Hilbert spaces of Dirichlet Series and their Multipliers

John E. McCarthy ^{∗} Washington University St. Louis, Missouri 63130

December 7 2001

Abstract We consider various Hilbert spaces of Dirichlet series whose norms are given by weighted ℓ^{2} norms of the Dirichlet coefficients. We describe the multiplier algebras of these spaces. The functions in the multiplier algebra may or may not extend to be analytic on a larger half-plane than the functions in the Hilbert space.

0 Introduction

Let w = {w_{n}}_{n=n_{0}}^{∞} be a sequence of positive numbers. In this paper we are concerned with Hilbert spaces of functions representable by Dirichlet series:

H_{w} = {f(s) = ∑_{n=n_{0}}^{∞} a_{n}n^{−s} | ∥f∥_{H_{w}}^{2} := ∑_{n=n_{0}}^{∞} |a_{n}|^{2}w_{n} < ∞} . (0.1)

The prototypical case, where w_{n} ≡ 1, n ≥ 1, was first studied in a beautiful paper by H. Hedenmalm, P. Lindqvist and K. Seip [9]. Among other results, they characterized the multipliers of the space (see Theorem 1.11 below for a statement of their result).

∗Partially supported by National Science Foundation Grant DMS 0070639

One purpose of this paper is to consider the scale of spaces obtained from the weight sequences w^{α}, defined for n ≥ 2 by

w_{n}^{α} = (log n)^{α}.  (0.2)

For brevity, we shall write H_{α} for the space H_{w^{α}}; specifically

H_{α} = {f(s) = ∑_{n=2}^{∞} a_{n}n^{−s} | ∑_{n=2}^{∞} |a_{n}|^{2}(log n)^{α} < ∞} .

(When α = 0, it is more natural to let n_{0} = 1 and to include the constant functions in H_{0}. It is not essential to any of the issues we discuss here).

Before going further, let us remind the reader of some basic facts about Dirichlet series. A nice treatment can be found in Titchmarsh’s book [16]. We shall follow the convention of writing the complex variable s = σ + it. A Dirichlet series is a series of the form

∑_{n=1}^{∞} a_{n}n^{−s}.  (0.3)

Such a series may converge for no values of s; if it converges for any particular s_{0}, then it converges for all s with σ > ℜ(s_{0}). Therefore the largest open set in which a series (0.3) converges is a half-plane (at what points on the boundary of the half-plane the series converges is, in general, a delicate question). Let us adopt the notation, for ρ a real number, Ω_{ρ} is the half-plane

Ω_{ρ} = {s ∈ ℂ | σ > ρ}.

Let

σ_{c} = inf{ℜ(s) : ∑_{n=1}^{∞} a_{n}n^{−s} converges};

this is called the abscissa of convergence of the series. The largest domain of convergence of the series is Ω_{σ_{c}}. There are three other abscissae associated with the series (0.3) which we shall need. The first is the abscissa of absolute convergence, σ_{a}, defined by

σ_{a} = inf{ℜ(s) : ∑_{n=1}^{∞} a_{n}n^{−s} converges absolutely}.

Obviously σ_{a} ≥ σ_{c}; it is straightforward that σ_{a} ≤ σ_{c} + 1, because if ∑_{n=1}^{∞} a_{n}n^{−s} converges, then |a_{n}|n^{−σ} = o(1). The second is the abscissa of boundedness σ_{b}, defined by

σ_{b} = inf{ρ : ∑_{n=1}^{∞} a_{n}n^{−s} converges to a bounded function in Ω_{ρ}}.

The third abscissa is the abscissa of uniform convergence σ_{u}, defined by

σ_{u} = inf{ρ : ∑_{n=1}^{∞} a_{n}n^{−s} converges uniformly in Ω_{ρ}}.

Clearly σ_{u} ≥ σ_{b}; H. Bohr proved that σ_{u} = σ_{b}, and σ_{a} ≤ σ_{b} + \frac{1}{2} [7]. Note that the series does not necessarily define a bounded function in Ω_{σ_{b}}, but the function it represents is bounded in all strictly smaller half-planes. If all the coefficients a_{n} are positive, then all three of σ_{c}, σ_{b}, σ_{a} coincide.

We shall let D denote the set of functions that can be represented in some half-plane by a Dirichlet series.

Let f(s) be holomorphic in the half-plane Ω_{ρ}. Let ε > 0. A real number τ is called an ε translation number of f if

sup_{s∈Ω_{ρ}} |f(s + iτ) − f(s)| ≤ ε.

