Dirichlet–Drury–Arveson Spaces

• Dirichlet–Drury–Arveson-type spaces are reproducing kernel Hilbert/Banach spaces of holomorphic functions defined via weighted sequence norms that generalize classical function spaces.
• These spaces leverage complete Pick kernels and universal embeddings to connect finite and infinite-dimensional domains with practical applications in operator theory.
• They bridge classical one-variable spaces with multi-variable and Dirichlet series contexts, offering insights into cyclicity, multiplier algebras, and spectral analysis.

A Dirichlet–Drury–Arveson-type space is a reproducing kernel Hilbert or Banach space of holomorphic functions on a multi-variable domain (typically a polydisc or an infinite-dimensional unit ball) defined via weighted sequence spaces, generalizing classical Dirichlet, Drury–Arveson, Hardy, and Besov–Sobolev spaces. These spaces are pivotal in function theory, operator theory, and the paper of Hilbert spaces of Dirichlet series, featuring intricate relationships with complete Pick kernels and multiplier algebras. The infinite-dimensional context, cyclicity properties, and universality phenomena differentiate the Dirichlet–Drury–Arveson-type settings from their classical finite-dimensional analogs.

1. Definition and Kernel Structure

Let $N\geq1$ (possibly infinite) and fix non-negative parameters $s,t\geq0$. For multi-indices $\alpha=(\alpha_1,\dots,\alpha_N)\in\mathbb{Z}_+^N$, set $|\alpha|=\alpha_1+\cdots+\alpha_N$ and $\alpha!=\alpha_1!\cdots\alpha_N!$. Define the sequence weight

$$w_\alpha^{(s,t)} = \left(\frac{\alpha!}{|\alpha|!}\right)^t\cdot (1+|\alpha|)^s.$$

For $1\leq p<\infty$, the Banach space $\ell^p(\mathbb{Z}_+^N, w^{(s,t)})$ consists of all $x=(x_\alpha)$ with

$$\|x\|^p = \sum_\alpha |x_\alpha|^p (w_\alpha^{(s,t)})^p < \infty.$$

Under the Bohr (Fourier–Bohr) transform, $$x\mapsto Bx(\zeta) = \sum_{\alpha} x_\alpha \zeta^\alpha$$ defines holomorphic functions on the Reinhardt domain $D_{X'}^N$, where $X'$ is the Köthe dual; for $N=\infty$, $D_{X'}^\infty$ is the polydisc or infinite unit ball depending on $(s,t,p)$ (Nikolski, 22 Nov 2025).

The corresponding reproducing kernel (for $p=2$) is

$$K(\zeta,\eta) = \sum_{\alpha\in\mathbb{Z}_+^N} \frac{\zeta^\alpha \overline{\eta}^\alpha}{[w_\alpha^{(s,t)}]^2}.$$

For $(s=0, t=0)$ and $N=1$ this reduces to the Hardy kernel $K(z,w)=1/(1-z\overline{w})$. For the Drury–Arveson case $(s=0, t=1/2)$ it yields

$$K(\zeta,\eta) = \sum_{n=0}^\infty \langle\zeta,\eta\rangle^n = \frac{1}{1-\langle\zeta,\eta\rangle}$$

on the unit ball $B_2^N$ (Nikolski, 22 Nov 2025).

2. Complete Pick Spaces and Universal Embedding

A central principle is the universality of the Drury–Arveson space $H^2_d$ for complete Pick kernels: every RKHS with the complete Pick property can be realized as a quotient of some $H^2_d$ via composition with a suitable map $b:X\to B_d$. The Agler–McCarthy theorem guarantees that for a RKHS $\mathcal{H}$ with kernel $K$ having the Pick property, there exists $d\in\mathbb{N}\cup\{\infty\}$ and $b:X\to B_d$ so that

$K(z,w) = (1-\langle b(z), b(w)\rangle)^{-1}$

(after absorbing weights into a renorming function) (Hartz, 2021).

For the classical Dirichlet space $\mathcal{D}$, this embedding necessarily occurs with $d=\infty$: the explicit map

$$b(z) = (\sqrt{c_1}\, z,\, \sqrt{c_2}\, z^2,\, \sqrt{c_3}\, z^3,\dots) \in \ell^2$$

exhibits the infinite-dimensionality, since the Dirichlet kernel cannot be embedded into any finite $B_d$ while preserving the surjectivity of the induced homomorphism on multiplier algebras (Hartz, 2021, Rochberg, 2016).

3. Dirichlet Series and Infinite-Dimensional Polydisks

Spaces of Dirichlet series with weighted norms (arising from $\sum a_n n^{-s}$ expansions) can be identified with holomorphic function spaces on $D^\infty$ or $B_\infty$ via the Bohr lift: writing $n=p_1^{\nu_1} \cdots p_k^{\nu_k}$ and $z^\nu=z_1^{\nu_1}\cdots z_k^{\nu_k}$, the map $$f(s)=\sum a_n n^{-s} \mapsto \tilde{f}(z)=\sum a_n z^\nu$$ is an isometric isomorphism (Olsen, 2010). The kernel structure and weight asymptotics determine local boundary behavior, Carleson measures, and interpolation properties, all governed by the order $(\alpha,\beta)$ of the partial sums $$W(X)=\sum_{n\leq X} w_n \sim X^\alpha (\log X)^\beta$$.

