Defect operators, defect functions and defect indices for analytic submodules

This paper mainly concerns defect operators and defect functions of Hardy submodules, Bergman submodules over the unit ball, and Hardy submodules over the polydisk. The defect operator (function) carries key information about operator theory (function theory) and structure of analytic submodules. The problem when a submodule has finite defect is attacked for both Hardy submodules and Bergman submodules. Our interest will be in submodules generated by polynomials. The reason for choosing such submodules is to understand the interaction of operator theory, function theory and algebraic geometry.     

Properties of Beurling-type submodules via Agler decompositions

In this paper, we study operator-theoretic properties of the compressed shift operators $S_{z_1}$ and $S_{z_2}$ on complements of submodules of the Hardy space over the bidisk $H^2({\mathbb{D}}^2)$. Specifically, we study Beurling-type submodules – namely submodules of the form $\theta H^2({\mathbb{D}}^2)$ for θ inner – using properties of Agler decompositions of θ to deduce properties of $S_{z_1}$ and $S_{z_2}$ on model spaces $H^2({\mathbb{D}}^2)?\theta H^2({\mathbb{D}}^2)$. Results include characterizations (in terms of θ) of when a commutator $[S_{z_j}^{?},S_{z_j}]$ has rank n and when subspaces associated to Agler decompositions are reducing for $S_{z_1}$ and $S_{z_2}$. We include several open questions.     

The Dirichlet-to-Neumann operator on rough domains

We consider a bounded connected open set Ω⊂Rd whose boundary Γ has a finite (d−1)-dimensional Hausdorff measure. Then we define the Dirichlet-to-Neumann operator D₀ on L₂(Γ) by form methods. The operator −D₀ is self-adjoint and generates a contractive C₀-semigroup S=(St)_t>0 on L₂(Γ). We show that the asymptotic behaviour of St as t→∞ is related to properties of the trace of functions in H¹(Ω) which Ω may or may not have.     

On Two Variable Jordan Block (II)

On the Hardy space over the bidisk H²(D²), the Toeplitz operators and are unilateral shifts of infinite multiplicity. A closed subspace M is called a submodule if it is invariant for both and . The two variable Jordan block (S₁, S₂) is the compression of the pair to the quotient H²(D²) ⊖M. This paper defines and studies its defect operators. A number of examples are given, and the Hilbert-Schmidtness is proved with good generality. Applications include an extension of a Douglas-Foias uniqueness theorem to general domains, and a study of the essential Taylor spectrum of the pair (S₁, S₂). The paper also estabishes a clean numerical estimate for the commutator [S₁*, S₂] by some spectral data of S₁ or S₂. The newly-discovered core operator plays a key role in this study.     

The trisecant identity and operator theory

For the special case of a Riemann surface which arises as the double of a planar domainR, the trisecant identity has a natural interpretation as a relation among reproducing kernels for subspaces of the Hardy spaceH ² (R). This relation and Riemann's theorem on the vanishing of the theta function is applied to Nevanlinna-Pick interpolation onR.     

Reducing subspaces of compressed analytic Toeplitz operators on the Hardy space

In this paper we discuss necessary conditions and sufficient conditions for the compression of an analytic Toeplitz operator onto a shift coinvariant subspace to have nontrivial reducing subspaces. We give necessary and sufficient conditions for the kernel of a Toeplitz operator whose symbol is the quotient of two inner functions to be nontrivial and obtain examples of reducing subspaces from these kernels. Motivated by this result we give necessary conditions and sufficient conditions for the kernel of a Toeplitz operator whose symbol is the quotient of two inner functions to be nontrivial in terms of the supports of the two inner functions. By studying the commutant of a compression, we are able to give a necessary condition for the existence of reducing subspaces on certain shift coinvariant subspaces.     

Quasi-Free Resolutions Of Hilbert Modules

The notion of a quasi-free Hilbert module over a function algebra $\mathcal{A}$ consisting of holomorphic functions on a bounded domain $\Omega$ in complex m space is introduced. It is shown that quasi-free Hilbert modules correspond to the completion of the direct sum of a certain number of copies of the algebra $\mathcal{A}$. A Hilbert module is said to be weakly regular (respectively, regular) if there exists a module map from a quasi-free module with dense range (respectively, onto). A Hilbert module $\mathcal{M}$ is said to be compactly supported if there exists a constant $\beta$ satisfying $\|\varphi f\| \leq \beta \ |\varphi \| \textsl{X} \|f \|$ for some compact subset X of $\Omega$ and $\varphi$ in $\mathcal{A}$, f in $\mathcal{M}$. It is shown that if a Hilbert module is compactly supported then it is weakly regular. The paper identifies several other classes of Hilbert modules which are weakly regular. In addition, this result is extended to yield topologically exact resolutions of such modules by quasi-free ones.     

Quasi-multipliers of operator spaces

We use the injective envelope to study quasi-multipliers of operator spaces. We prove that all representable operator algebra products that an operator space can be endowed with are induced by quasi-multipliers. We obtain generalizations of the Banach-Stone theorem.     