The function f(s) is called uniformly almost periodic in the half-plane Ω_{ρ} if, for every ε > 0, there exists a positive real number M such that every interval in ℝ of length M contains at least one ε translation number of f.

We shall need the following theorem. The proof can be found in [6, p. 144].

Theorem 0.4 Suppose that f(s) is represented by a Dirichlet series that converges uniformly in the half-plane Ω_{ρ}. Then f is uniformly almost periodic in Ω_{ρ}.

Returning to the spaces H_{α}, it follows from the Cauchy-Schwarz inequality that any function in any space H_{α} has σ_{a} ≤ \frac{1}{2}. Moreover, for all ε > 0, the function^{1}

ζ(\frac{1}{2} + ε + s) − 1 = ∑_{n=2}^{∞} \frac{1}{n^{\frac{1}{2} +ε}} n^{−s}

is in every H_{α} and has a pole at \frac{1}{2} − ε, so the largest common domain of analyticity of the functions in any H_{α} is Ω_{1/2}. The reproducing kernel for H_{w} is

k(s, u) = ∑_{n=n_{0}}^{∞} \frac{1}{w_{n}} n^{−s−ū};  (0.5)

for the spaces H_{α} this is essentially a fractional derivative or integral of the ζ function at s + ū.

We claim that the scale of spaces H_{α} is in many ways analogous to the scale of spaces of holomorphic functions in the unit disk defined by

K_{α} = {g(z) = ∑_{n=0}^{∞} a_{n}z^{n} | ∑_{n=0}^{∞} |a_{n}|^{2}(n + 1)^{α} < ∞} .  (0.6)

Thus the space H_{0} corresponds to the Hardy space, the space H_{−1} corresponds to the Bergman space, and the space H_{1} corresponds to the Dirichlet space in the setting of the disk.

Notice first that differentiation is a unitary map from H_{α} to H_{α−2}, and fractional differentiation/integration maps any H_{α} to any H_{β}. Although this is not exactly true for the spaces K_{α} as we defined them, it is essentially true. It fails only because (a) differentiation annihilates the constant functions, and (b) the choice of norms in (0.6) does not quite render differentiation isometric.

1We use ζ to denote the Riemann zeta-function: ζ(s) = ∑_{n=1}^{∞} n^{−s}.

For example, the norm of z^{n} in the Dirichlet space is \sqrt{n + 1}, and the norm of nz^{n−1} in the Bergman space is \sqrt{n}. Asymptotically these agree, and it is true as function spaces that the Dirichlet space is the set of functions whose derivatives are in the Bergman space.

The main reason for our analogy is the Plancherel formula (Theorem 1.5 below) which gives the norm of a function in H_{α}, for α > 0, in terms of a weighted area integral. As α tends to 0, the measure tends to a line integral on the boundary.

Why study the H_{α} spaces? We offer three considerations. First, a Plancherel formula calculating the norm of a function in two different ways frequently has serendipitous consequences (see e.g. [9], where they exploit it to answer an old question of A. Beurling). Second, the kernel functions for these spaces are formed from the zeta function and its relatives (0.5), and anything connected with the zeta function is, in the author’s opinion, automatically interesting. Third, much of the interesting function theory on the disk is obtained by restricting functions to various spaces. These spaces are often naturally included in an L^{p}-scale in one direction (e.g. the Hardy scale H^{p}), and in another scale obtained by changing the measure (e.g. K_{α}). The function theory of Dirichlet series is not nearly as advanced as the function theory of the unit disk, so the natural scale of Hilbert spaces seems to be a good starting point. In this paper we do not consider the L^{p}-scale, but changing the L^{2}-norm on the right-hand side of the Plancherel formula (1.6) gives the obvious candidate for the H^{p}-like norms. For some interesting results on H^{p} spaces of Dirichlet series, we refer the reader to [5].

In Section 1, we study H_{α} for α ≤ 0. These spaces are Bergman-like. The norm can be given as (a limit of) an L^{2}-norm of the function. We extend the theorem of Hedenmalm, Lindqvist and Seip [9] to show that for all these spaces the multiplier algebra is the same, viz. the space of all functions representable by Dirichlet series and bounded in Ω_{0} (Theorem 1.11).

In Section 2, we study H_{α} for α > 0. These are Dirichlet-like spaces: functions whose derivatives of some fixed order are in a Bergman-like space. We give an incomplete characterization of the multiplier algebra (Corollary 2.4)..

In Section 3, we consider a different choice of weight sequence. We let w_{n} = \frac{1}{F (n)}, where F (n) is the number of ways n can be factored with integer factors greater than 1. The reproducing kernel is

k(s, u) = \frac{1}{2 − ζ(s + ū)}.

