Drury Arveson Space Research Articles

Cyclicity in the Drury-Arveson space and other weighted Besov spaces

Let H \mathcal {H} be a space of analytic functions on the unit ball B d {\mathbb {B}_d} in C d \mathbb {C}^d with multiplier algebra M u l t ( H ) \mathrm {Mult}(\mathcal {H}) . A function f ∈ H f\in \mathcal {H} is called cyclic if the set [ f ] [f] , the closure of { φ f : φ ∈ M u l t ( H ) } \{\varphi f: \varphi \in \mathrm {Mult}(\mathcal {H})\} , equals H \mathcal {H} . For multipliers we also consider a weakened form of the cyclicity concept. Namely for n ∈ N 0 n\in \mathbb {N}_0 we consider the classes C n ( H ) = { φ ∈ M u l t ( H ) : φ ≠ 0 , [ φ n ] = [ φ n + 1 ] } . \begin{equation*} \mathcal {C}_n(\mathcal {H})=\{\varphi \in \mathrm {Mult}(\mathcal {H}):\varphi \ne 0, [\varphi ^n]=[\varphi ^{n+1}]\}. \end{equation*} Many of our results hold for N N :th order radially weighted Besov spaces on B d {\mathbb {B}_d} , H = B ω N \mathcal {H}= B^N_\omega , but we describe our results only for the Drury-Arveson space H d 2 H^2_d here. Letting C s t a b l e [ z ] \mathbb {C}_{stable}[z] denote the stable polynomials for B d {\mathbb {B}_d} , i.e. the d d -variable complex polynomials without zeros in B d {\mathbb {B}_d} , we show that a m p ; if d is odd, then C s t a b l e [ z ] ⊆ C d − 1 2 ( H d 2 ) , and a m p ; if d is even, then C s t a b l e [ z ] ⊆ C d 2 − 1 ( H d 2 ) . \begin{align*} &\text { if } d \text { is odd, then } \mathbb {C}_{stable}[z]\subseteq \mathcal {C}_{\frac {d-1}{2}}(H^2_d), \text { and }\\ &\text { if } d \text { is even, then } \mathbb {C}_{stable}[z]\subseteq \mathcal {C}_{\frac {d}{2}-1}(H^2_d). \end{align*} For d = 2 d=2 and d = 4 d=4 these inclusions are the best possible, but in general we can only show that if 0 ≤ n ≤ d 4 − 1 0\le n\le \frac {d}{4}-1 , then C s t a b l e [ z ] ⊈ C n ( H d 2 ) \mathbb {C}_{stable}[z]\nsubseteq \mathcal {C}_n(H^2_d) . For functions other than polynomials we show that if f , g ∈ H d 2 f,g\in H^2_d such that f / g ∈ H ∞ f/g\in H^\infty and f f is cyclic, then g g is cyclic. We use this to prove that if f , g f,g extend to be analytic in a neighborhood of B d ¯ \overline {{\mathbb {B}_d}} , have no zeros in B d {\mathbb {B}_d} , and the same zero sets on the boundary, then f f is cyclic in ∈ H d 2 \in H^2_d if and only if g g is. Furthermore, if the boundary zero set of f ∈ H d 2 ∩ C ( B d ¯ ) f\in H^2_d\cap C(\overline {{\mathbb {B}_d}}) embeds a cube of real dimension ≥ 3 \ge 3 , then f f is not cyclic in the Drury-Arveson space.

A new formalization of Dirichlet-type spaces

Applications and findings involving the norm of the weighted Dirichlet space Dα are steadily being discovered. There are notable characterizations of Dα through estimations of its norm. Applications include determining the conditions for boundedness of a Bergman type operator, and characterizing functions in the Drury–Arveson space that induce bounded Hankel operators. This paper improves the norm estimation by extending the domain of relevant parameters, and establishing the asymptotic formula with precise constants.

Perturbations of principal submodules in the Drury–Arveson space

We study the geometry in the perturbations of principal submodules in the Drury–Arveson space. We show that the perturbations give rise to smooth vector bundles of Hilbert spaces which are equipped with natural Hermitian connections. We compute the associated parallel transport operators and explore properties of the monodromy.

