phys., 47 (1987), 149–171.

[36] D. H. Mayer & G. Roepstorff, On the relaxation time of Gauss’ continuedfraction map. II: The Banach space approach (Transfer operator method), J. Stat. Phys., 50 (1988), 331–344.

[37] H. Mueller, Uber die meromorphe Fortsetzung einer Klasse verallgemeinter Zetafunktionen, Arch. Math., 58 (1992), 265-275.

[38] S. Rolewicz, Metric linear spaces, PWN - Polish Scientific Publishers, 1972.

[39] D. Ruelle, Zeta functions for expanding maps and Anosov flows, Invent. Math., 34 (1976), 231–242.

[40] D. Ruelle, Thermodynamic Formalism, Wiley, New York, 1978.

[41] D. Ruelle, Resonances of chaotic dynamical systems, Phys. Rev. Lett., 56 (1986), 405–407.

[42] D. Ruelle, Resonances for Axiom A flows, J. Diff. Geom., 25 (1987), 117–137.

[43] D. Ruelle, An extension of the theory of Fredholm determinants, Publ. Math. (IHES), 72 (1990), 175–193.

[44] H-H. Rugh, On the asymptotic form and the reality of spectra of Perron-Frobenius operators, Nonlinearity, 7 (1994), 1055-1066.

[45] J. Shallit, Real numbers with bounded partial quotients:a survey, Enseign. Math., 38 (1992), 151–187.

[46] B. Simon, Trace ideals and their applications, LMS Lecture Note Series 35, Cambridge University Press, Cambridge, 1979.

[47] A. E. Taylor, An introduction to functional analysis, John Wiley, New York, 1964.

[48] S. Thangavelu, Lectures on Hermite and Laguerre expansions, Princeton University Press, 1993.

[49] E.C. Titchmarsh, The Theory of Functions, 2nd ed., Oxford, Oxford University Press 1939.

[50] A. Voros, Spectral functions, special functions, and the Selberg zeta function, Comm. Math. Phys., 110 (1987), 439–465.

[51] G. N. Watson, A treatise on the theory of Bessel functions, Cambridge University Press, 1922.

[52] E. Wirsing, On the theorem of Gauss-Kuzmin-Levy and a Frobenius type theorem for function spaces, Acta Arithm., 24 (1974), 507–528.

Oliver Jenkinson; School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, UK

omj@maths.qmul.ac.uk, http://www.maths.qmul.ac.uk/∼omj

Luis Felipe Gonzalez; Institute of Mathematics, University of Warwick, Coventry CV4 7AL, UK and Department of Mathematics, University of North Texas, Denton, TX 76203-1430, USA

lfgonz@unt.edu

Mariusz Urbański; Department of Mathematics, University of North Texas, Denton, TX 76203-1430, USA

urbanski@unt.edu, http://www.math.unt.edu/∼urbanski