This has the interesting property of being a complete Pick kernel, and several properties flow from this (see e.g. [2] for a discussion). We show that, unlike for the spaces H_{α}, the multipliers do not extend to be analytic on a larger domain than the domain of analyticity of the Hilbert space, by constructing an interpolating sequence for the multiplier algebra that accumulates on the boundary of the half-plan of convergence of the Hilbert space (Theorem 3.6).

1 Bergman-like Spaces

Throughout this section, μ will be a positive Radon measure on [0, ∞) for which

∫_{0}^{∞} n_{0}^{−2σ}dμ(σ) = ∫_{0}^{∞} e^{−2(log n_{0}) σ}dμ(σ) < ∞  (1.1)

for some positive integer n_{0}. We also assume that

0 is in the support of μ.  (1.2)

We define w_{n}, for n ≥ n_{0}, by

w_{n} := ∫_{0}^{∞} n^{−2σ}dμ(σ).  (1.3)

Letting μ_{α} be the measure

dμ_{α}(σ) = \frac{2^{−α}}{Γ(−α)} σ^{−1−α}dσ

gives the weights from (0.2) for α < 0:

∫_{0}^{∞} n^{−2σ}dμ_{α}(σ) = (log n)^{α}, n ≥ 2, α < 0.

We let μ_{0} be the unit point mass at 0, which has all of its moments equal to 1.

For any measure satisfying (1.1) and (1.2), the moments w_{n} are a decreasing sequence that decays more slowly than any negative power of n: for all ε > 0, there exists c > 0 so that

w_{n} > c n^{−ε}.  (1.4)

Therefore, every space H_{w} consists of functions analytic in Ω_{1/2}, and contains functions that are not analytically extendable to any larger domain. Nonetheless there is a dense subspace of functions in H_{w} whose norms can be obtained by evaluating suitable integrals over the larger half-plane Ω_{0}. For the case of H_{0}, the following theorem is due to F. Carlson [8]. Note that we do not assume that either side of (1.6) is finite.

Theorem 1.5 Let f(s) = ∑_{n=n_{0}}^{∞} a_{n}n^{−s} be a function in D that has σ_{b} = 0. Let μ satisfy (1.1) and (1.2), and let w_{n} be given by (1.3). Then

∑_{n=n_{0}}^{∞} |a_{n}|^{2}w_{n} = lim_{c→0^{+}} lim_{T →∞} \frac{1}{2T} ∫_{−T}^{T} dt ∫_{0}^{∞} dμ(σ)|f(s + c)|^{2}.  (1.6)

Proof: Fix 0 < c < 1, and let ε > 0. Let δ be given by

δ = \frac{ε}{(1 + μ[c, \frac{1}{c} + c]) (1 + ∥f∥_{Ω_{c}})}.

By Bohr’s theorem, the Dirichlet series of f converges uniformly in \overline{Ω_{c}}, so there exists N so that

| ∑_{n=n_{0}}^{N′} a_{n}n^{−s} − f(s)| < δ ∀ s ∈ \overline{Ω_{c}}, ∀ N′ ≥ N.

Then

lim_{T →∞} \frac{1}{2T} ∫_{−T}^{T} dt ∫_{0}^{1/c} dμ(σ)|f(s + c)|^{2} =

lim_{T →∞} \frac{1}{2T} ∫_{−T}^{T} dt ∫_{0}^{1/c} dμ(σ)| ∑_{n=n_{0}}^{N′} a_{n}n^{−(s+c)}|^{2} + O(ε). (1.7)

As

lim_{T →∞} \frac{1}{2T} ∫_{−T}^{T} dt n^{−(σ+it)}m^{−(σ−it)} = δ_{m n} n^{−2σ},

we get from (1.7) that

lim_{T →∞} \frac{1}{2T} ∫_{−T}^{T} dt ∫_{0}^{1/c} dμ(σ)|f(s + c)|^{2} =

∑_{n=n_{0}}^{N′} |a_{n}|^{2} ∫_{0}^{1/c} dμ(σ)n^{−2σ−2c} + O(ε)  (1.8)

for all N′ ≥ N. Taking the limit in (1.8) as c decreases to 0, and noting that ε can be made arbitrarily small for N large enough, we get

∑_{n=n_{0}}^{∞} |a_{n}|^{2}w_{n} = lim_{c→0^{+}} lim_{T →∞} \frac{1}{2T} ∫_{−T}^{T} dt ∫_{0}^{1/c} dμ(σ)|f(s + c)|^{2}.