The common range of co-analytic Toeplitz operators on the Drury-Arveson space

We characterize the common range of the adjoints of cyclic multiplication operators on the Drury-Arveson space. We show that a function belongs to this common range if and only if its Taylor coefficients satisfy a simple decay condition. To achieve this, we introduce the uniform Smirnov class on the ball and determine its dual space. We show that the dual space of the uniform Smirnov class equals the dual space of the strictly smaller Smirnov class of the Drury-Arveson space, and that this in turn equals the common range of the adjoints of cyclic multiplication operators.

Interpolation and duality in algebras of multipliers on the ball

We study the multiplier algebras A(\mathcal{H}) obtained as the closure of the polynomials on certain reproducing kernel Hilbert spaces \mathcal{H} on the ball \mathbb{B}_d of \mathbb{C}^d . Our results apply, in particular, to the Drury–Arveson space, the Dirichlet space and the Hardy space on the ball. We first obtain a complete description of the dual and second dual spaces of A(\mathcal H) in terms of the complementary bands of Henkin and totally singular measures for \operatorname{Mult}(\mathcal{H}) . This is applied to obtain several definitive results in interpolation. In particular, we establish a sharp peak interpolation result for compact \operatorname{Mult}(\mathcal{H}) -totally null sets as well as a Pick and peak interpolation theorem. Conversely, we show that a mere interpolation set is \operatorname{Mult}(\mathcal{H}) -totally null.

Non-commutative measure theory: Henkin and analytic functionals on $$\mathrm {C}^*$$-algebras

Henkin functionals on non-commutative $$\mathrm {C}^*$$ -algebras have recently emerged as a pivotal link between operator theory and complex function theory in several variables. Our aim in this paper is characterize these functionals through a notion of absolute continuity, inspired by a seminal theorem of Cole and Range. To do this, we recast the problem as a question in non-commutative measure theory. We develop a Glicksberg–König–Seever decomposition of the dual space of a $$\mathrm {C}^*$$ -algebra into an absolutely continuous part and a singular part, relative to a fixed convex subset of states. Leveraging this tool, we show that Henkin functionals are absolutely continuous with respect to the so-called analytic functionals if and only if a certain compatibility condition is satisfied by the ambient weak- $$*$$ topology. In contrast with the classical setting, the issue of stability under absolute continuity is not automatic in this non-commutative framework, and we illustrate its key role in sharpening our description of Henkin functionals. Our machinery yields new insight when specialized to the multiplier algebras of the Drury–Arveson space and of the Dirichlet space, and to Popescu’s noncommutative disc algebra. As another application, we make a contribution to the theory of non-commutative peak and interpolation sets.

Multiplier tests and subhomogeneity of multiplier algebras

Multipliers of reproducing kernel Hilbert spaces can be characterized in terms of positivity of $n \times n$ matrices analogous to the classical Pick matrix. We study for which reproducing kernel Hilbert spaces it suffices to consider matrices of bounded size $n$. We connect this problem to the notion of subhomogeneity of non-selfadjoint operator algebras. Our main results show that multiplier algebras of many Hilbert spaces of analytic functions, such as the Dirichlet space and the Drury-Arveson space, are not subhomogeneous, and hence one has to test Pick matrices of arbitrarily large matrix size $n$. To treat the Drury-Arveson space, we show that multiplier algebras of certain weighted Dirichlet spaces on the disc embed completely isometrically into the multiplier algebra of the Drury-Arveson space.

Multipliers and operator space structure of weak product spaces

In the theory of reproducing kernel Hilbert spaces, weak product spaces generalize the notion of the Hardy space $H^1$. For complete Nevanlinna-Pick spaces $\mathcal H$, we characterize all multipliers of the weak product space $\mathcal H \odot \mathcal H$. In particular, we show that if $\mathcal H$ has the so-called column-row property, then the multipliers of $\mathcal H$ and of $\mathcal H \odot \mathcal H$ coincide. This result applies in particular to the classical Dirichlet space and to the Drury-Arveson space on a finite dimensional ball. As a key device, we exhibit a natural operator space structure on $\mathcal H \odot \mathcal H$, which enables the use of dilations of completely bounded maps.