Infinite-variable analogs include:

• Hardy on $D^\infty$ ($w_n\equiv 1$)
• Bergman on $D^\infty$ (coefficients $w_n=d_{\beta+1}(n)$)
• Besov–Sobolev on $B_\infty$ (weights $w_n =$ multinomial coefficients)
• Drury–Arveson on $B_\infty$ (kernel $1/(1-\langle z,\zeta\rangle)$) (Olsen, 2010, Nikolski, 22 Nov 2025)

4. Multiplier Algebras and Quotient Structures

The multiplier algebra $\mathrm{Mult}(\mathcal{H})$ of a Dirichlet–Drury–Arveson-type space is tightly linked to its embedding dimension. For $\mathcal{D}$, the induced homomorphism $$\Phi: \mathrm{Mult}(H^2_d)\to \mathrm{Mult}(\mathcal{D}),\ \varphi\mapsto \varphi\circ b$$, is a surjective, contractive (and in fact isometric) quotient map only for $d=\infty$. The finite-dimensional Drury–Arveson multiplier algebra $\mathrm{Mult}(H^2_d)\simeq H^\infty(B_d)$ is tractable and has maximal-ideal fibers over $\overline{B}_d$; by contrast, $\mathrm{Mult}(\mathcal{D})$ inherits complexity (corona phenomena) from the non-metrizable $\beta \mathbb{N}\setminus \mathbb{N}$ component forced by $d=\infty$ (Hartz, 2021).

More generally, for Dirichlet series RKHSs, if the embedding is $f(s)=(b_1 n_1^{-s}, b_2 n_2^{-s}, \ldots)$, then rational independence of the $\log n_k$ implies the closure of the image is all of $B_d$ and the multiplier algebra is unitarily, isometrically isomorphic to $\mathrm{Mult}(H^2_d)$; otherwise, it is a proper quotient (McCarthy et al., 2015). Every complete Pick algebra is a quotient of such a Dirichlet–Drury–Arveson-type multiplier algebra.

5. Cyclicity, Dilated Systems, and Factorization

The cyclicity of dilated systems (i.e., when the family $\{D_n x:n\geq1\}$ spans the space) in Dirichlet–Drury–Arveson-type sequence lattices is closely linked to multiplicative structure, invertibility in associated power-series algebras, and the notion of dominating free term. Under the Bohr transform, dilations correspond to monomial multiplication in function spaces on $D_{X'}^\infty$ or $B_{p'}^N$. For totally multiplicative sequences, cyclicity is guaranteed by the extended Haar lemma, and functions with sufficiently large free term (in $\ell^1$ Banach algebras) are invertible and thus cyclic (Nikolski, 22 Nov 2025).

Polynomials with linear factorization—splitting into products of affine factors—are cyclic if they have no zeros on the spectrum. Cyclicity results vary with underlying domain and parameter values: in Drury–Arveson or Hilbertian Dirichlet-type cases, generic cyclicity may require deeper boundary conditions, especially in finite dimensions (Nikolski, 22 Nov 2025).

6. Operator Theory and Function-Theoretic Consequences

The infinite embedding dimension yields significant implications for operator models. The Dirichlet shift cannot be modeled by any finite commuting $d$-tuple of weighted shifts on the ball, precluding finite-dimensional function-theoretic realizations for certain Dirichlet-type operators (Hartz, 2021, Rochberg, 2016). In Drury–Arveson spaces, the canonical Gleason problem admits contractive multi-operator solutions, and de Branges–Rovnyak-type subspaces may or may not be invariant under coordinate multipliers, depending on quasi-extremality—through the noncommutative Herglotz representation and GNS constructions (Jury, 2013).

Boundary behavior, spectral analysis, and Clark theory extend into the multivariable and infinite-variable context, requiring new operator-algebraic machinery, such as Cuntz–Toeplitz algebras and noncommutative Fantappiè transforms.

7. Connections to Classical Scales and Invariant Theory

Dirichlet–Drury–Arveson-type spaces interpolate between the classical Hardy, Bergman, Dirichlet, Besov–Sobolev, and de Branges–Rovnyak settings. The table below summarizes parameter regimes and space identification (adapted from (Olsen, 2010, Nikolski, 22 Nov 2025)):

$(\alpha, \beta)$	Weight $W(X)$	Local Model	Infinite-var. space
$(0,0)$	$X$	$H^2(C_{1/2})$	Dirichlet–Hardy $H^2(D^\infty)$
$(1,0)$	$X/\ln X$	$D_0(C_{1/2})$	Drury–Arveson $H^2(B_\infty)$
$(1,\beta>0)$	$X(\ln X)^\beta$	$D_{-\beta}(C_{1/2})$	Bergman $A_\beta(D^\infty)$
$(\alpha>0,0)$	$X^\alpha$	$D_{1-\alpha}(C_{1/2})$	Besov–Sobolev $B_2^\alpha(B_\infty)$

This framework enables a unified treatment of cyclicity and multiplier algebra structure across domains and parameter regimes, showing the precise specialization to one-variable or multi-variable operator theory, and highlighting the essential difference between finite and infinite-dimensional spaces with respect to universality, metric geometry, and functional models (Olsen, 2010, McCarthy et al., 2015, Nikolski, 22 Nov 2025).