But this is the same limit as (1.6). □

Note that the integrals

\frac{1}{2T} ∫_{−T}^{T} dt|f(σ + it)|^{2}

are monotonically decreasing as a function of σ, so if μ({0}) = 0, the monotone convergence theorem yields:

Corollary 1.9 Assume the hypotheses of Theorem 1.5, and also that μ({0}) = 0. Then

∑_{n=n_{0}}^{∞} |a_{n}|^{2}w_{n} = lim_{T →∞} \frac{1}{2T} ∫_{−T}^{T} dt ∫_{0}^{∞} dμ(σ)|f(s)|^{2}.  (1.10)

By a multiplier of H_{w} we mean a function ϕ with the property that ϕf is in H_{w} for every f in H_{w}. It follows from the closed graph theorem that for any multiplier ϕ, the operator of multiplication by ϕ, which we denote M_{ϕ}, is bounded. It is somewhat surprising that, although the spaces H_{w} consist of functions analytic in Ω_{1/2}, the multipliers are somehow forced to extend to be analytic on all of Ω_{0}. For the case μ = μ_{0}, the following theorem is due to Hedenmalm, Lindqvist and Seip [9]. The main idea of our proof is constructing an increasing sequence of semi-invariant subspaces for multiplication operators (Equation 1.15).

Theorem 1.11 Let μ satisfy (1.1) and (1.2), and let w_{n} be given by (1.3). Then the multiplier algebra of H_{w} is isometrically isomorphic to H^{∞}(Ω_{0})∩D, where the norm on H^{∞}(Ω_{0}) ∩ D is the supremum of the absolute value on Ω_{0}.

Proof: It is clear that any multiplier ϕ must be in D, just by considering ϕ(s) · n_{0}^{−s}. We shall prove the theorem in two parts: (A) Show that if ϕ ∈ D has σ_{b} = 0, then

∥M_{ϕ}∥ = ∥ϕ∥_{Ω_{0}}.  (1.12)

(B) Show that if ϕ is a multiplier of H_{w}, then ϕ is analytic and bounded in Ω_{0}.

Proof of (A): Suppose ϕ(s) = ∑_{n=1}^{∞} b_{n}n^{−s} is bounded in all half-planes strictly smaller than Ω_{0}. Let f(s) = ∑_{n=n_{0}}^{N} a_{n}n^{−s} be a finite sum in H_{w}. Then ϕf has σ_{b} = 0, so by Theorem 1.5

∥M_{ϕ} f∥^{2} = lim_{c→0^{+}} lim_{T →∞} \frac{1}{2T} ∫_{−T}^{T} dt ∫_{0}^{∞} dμ(σ)|ϕ(σ + c) f(σ + c)|^{2}
≤ ∥ϕ∥_{Ω_{0}}^{2} ∥f∥^{2}.

So if ∥ϕ∥_{Ω_{0}} is finite, then M_{ϕ} is bounded on the dense subspace of H_{w} consisting of finite sums, so extends by continuity to be a multiplier of the whole space.

We must show that ∥M_{ϕ}∥ equals ∥ϕ∥_{Ω_{0}}. So let us assume that ∥M_{ϕ}∥ = 1 and ∥ϕ∥_{Ω_{0}} > 1, and derive a contradiction. (We are not assuming that ∥ϕ∥_{Ω_{0}} is necessarily finite).

For each σ > 0, let

N_{σ} = sup_{t} |ϕ(σ + it)|.

By the Phragmén-Lindelöf theorem, N_{σ} is a strictly decreasing function of σ. Indeed, for σ very large, N_{σ} tends to |b_{1}| which is less than or equal to ∥M_{ϕ}∥ < ∥ϕ∥_{Ω_{0}}, so the conclusion follows by applying the Phragmén-Lindelöf theorem to the function e^{εs}ϕ(s) for an appropriate choice of ε on a vertical strip.

Moreover, in each half-plane Ω_{c} for c > 0, the Dirichlet series of ϕ converges uniformly to ϕ by Bohr’s theorem, so by Theorem 0.4, ϕ is uniformly almost periodic in Ω_{c}. Therefore there exists ε_{1}, ε_{2}, ε_{3}, ε_{4} > 0 so that, for large enough T,

|{t : |ϕ(σ + it)| > 1 + ε_{1}, −T ≤ t ≤ T }| ≥ ε_{2}(2T) ∀ ε_{3} ≤ σ ≤ ε_{3} + ε_{4}.