Shift operators on harmonic Hilbert function spaces on real balls and von Neumann inequality

On harmonic function spaces, we define shift operators using zonal harmonics and partial derivatives, and develop their basic properties. These operators turn out to be multiplications by the coordinate variables followed by projections on harmonic subspaces. This duality gives rise to a new identity for zonal harmonics. We introduce large families of reproducing kernel Hilbert spaces of harmonic functions on the unit ball of Rn and investigate the action of the shift operators on them. We prove a dilation result for a commuting row contraction which is also what we call harmonic type. As a consequence, we show that the norm of one of our spaces G˘ is maximal among those spaces with contractive norms on harmonic polynomials. We then obtain a von Neumann inequality for harmonic polynomials of a commuting harmonic-type row contraction. This yields the maximality of the operator norm of a harmonic polynomial of the shift on G˘ making this space a natural harmonic counterpart of the Drury-Arveson space.

Distance between reproducing kernel Hilbert spaces and geometry of finite sets in the unit ball

In this paper we study the relationships between a reproducing kernel Hilbert space, its multiplier algebra, and the geometry of the point set on which they live. We introduce a variant of the Banach-Mazur distance suited for measuring the distance between reproducing kernel Hilbert spaces, that quantifies how far two spaces are from being isometrically isomorphic as reproducing kernel Hilbert spaces. We introduce an analogous distance for multiplier algebras, that quantifies how far two algebras are from being completely isometrically isomorphic. We show that, in the setting of finite dimensional quotients of the Drury-Arveson space, two spaces are “close” to one another if and only if their multiplier algebras are “close”, and that this happens if and only if one of the underlying point sets is close to an image of the other under a biholomorphic automorphism of the unit ball. These equivalences are obtained as corollaries of quantitative estimates that we prove.

Cyclicity Preserving Operators on Spaces of Analytic Functions in $${\mathbb {C}}^n$$

For spaces of analytic functions defined on an open set in $\mathbb{C}^n$ that satisfy certain nice properties, we show that operators that preserve shift-cyclic functions are necessarily weighted composition operators. Examples of spaces for which this result holds true consist of the Hardy space $H^p(\mathbb{D}^n) \, (0 < p < \infty)$, the Drury-Arveson space $\mathcal{H}^2_n$, and the Dirichlet-type space $\mathcal{D}_{\alpha} \, (\alpha \in \mathbb{R})$. We focus on the Hardy spaces and show that when $1 \leq p < \infty$, the converse is also true. The techniques used to prove the main result also enable us to prove a version of the Gleason-Kahane-\.Zelazko theorem for partially multiplicative linear functionals on spaces of analytic functions in more than one variable.

Weak products of complete Pick spaces

Let H be the Drury-Arveson or Dirichlet space of the unit ball of Cd. The weak product H ☉ H of H is the collection of all functions h that can be written as h =∑∞n=1 fngn, where ∑∞n=1 ||fn|| ||gn|| < ∞. We show that H ☉ H is contained in the Smirnov class of H; that is, every function in H ☉ H is a quotient of two multipliers of H, where the function in the denominator can be chosen to be cyclic in H . As a consequence, we show that the map N → closH ☉H N establishes a one-to-one and onto correspondence between the multiplier invariant subspaces of H and of H ☉ H . The results hold for many weighted Besov spaces H in the unit ball of Cd provided the reproducing kernel has the complete Pick property. One of our main technical lemmas states that, for weighted Besov spaces H that satisfy what we call the multiplier inclusion condition, any bounded column multiplication operator H → ⊕∞n=1 H induces a bounded row multiplication operator ⊕∞n=1 H → H . For the Drury-Arveson space Hd2 this leads to an alternate proof of the characterization of interpolating sequences in terms of weak separation and Carleson measure conditions. (Less)

Gelfand transforms and boundary representations of complete Nevanlinna–Pick quotients

The main objects under study are quotients of multiplier algebras of certain complete Nevanlinna–Pick spaces, examples of which include the Drury–Arveson space on the ball and the Dirichlet space on the disc. We are particularly interested in the non-commutative Choquet boundaries for these quotients. Arveson’s notion of hyperrigidity is shown to be detectable through the essential normality of some natural multiplication operators, thus extending previously known results on the Arveson–Douglas conjecture. We also highlight how the non-commutative Choquet boundaries of these quotients are intertwined with their Gelfand transforms being completely isometric. Finally, we isolate analytic and topological conditions on the so-called supports of the underlying ideals that clarify the nature of the non-commutative Choquet boundaries.

