Drury Arveson Space Research Articles

The Hopf Structure of Some Dual Operator Algebras

We study the Hopf structure of a class of dual operator algebras corresponding to certain semigroups. This class of algebras arises in dilation theory, and includes the noncommutative analytic Toeplitz algebra and the multiplier algebra of the Drury–Arveson space, which correspond to the free semigroup and the free commutative semigroup respectively. The preduals of the algebras in this class naturally form Hopf (convolution) algebras. The original algebras and their preduals form (non-self-adjoint) dual Hopf algebras in the sense of Effros and Ruan. We study these algebras from this perspective, and obtain a number of results about their structure.

Contractively Included Subspaces of Pick Spaces

Pick spaces are a class of reproducing kernel Hilbert spaces that generalize the classical Hardy space and the Drury–Arveson reproducing kernel spaces. We give characterizations of certain contractively included subspaces of Pick spaces. These generalize the characterization of closed invariant subspaces of Trent and McCullough, as well as results for the Drury–Arveson space obtained by Ball, Bolotnikov and Fang.

Clark theory in the Drury–Arveson space

We extend the basic elements of Clark's theory of rank-one perturbations of backward shifts, to row-contractive operators associated to de Branges–Rovnyak type spaces H(b) contrastively contained in the Drury–Arveson space on the unit ball in Cd. The Aleksandrov–Clark measures on the circle are replaced by a family of states on a certain noncommutative operator system, and the backward shift is replaced by a canonical solution to the Gleason problem in H(b). In addition we introduce the notion of a “quasi-extreme” multiplier of the Drury–Arveson space and use it to characterize those H(b) spaces that are invariant under multiplication by the coordinate functions.

On the Isomorphism Question for Complete Pick Multiplier Algebras

Every multiplier algebra of an irreducible complete Pick kernel arises as the restriction algebra $${\mathcal{M}_V = \{f \big|_V : f \in \mathcal{M}_d\}}$$ , where d is some integer or $${\infty, \mathcal{M}_d}$$ is the multiplier algebra of the Drury-Arveson space $${H^2_d}$$ , and V is a subvariety of the unit ball. For finite dimensional d it is known that, under mild assumptions, every isomorphism between two such algebras $${\mathcal{M}_V}$$ and $${\mathcal{M}_W}$$ is induced by a biholomorphism between W and V. In this paper we consider the converse, and obtain positive results in two directions. The first deals with the case where V is the proper image of a finite Riemann surface. The second deals with the case where V is a disjoint union of varieties.

The defect sequence for contractive tuples

We introduce the defect sequence for a contractive tuple of Hilbert space operators and investigate its properties. The defect sequence is a sequence of numbers, called defect dimensions associated with a contractive tuple. We show that there are upper bounds for the defect dimensions. The tuples for which these upper bounds are obtained, are called maximal contractive tuples. The upper bounds are different in the non-commutative and in the commutative case. We show that the creation operators on the full Fock space and the co-ordinate multipliers on the Drury–Arveson space are maximal. We also study pure tuples and see how the defect dimensions play a role in their irreducibility.

Topological isomorphisms for some universal operator algebras

Let I⊂C[z1,…,zd] be a radical homogeneous ideal, and let AI be the norm-closed non-selfadjoint algebra generated by the compressions of the d-shift on Drury–Arveson space Hd2 to the co-invariant subspace Hd2⊖I. Then AI is the universal operator algebra for commuting row contractions subject to the relations in I. We ask under which conditions are there topological isomorphisms between two such algebras AI and AJ? We provide a positive answer to a conjecture of Davidson, Ramsey and Shalit: AI and AJ are topologically isomorphic if and only if there is an invertible linear map A on Cd which maps the vanishing locus of J isometrically onto the vanishing locus of I. Most of the proof is devoted to showing that finite algebraic sums of full Fock spaces over subspaces of Cd are closed. This allows us to show that the map A induces a completely bounded isomorphism between AI and AJ.

Coincidence of Schur multipliers of the Drury-Arveson space

Coincidence of Schur multipliers of the Drury-Arveson space

Corrigendum to “Multipliers and essential norm on the Drury-Arveson space”

Corrigendum to “Multipliers and essential norm on the Drury-Arveson space”

Canonical transfer-function realization for Schur multipliers on the Drury-Arveson space and models for commuting row contractions

Canonical transfer-function realization for Schur multipliers on the Drury-Arveson space and models for commuting row contractions

A harmonic analysis approach to essential normality of principal submodules

Guo and the second author have shown that the closure [ I ] in the Drury–Arveson space of a homogeneous principal ideal I in C [ z 1 , … , z n ] is essentially normal. In this note, the authors extend this result to the closure of any principal polynomial ideal in the Bergman space. In particular, the commutators and cross-commutators of the restrictions of the multiplication operators are shown to be in the Schatten p-class for p > n . The same is true for modules generated by polynomials with vector-valued coefficients. Further, the maximal ideal space X I of the resulting C ⁎ -algebra for the quotient module is shown to be contained in Z ( I ) ∩ ∂ B n , where Z ( I ) is the zero variety for I, and to contain all points in ∂ B n that are limit points of Z ( I ) ∩ B n . Finally, the techniques introduced enable one to study a certain class of weight Bergman spaces on the ball.