(1.13)

(Just find some open set where |ϕ| is bigger than one, and by uniform almost periodicty, this will recur as one moves up in the imaginary direction). Since multiplication by ϕ is a contraction, so is multiplication by ϕ^{j} for any positive integer j. Therefore

∥ϕ^{j}(s) n_{0}^{−s}∥^{2} ≤ w_{n_{0}} ∀ j ∈ ℕ.

So by Theorem 1.5, we conclude that

w_{n_{0}} ≥ lim_{T →∞} \frac{1}{2T} ∫_{−T}^{T} dt ∫_{0}^{ε_{4}} dμ(σ)|ϕ^{j}(σ + ε_{3})|^{2}n_{0}^{−2(σ+ε_{3})}
≥ ε_{2} (1 + ε_{1})^{2j} n_{0}^{−4ε_{3}} μ([0, ε_{4}]).  (1.14)

As the right-hand side of (1.14) tends to infinity with j, we get a contradiction.

Proof of (B): Let p_{j} denote the j^{th} prime, and let ℕ_{N} denote the set of positive integers all of whose prime factors are in the set {p_{1}, …, p_{N}}:

ℕ_{N} = {p_{1}^{ν_{1}} ⋯ p_{N}^{ν_{N}} : ν_{1}, …, ν_{N} ∈ ℕ}.

For every positive integer N, let Q_{N} denote orthogonal projection from H_{w} onto the closed linear span of the functions

{n^{−s} : n ∈ ℕ_{N}, n ≥ n_{0}}.

Suppose ϕ is a multiplier of H_{w}. Then we have

Q_{N}M_{ϕ}Q_{N} = M_{Q_{N} (ϕ)}Q_{N} = Q_{N}M_{ϕ}.  (1.15)

Moreover, by a truncated version of the Euler product formula, we have that if f(s) = ∑ a_{n}n^{−s} is any function in H_{w}, then

|Q_{N}(f)(s)| = | ∑_{n∈ℕ_{N}} a_{n}n^{−s}|
≤ (sup |a_{n}|) ∏_{j=1}^{N}(1 − p_{j}^{−σ})^{−1}.

So if the coefficients of f are bounded, then Q_{N}(f) is a bounded function in Ω_{c} for every c > 0. As (Q_{N}f)(s + ε) = Q_{N}(f(s + ε)), if the coefficients of f are O(n^{ε}) for every ε > 0, we have that σ_{b}(Q_{N}f) ≤ ε for every ε > 0, i.e. σ_{b}(Q_{N}f) ≤ 0. By (1.4), as the weights decay more slowly than any negative power of n, it follows that for every f in H_{w}, the coefficients of f are indeed o(n^{ε}) for every ε > 0, and a fortiori this hypothesis holds for every ϕ in the multiplier algebra of H_{w} (since ϕ(s)n_{0}^{−s} is in H_{w}).

Therefore we can conclude that

σ_{b}(Q_{N} ϕ) ≤ 0 ∀ N ≥ 1.

Moreover, by (1.15), multiplication by Q_{N}ϕ on Q_{N}H_{w} is a compression of M_{ϕ}, so

∥M_{Q_{N} ϕ}∥_{Q_{N} H_{w}} ≤ ∥M_{ϕ}∥_{H_{w}} < ∞.

By repeating the argument in Part (A) and estimating

∥(Q_{N} ϕ)^{j} 2^{−νs}∥

for 2^{ν} ≥ n_{0}, we therefore conclude that

∥Q_{N} ϕ∥_{Ω_{0}} ≤ ∥M_{ϕ}∥_{H_{w}} ∀ N.

By a normal families argument, some subsequence of Q_{N} ϕ converges uniformly on compact subsets of Ω_{0} to some H^{∞}(Ω_{0}) function, ψ say. On compact subsets of Ω_{1}, where the Dirichlet series for ϕ converges absolutely, Q_{N} ϕ converges uniformly to ϕ. Therefore ϕ = ψ, and so ϕ must be bounded and analytic in all of Ω_{0}. □

It is a theorem of Khintchine and Kolmogorov that if the series ∑ |c_{n}|^{2} is finite, then almost every series ∑ ±c_{n} converges (see e.g. [10] for a proof). It follows that if ∑ a_{n}n^{−s} is in H_{w}, then for almost every choice of signs, ∑ ±a_{n}n^{−s} will converge in Ω_{0} (and in cl(Ω_{0}) for H_{α} with α ≥ 0).

2 Dirichlet-like Spaces

Throughout this section, let μ be a measure satisfying conditions (1.1) and (1.2) of Section 1, and let w_{n} be defined by (1.3) for n ≥ 2. Define another weight sequence w^{♭} by

w_{n}^{♭} = (log n)^{2} w_{n}.