Row Contractions Annihilated by Interpolating Vanishing Ideals

Abstract We study similarity classes of commuting row contractions annihilated by what we call higher-order vanishing ideals of interpolating sequences. Our main result exhibits a Jordan-type direct sum decomposition for these row contractions. We illustrate how the family of ideals to which our theorem applies is very rich, especially in several variables. We also give two applications of the main result. First, we obtain a purely operator theoretic characterization of interpolating sequences for the multiplier algebra of the Drury–Arveson space. Second, we classify certain classes of cyclic commuting row contractions up to quasi-similarity in terms of their annihilating ideals. This refines some of our recent work on the topic. We show how this classification is sharp: in general quasi-similarity cannot be improved to similarity. The obstruction to doing so is a scarcity of norm-controlled similarities between commuting tuples of nilpotent matrices, and we investigate this question in detail.

Commutant Lifting and Nevanlinna–Pick Interpolation in Several Variables

This paper concerns a commutant lifting theorem and a Nevanlinna–Pick type interpolation result in the setting of multipliers from vector-valued Drury–Arveson space to a large class of vector-valued reproducing kernel Hilbert spaces over the unit ball in \({\mathbb {C}}^n\). The special case of reproducing kernel Hilbert spaces includes all natural examples of Hilbert spaces like Hardy space, Bergman space and weighted Bergman spaces over the unit ball.

A closer look at a Poisson-like condition on the Drury-Arveson space

A closer look at a Poisson-like condition on the Drury-Arveson space

Committee Spaces and the Random Column–Row Property

A committee space is a Hilbert space of power series, perhaps in several or noncommuting variables, such that $\|z^\alpha\|\|z^\beta\| \geq \|z^{\alpha+\beta}\|.$ Such a space satisfies the true column-row property when ever the map transposing a column multiplier to a row multiplier is contractive. We describe a model for random multipliers and show that such random multipliers satisfy the true column-row property. We also show that the column-row property holds asymptotically for large random Nevanlinna-Pick interpolation problems where the nodes are chosen identically and independently. These results suggest that finding a violation of the true column-row property for the Drury-Arveson space via na\"ive random search is unlikely.

Operators affiliated to the free shift on the Free Hardy Space

The Smirnov class for the classical Hardy space is the set of ratios of bounded analytic functions in the open complex unit disk with outer denominators. This definition extends naturally to the commutative and non-commutative multi-variable settings of the Drury-Arveson space and the full Fock space over Cd. Identifying the Fock space with the free multi-variable Hardy space of non-commutative or free holomorphic functions in a non-commutative open unit ball in several matrix-variables, we prove that any closed, densely-defined operator affiliated to the right free multiplier algebra of the full Fock space acts as right multiplication by a non-commutative function in the right free Smirnov class (and analogously, replacing ‘right’ with ‘left’).

Paley–Wiener Theorems on the Siegel Upper Half-Space

In this paper we study spaces of holomorphic functions on the Siegel upper half-space $\mathcal U$ and prove Paley-Wiener type theorems for such spaces. The boundary of $\mathcal U$ can be identified with the Heisenberg group $\mathbb H_n$. Using the group Fourier transform on $\mathbb H_n$, Ogden-Vagi proved a Paley-Wiener theorem for the Hardy space $H^2(\mathcal U)$. We consider a scale of Hilbert spaces on $\mathcal U$ that includes the Hardy space, the weighted Bergman spaces, the weighted Dirichlet spaces, and in particular the Drury-Arveson space, and the Dirichlet space $\mathcal D$. For each of these spaces, we prove a Paley-Wiener theorem, some structure theorems, and provide some applications. In particular we prove that the norm of the Dirichlet space modulo constants $\dot{\mathcal D}$ is the unique Hilbert space norm that is invariant under the action of the group of automorphisms of $\mathcal U$.

Hyponormality of Toeplitz operators in several variables by the weighted shifts approach

ABSTRACT In this paper, we characterize hyponormality of a class of operators related to a tuple of weighted shifts. As applications we construct examples of hyponormal Toeplitz operators on Hilbert spaces of holomorphic functions on the polydisk or the unit ball which include the Bergman space of the polydisk, the Hardy space of the unit ball and the Drury-Arveson space.