Local properties of Hilbert spaces of Dirichlet series

We show that the asymptotic behavior of the partial sums of a sequence of positive numbers determine the local behavior of the Hilbert space of Dirichlet series defined using these as weights. This extends results recently obtained describing the local behavior of Dirichlet series with square summable coefficients in terms of local integrability, boundary behavior, Carleson measures and interpolating sequences. As these spaces can be identified with functions spaces on the infinite-dimensional polydisk, this gives new results on the Dirichlet and Bergman spaces on the infinite-dimensional polydisk, as well as the scale of Besov–Sobolev spaces containing the Drury–Arveson space on the infinite-dimensional unit ball. We use both techniques from the theory of sampling in Paley–Wiener spaces, and classical results from analytic number theory.

Multipliers and essential norm on the Drury-Arveson space

It is well known that for multipliers $f$ of the Drury-Arveson space $H_{n}^{2}$, $\|f\|_{\infty }$ does not dominate the operator norm of $M_{f}$. We show that in general $\|f\|_{\infty }$ does not even dominate the essential norm of $M_{f}$. A consequence of this is that there exist multipliers $f$ of $H_{n}^{2}$ for which $M_f$ fails to be essentially hyponormal; i.e., if $K$ is any compact, self-adjoint operator, then the inequality $M_f^\ast M_f - M_fM_f^\ast + K \geq 0$ does not hold.

Commutators and localization on the Drury–Arveson space

Let f be a multiplier for the Drury–Arveson space H n 2 of the unit ball, and let ζ 1 , … , ζ n denote the coordinate functions. We show that for each 1 ⩽ i ⩽ n , the commutator [ M f ⁎ , M ζ i ] belongs to the Schatten class C p , p > 2 n . This leads to a localization result for multipliers.

Free holomorphic functions on the unit ball of [formula omitted], II

In this paper we continue the study of free holomorphic functions on the noncommutative ball [ B ( H ) n ] 1 : = { ( X 1 , … , X n ) ∈ B ( H ) n : ‖ X 1 X 1 ∗ + ⋯ + X n X n ∗ ‖ 1 / 2 < 1 } , where B ( H ) is the algebra of all bounded linear operators on a Hilbert space H , and n = 1 , 2 , … or n = ∞ . Several classical results from complex analysis have free analogues in our noncommutative setting. We prove a maximum principle, a Naimark type representation theorem, and a Vitali convergence theorem, for free holomorphic functions with operator-valued coefficients. We introduce the class of free holomorphic functions with the radial infimum property and study it in connection with factorizations and noncommutative generalizations of some classical inequalities obtained by Schwarz and Harnack. The Borel–Carathéodory theorem is extended to our noncommutative setting. Using a noncommutative generalization of Schwarz's lemma and basic facts concerning the free holomorphic automorphisms of the noncommutative ball [ B ( H ) n ] 1 , we obtain an analogue of Julia's lemma for free holomorphic functions F : [ B ( H ) n ] 1 → [ B ( H ) m ] 1 . We also obtain Pick–Julia theorems for free holomorphic functions with operator-valued coefficients. We provide a noncommutative generalization of a classical inequality due to Lindelöf, which turns out to be sharper then the noncommutative von Neumann inequality. Finally, we introduce a pseudohyperbolic metric on [ B ( H ) n ] 1 which is invariant under the action of the free holomorphic automorphism group of [ B ( H ) n ] 1 and turns out to be a noncommutative extension of the pseudohyperbolic distance on B n , the open unit ball of C n . In this setting, we obtain a Schwarz–Pick type lemma. We also provide commutative versions of these results for operator-valued multipliers of the Drury–Arveson space.

Interpolation Problems for Schur Multipliers on the Drury-Arveson Space: from Nevanlinna-Pick to Abstract Interpolation Problem

We survey various increasingly more general operator-theoretic formulations of generalized left-tangential Nevanlinna-Pick interpolation for Schur multipliers on the Drury-Arveson space. An adaptation of the methods of Potapov and Dym leads to a chain-matrix linear-fractional parametrization for the set of all solutions for all but the last of the formulations for the case where the Pick operator is invertible. The last formulation is a multivariable analogue of the Abstract Interpolation Problem formulated by Katsnelson, Kheifets and Yuditskii for the single-variable case; we obtain a Redheffer-type linear-fractional parametrization for the set of all solutions (including in degenerate cases) via an adaptation of ideas of Arov and Grossman.

Hyperbolic geometry on the unit ball of $B(\mathcal{H})^{n}$ and dilation theory

In this paper we continue our investigation concerning the hyperbolic geometry on the noncommutative ball $[B(H)^n]_1^-$, where $B(H)$ is the algebra of all bounded linear operators on a Hilbert space $H$, and its implications to noncommutative function theory. The central object is an intertwining operator $L_{B,A}$ of the minimal isometric dilations of $A, B\in [B(H)^n]_1^-$, which establishes a strong connection between noncommutative hyperbolic geometry on $[B(H)^n]_1^-$ and multivariable dilation theory. The goal of this paper is to study the operator $L_{B,A}$ and its connections to the hyperbolic metric $\delta$ on the Harnack parts of $[B(H)^n]_1^-$. We study the geometric structure of the operator $L_{B,A}$ and obtain new characterizations for the Harnack domination (resp. equivalence) in $[B(H)^n]_1^-$. We express $\|L_{B,A}\|$ in terms of the reconstruction operators $R_A$ and $R_B$, and obtain a Schwartz-Pick lemma for contractive free holomorphic functions on $[B(H)^n]_1$ with respect to the intertwining operator $L_{B,A}$. As a consequence, we deduce a Schwartz-Pick lemma for operator-valued multipliers of the Drury-Arveson space, with respect to the hyperbolic metric.