The space H_{w^{♭}} is exactly the set of functions whose derivatives are in H_{w}, and is analogous to the Dirichlet space. We shall prove that the multipliers of H_{w^{♭}} are contained in the multipliers of H_{w}. One can prove a similar result for higher order derivatives, but for simplicity we stick to the case of a single derivative.

Theorem 2.1 With notation as above, the multipliers of H_{w^{♭}} are contractively contained in the multipliers of H_{w}.

Proof: We shall boot-strap from the following claim.

Claim: There is a constant K < ∞ such that, if ϕ is a multiplier of H_{w^{♭}} of norm one, and both ϕ and ϕ′ have σ_{b} ≤ 0, then ϕ is a multiplier of H_{w} of norm at most K.

Suppose the claim were proved. Let ψ be any multiplier of H_{w^{♭}} of norm one. Then for every N, Q_{N} ψ satisfies the hypotheses of the claim, so is a multiplier of H_{w} of norm at most K. By taking the weak-star limit of a subsequence of Q_{N} ψ, we can conclude that ψ is a multiplier of H_{w} of norm at most K.

To show K must be 1, assume it were greater. Then there would be a multiplier ϕ of H_{w^{♭}} of norm one, which has norm greater than \sqrt{K} as a multiplier of H_{w}. By Theorem 1.11,

∥M_{ϕ^{2}}∥_{H_{w}} = ∥M_{ϕ}∥_{H_{w}}^{2}.

Then ϕ^{2} would be a multiplier of norm one of H_{w^{♭}}, and have norm greater than K as a multiplier of H_{w}, a contradiction.

Proof of Claim: We shall prove the claim with K = \sqrt{2}. Suppose the claim is false. Then there is some finite Dirichlet series

f(s) = ∑_{n=n_{0}}^{N} a_{n}n^{s}

in H_{w} of norm 1 such that ∥ϕf∥ > K. Let

g(s) = ∑_{n=n_{0}}^{N} a_{n} \frac{1}{log n}n^{s}

be the primitive of f, which is of norm one in H_{w^{♭}}. Let

B = {s ∈ Ω_{0} : |ϕ(s)| > 1}.

By Theorem 1.5, there exists c > 0 such that

lim_{T →∞} \frac{1}{2T} ∫_{−T}^{T} dt ∫_{0}^{∞} dμ(σ)|ϕ(s + c)f(s + c)|^{2}χ_{B}(s + c) > K^{2} − 1. (2.2)

As ϕ^{j} is a multiplier of H_{w^{♭}} of norm at most one for every positive integer j, we have

1 ≥ ∥ϕ^{j}g∥_{H_{w^{♭}}} = ∥ϕ^{j}f + jϕ^{j−1}ϕ′g∥_{H_{w}}.

Therefore

1 ≥ lim_{T →∞} \frac{1}{2T} ∫_{−T}^{T} dt ∫_{0}^{∞} dμ(σ)|ϕ(s + c)|^{2(j−1)} |ϕ(s + c)f(s + c)
+ jϕ′(s + c)g(s + c)|^{2}χ_{B}(s + c)
≥ lim_{T →∞} \frac{1}{2T} ∫_{−T}^{T} dt ∫_{0}^{∞} dμ(σ)|ϕ(s + c)f(s + c)
+ jϕ′(s + c)g(s + c)|^{2}χ_{B}(s + c)  (2.3)

By subtracting ϕf + ϕ′g from ϕf + jϕ′g and using Minkowski’s inequality on (2.3), we get

4 ≥ (j − 1)^{2} lim_{T →∞} \frac{1}{2T} ∫_{−T}^{T} dt ∫_{0}^{∞} dμ(σ)|ϕ′(s + c)g(s + c)|^{2}χ_{B}(s + c)

for all j, and so the limit is zero. Therefore by Cauchy-Schwarz, (2.3) becomes

1 ≥ lim_{T →∞} \frac{1}{2T} ∫_{−T}^{T} dt ∫_{0}^{∞} dμ(σ)|ϕ(s + c)f(s + c)|^{2}χ_{B}(s + c).

This contradicts (2.2) if K ≥ \sqrt{2}. □

Let us say a Radon measure ν supported in cl(Ω_{0}) is an α-Carleson measure if there exists some constant C such that

lim_{T →∞} \frac{1}{2T} ∫_{|ℑs|≤T} |f(s)|^{2}dν(s) ≤ C∥f∥_{H_{α}}^{2}

for every finite Dirichlet series f(s) = ∑_{n=2}^{N} a_{n}n^{−s}. Then we have

Corollary 2.4 For 0 < α < 2, the function ϕ is a multiplier of H_{α} if and only if (i) ϕ is in D ∩ H^{∞}(Ω_{0}). and (ii) The measure |ϕ′(s)|^{2}dμ_{α−2}(σ)dt is α-Carleson.