Essentially normal Hilbert modules and <Emphasis Type="Italic">K</Emphasis>-homology

This paper mainly concerns the essential normality of graded submodules. Essentially all of the basic Hilbert modules that have received attention over the years are p-essentially normal—including the d-shift Hilbert module, the Hardy and Bergman modules of the unit ball. Arveson conjectured graded submodules over the unit ball inherit this property and provided motivations to seek an affirmative answer. Some positive results have been obtained by Arveson and Douglas. However, the problem has been resistant. In dimensions d = 2, 3, this paper shows that the Arveson’s conjecture is true. In any dimension, the paper also gives an affirmative answer in the case of the graded principal submodule. Finally, the paper is associated with K-homology invariants arising from graded quotient modules, by which geometry of the quotient modules and geometry of algebraic varieties are connected. In dimensions d = 2, 3, it is shown that K-homology invariants determined by graded quotients are nontrivial. The paper also establishes results on p-smoothness of K-homology elements, and gives an explicit expression for K-homology invariant in dimension d = 2.     

A completely bounded characterization of operator algebras

Supported by a grant from the NSF     

Toeplitz algebras, subnormal tuples and rigidity on reproducing C[z1,…,zd]-modules

This paper mainly considers Toeplitz algebras, subnormal tuples and rigidity concerning reproducing C[z₁,…,z_d]-modules. By making use of Arveson's boundary representation theory, we find there is more rigidity in several variables than there is in single variable. We specialize our attention to reproducing C[z₁,…,z_d]-modules with -invariant kernels by examining the spectrum and the essential spectrum of the d-tuple {M_z1,…,M_zd}, and deducing an exact sequence of C∗-algebras associated with Toeplitz algebra. Finally, we deal with Toeplitz algebras defined on Arveson submodules and rigidity of Arveson submodules.     

Joint similarity problems and the generation of operator algebras with bounded length

There is a pair of commuting operators (T ₁,T ₂) on Hilbert space such that eachT ₁ andT ₂ is similar to a contraction but the pair (T ₁,T ₂) is not similar to a pair of contractions. There is a pair of commuting unitarizable representations (₁,₂) on the free group withN2 generators such that (₁,₂) is not similar to a pair of unitary representations. In connection with these examples, we introduce and study a notion of length for aC ^*-algebra (or an operator algebra) generated by two subalgebras, which is analogous to the minimum length of a word in the generators of a group.Partially supported by the N.S.F.     

Quasi-homogeneous Hilbert Modules

This paper is to study the quasihomogeneous Hilbert modules and generalize a result of Arveson [3] which relates the curvature invariant to the index of the Dirac operator. This work was partially supported by NKBRPC (#2006CB805905) and SRFDP.     

Minimal isometric dilations of operator cocycles

We obtain existence, uniqueness results for minimal isometric dilations of contractive cocycles of semigroups of unital *-endomorphisms ofB(H. This generalizes the result of Sz. Nagy on minimal isometric dilations of semigroups of contractive operators on a Hilbert space. In a similar fashion we explore results analogus to Sarason's characterization that subspaces to which compressions of semigroups are again semigroups are semi-invariant subspaces, in the context of cocycles and quantum dynamical semigroups.This research is supported by the Indian National Science Academy under Young Scientist Project.     

Ersatz Commutant Lifting with Test Functions

This note presents a commutant lifting theorem (CLT) with initial data a finite set of (test) functions and a compatible reproducing kernel k on a set X. This covers the CLT of Ball, Li, Timotin, and Trent [9] for the polydisc, but in general no analyticity is required, rather statements and proofs use the language and techniques of reproducing kernel Hilbert spaces. Uniqueness of the de Branges–Rovnyak construction like found in Agler [1] and Ambrozie, Englis, and Müller [5] and an abstract Beurling Theorem in the present context are of independent interest. Received: October 12, 2006. Accepted: May 8, 2007.     

Discrete time-variant interpolation as classical interpolation with an operator argument

Necessary and sufficient conditions are derived for the existence of solutions to discrete time-variant interpolation problems of Nevanlinna-Pick and Nudelman type. The proofs are based on a reduction scheme which allows one to treat these time-variant interpolation problems as classical interpolation problems for operator-valued functions with operator arguments. The latter ones are solved by using the commutant lifting theorem.     

Characteristic Functions for Ergodic Tuples

Motivated by a result on weak Markov dilations, we define a notion of characteristic function for ergodic and coisometric row contractions with a one-dimensional invariant subspace for the adjoints. This extends a definition given by G. Popescu. We prove that our characteristic function is a complete unitary invariant for such tuples and show how it can be computed.     

Banach operator pairs and common fixed points in hyperconvex metric spaces

The purpose of this paper is to establish DeMarr’s well-known theorem for an arbitrary family of symmetric Banach operator pairs in hyperconvex metric spaces without the compactness assumption. We also give necessary and sufficient criteria for the existence of a common fixed point of a semigroup of isometric mappings. As an application, several results on the invariant best approximation are proved.