Proof: The necessity of Condition (i) follows from Theorem 2.1. For Condition (ii), observe that by Cauchy’s theorem, if ϕ is in D ∩ H^{∞}(Ω_{0}), then σ_{b}(ϕ′) ≤ 0.

The function ϕ is a multiplier if and only if

∥ϕf′ + ϕ′f∥_{H_{α−2}} ≤ C∥f∥_{H_{α}}

for every finite Dirichlet series f. If ϕ satisfies Condition (i), then

∥ϕf′∥_{H_{α−2}} ≤ ∥ϕ∥_{Ω_{0}} ∥f∥_{H_{α}}.

So such a ϕ is a multiplier if and only if

∥ϕ′f∥_{H_{α−2}}^{2} = lim_{T →∞} \frac{1}{2T} ∫_{−T}^{T} ∫_{0}^{∞} |ϕ′(s)|^{2}|f(s)|^{2}dμ_{α−2}(σ)dt

is controlled by ∥f∥_{H_{α}}^{2}, i.e. if and only if Condition (ii) holds. □

Of course, Corollary 2.4 does not answer the question of what the multipliers of H_{α} are. For this one needs a characterization of those functions ϕ in D that satisfy Condition (ii), as D. Stegenga did for K_{α} in [15]. We do not know what this condition should be.

3 A Space of Dirichlet series with the Pick property

Let H be a Hilbert function space on a set X with reproducing kernel k. We say H has the Pick property if, given any distinct points λ_{1}, …, λ_{N} in X and any complex numbers z_{1}, …, z_{n}, then a necessary and sufficient condition for the existence of a function ϕ in the closed unit ball of the multiplier algebra of H that has the value z_{i} at each λ_{i} is that the matrix

[k(λ_{i}, λ_{j}) (1 − z_{i}z̄_{j})]_{i,j=1}^{N}

be positive semi-definite. We say H has the complete Pick property if, for any positive integer s, any distinct points λ_{1}, …, λ_{N} in X and any s-by-s matrices Z_{1}, …, Z_{n}, then a necessary and sufficient condition for the existence of a function ϕ in the closed unit ball of the multiplier algebra of H ⊗ ℂ^{s} that has the value Z_{i} at each λ_{i} is that the Ns-by-Ns matrix

[k(λ_{i}, λ_{j}) (I − Z_{i}Z_{j}^{∗})]_{i,j=1}^{N}

be positive semi-definite. See [2] for a treatment of complete Pick kernels.

For every integer n ≥ 2, let F (n) be the number of ways n can be factored into a product of integers greater than 1, where the order matters (so e.g. F (6) = 3, because of the three factorings 6, 2 × 3, 3 × 2). Let F (1) = 1. Then the following identity holds [17, 1.2.15]

∑_{n=1}^{∞} \frac{F (n)}{n^{s}} = \frac{1}{2 − ζ(s)}.  (3.1)

For the rest of this section, we shall fix

w_{n} = \frac{1}{F (n)},  (3.2)

and consider the space H_{w} (with n_{0} = 1). The kernel function for H_{w} is then

k(s, u) = \frac{1}{2 − ζ(s + ū)}.  (3.3)

As the reciprocal of k has only one positive square, it follows from the McCullough-Quiggin theorem (see [11], [14], [1]), that k is a complete Pick kernel.

Let ρ = 0.86 … be the unique real number in (1/2, ∞) satisfying ζ(2ρ) = 2. Then the space H_{w} is analytic in the half-plane Ω_{ρ}, and in no larger domain because the kernel function blows up on the boundary of Ω_{ρ}.

As k is a complete Pick kernel, there is a realization formula for the multipliers of H_{w}.

Theorem 3.4 Let w_{n} be defined by (3.2). Then ϕ is in the unit ball of the multiplier algebra of H_{w} if and only if there is an auxiliary Hilbert space L, and a unitary U : ℂ ⊕ L → ℂ ⊕ ℓ^{2} ⊗ L such that, writing U as

U = ( \begin{matrix}& ℂ & L \\ ℂ & A & B \\ ℓ^{2} ⊗ L & C & D \end{matrix} ),

we have

ϕ(s) = A + B(I − E_{s}D)^{−1}E_{s}C.  (3.5)

Here ℓ^{2} is the space of square summable sequences (c_{n})_{n=2}^{∞}, and E_{s} : ℓ^{2}⊗L → L is the contractive linear operator defined for s ∈ Ω_{ρ} on elementary tensors by

E_{s} : (c_{n})_{n=2}^{∞} ⊗ ξ ↦ (∑_{n=2}^{∞} c_{n}n^{−s}) ξ.

Theorem 3.4 is a special case of the realization formula for general complete Pick kernels (see [4], [3], [18], [2]), and we omit the proof.

Notice that ϕ(s) defined by (3.5) is guaranteed to be holomorphic only if ∥E_{s}∥ = \sqrt{ζ(2σ) − 1} is less than 1, i.e. if s is in Ω_{ρ}. It turns out that Ω_{ρ} is indeed the maximal domain of analyticity of the multiplier algebra (which we shall denote Mult(H_{w})).

We shall prove this assertion via interpolating sequences. Let {λ_{i}}_{i=1}^{∞} be a sequence in Ω_{ρ}. We say {λ_{i}}_{i=1}^{∞} is an interpolating sequence for H_{w} if the linear operator

Λ : f ↦ (\frac{f(λ_{i})}{∥k_{λ_{i}}∥} )_{i=1}^{∞}

maps H_{w} to l^{2} surjectively. We say {λ_{i}}_{i=1}^{∞} is an interpolating sequence for Mult(H_{w}) if, for any bounded sequence (w_{i})_{i=1}^{∞}, there is a function ϕ in Mult(H_{w}) with ϕ(λ_{i}) = w_{i}.

The sequence {λ_{i}}_{i=1}^{∞} is an interpolating sequence for H_{w} if and only if the associated normalized Gram matrix

G_{ij} = \frac{⟨k_{λ_{j}}, k_{λ_{i}}⟩}{∥k_{λ_{j}}∥ ∥k_{λ_{i}}∥}

is bounded above and below (for a proof of this assertion, see e.g. [13] or [2]). Therefore, every sequence {λ_{i}}_{i=1}^{∞} that tends to the line {σ = ρ} has a subsequence that is an interpolating sequence for H_{w}: just choose the n^{th} element in the subsequence so that the first n − 1 entries in the n^{th} row and n^{th} column of G are smaller in modulus than 2^{−n}, and then the off-diagonal entries will have Hilbert-Schmidt norm smaller than 1.

It is a theorem of D. Marshall and C. Sundberg [12] that, for any space with the Pick property, the interpolating sequences for the Hilbert space and for its multiplier algebra coincide (the proof can also be found in [2]). Coupling this with the previous observation, we conclude that for every boundary point of Ω_{ρ}, there is a sequence that converges to that boundary point and that is interpolating for Mult(H_{w}). Choosing w_{i} to alternate 1 and −1, we conclude that there is a function in Mult(H_{w}) that cannot be continued analytically to any neighborhood of that boundary point. Indeed we can do more. Provided ℜ(λ_{i}) converges to ρ quickly enough, the sequence {λ_{i}} will be interpolating for H_{w} regardless of how ℑ(λ_{i}) is chosen. So we can choose a sequence that accumulates on the whole line {ℜ(s) = ρ}, and conclude that there is a single function ϕ in Mult(H_{w}) that has Ω_{ρ} as its maximal domain of analyticity. Thus we have proved:

Theorem 3.6 With notation as above, the largest common domain of analyticity of the multipliers of H_{w} is Ω_{ρ}.

Is there a better way to describe the interpolating sequences for Mult(H_{w})

(or H_{w}) than requiring that the normalized Grammian be bounded above and below?

Conjecture 3.7 Let w_{n} be given by (3.2). Then the sequence {λ_{i}}_{i=1}^{∞} is interpolating for Mult(H_{w}) if and only if

(i) There exists some constant C such that, for any i ≠ j, there is a function ϕ in Mult(H_{w}) of norm less than C with ϕ(λ_{i}) = 0 and ϕ(λ_{j}) = 1. and (ii) There exists some constant C such that, for any f in H_{w},

∑_{i=1}^{∞} \frac{|f(λ_{i})|^{2}}{∥k_{λ_{i}}∥^{2}} ≤ C ∥f∥_{H_{w}}^{2}.

Condition (i) is clearly necessay, and condition (ii) is equivalent to requiring that the normalized Grammian be bounded. So the question is whether weak separation (condition (i)) and the Carleson measure condition (ii) suffice. The conjecture is a special case of Question 9.57 in [2].

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