The core operator and congruent submodules

The Hardy spaces H²(D²) can be conveniently viewed as a module over the polynomial ring C[z₁,z₂]. Submodules of H²(D²) have connections with many areas of study in operator theory. A large amount of research has been carried out striving to understand the structure of submodules under certain equivalence relations. Unitary equivalence is a well-known equivalence relation in set of submodules. However, the rigidity phenomenon discovered in [Douglas et al., Algebraic reduction and rigidity for Hilbert modules, Amer. J. Math. 117 (1) (1995) 75-92] and some other related papers suggests that unitary equivalence, being extremely sensitive to perturbations of zero sets, lacks the flexibility one might need for a classification of submodules. In this paper, we suggest an alternative equivalence relation, namely congruence. The idea is motivated by a symmetry and stability property that the core operator possesses. The congruence relation effectively classifies the submodules with a finite rank core operator. Near the end of the paper, we point out an essential connection of the core operator with operator model theory.     

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.     

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.     

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.     

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.     

For which reproducing kernel Hilbert spaces is Pick's theorem true?

Pick's theorem tells us that there exists a function inH , which is bounded by 1 and takes given values at given points, if and only if a certain matrix is positive.H is the space of multipliers ofH ², and this theorem has a natural generalisation whenH is replaced by the space of multipliers of a general reproducing kernel Hilbert spaceH(K) (whereK is the reproducing kernel). J. Agler has shown that this generalised theorem is true whenH(K) is a certain Sobolev space or the Dirichlet space, so it is natural to ask for which reproducing kernel Hilbert spaces this generalised theorem is true. This paper widens Agler's approach to cover reproducing kernel Hilbert spaces in general, replacing Agler's use of the deep theory of co-analytic models by a relatively elementary, and more general, matrix argument. The resulting theorem gives sufficient (and usable) conditions on the kernelK, for the generalised Pick's theorem to be true forH(K), and these are then used to prove Pick's theorem for certain weighted Hardy and Sobolev spaces and for a functional Hilbert space introduced by Saitoh.     

Realizations for Schur upper triangular operators centered at an arbitrary point

Reproducing kernel spaces introduced by L. de Branges and J. Rovnyak provide isometric, coisometric and unitary realizations for Schur functions, i.e. for matrix-valued functions analytic and contractive in the open unit disk. In our previous paper [12] we showed that similar realizations exist in the nonstationary setting, i.e. when one considers upper triangular contractions (which appear in time-variant system theory as transfer functions of dissipative systems) rather than Schur functions and diagonal operators rather than complex numbers. We considered in [12] realizations centered at the origin. In the present paper we study realizations of a more general kind, centered at an arbitrary diagonal operator. Analogous realizations (centered at a point of the open unit disk) for Schur functions were introduced and studied in [3] and [4].     

Dynamics of Properties of Toeplitz Operators with Radial Symbols

In the case of radial symbols we study the behavior of different properties (boundedness, compactness, spectral properties, etc.) of Toeplitz operators T_a⁽⁾ acting on weighted Bergman spaces over the unit disk , in dependence on , and compare their limit behavior under with corresponding properties of the initial symbol a.     

On minimal representations of Chevalley groups of type D n , E n and G 2

Let G be a simply connected Chevalley group of type D _n , E _n or G₂. In this paper, we show that the minimal representation of G is unique for types D _n and E _n and it does not exist for the type G₂.     

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.     

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.     

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.     

Pontryagin spaces of entire functions I

We give a generalization of L.de Branges theory of Hilbert spaces of entire functions to the Pontryagin space setting. The aim of this-first-part is to provide some basic results and to investigate subspaces of Pontryagin spaces of entire functions. Our method makes strong use of L.de Branges's results and of the extension theory of symmetric operators as developed by M.G.Krein.     

Generalized resolvent matrices and spaces of analytic functions

We introduce triplet spaces for symmetric relations with defect index (1, 1) in a Pontryagin space. Representations of Pontryagin spaces by spaces of vector-valued analytic functions are investigated. These concepts are used to study 2×2-matrix valued analytic functions which satisfy a certain kernel condition.     

The functional model of a j-unitary node with a given j-inner characteristic matrix function

The known model of aj-unitary node with a givenj-inner characteristic matrix functionW is obtained from the known model of a unitary node with a given bi-inner characteristic matrix functionS, using the Potapov-Ginzburg transforms of the nodes and their characteristic functionsW andS. We show that some new properties of the L. de Branges reproducing kernal Hilbert spaces (W) which were discovered by H. Dym are characteristic properties of these spaces.     

A geometric construction of local representations of local Lie groups

Local transformation groups acting on a manifold X define a natural action of on a space D(X), of functions on X. The natural action induces a local representation of on a Hilbert subspace of the space of distributions on D(X).     

Matrix valued interpolation and truncated Hamburger moment problems

A solvability condition for matrix valued directional single-node interpolation problems of Loewner type is established, in terms of properties of Pick kernel. As a consequence, a solvability condition for matrix valued directional truncated Hamburger moment problems is obtained.     

J-inner matrix functions, interpolation and inverse problems for canonical systems, V: The inverse input scattering problem for Wiener class and rational p×q input scattering matrices

The general formulas developed in the fourth paper in this series are applied to solve the inverse input scattering problem for canonical integral systems in the special cases that the input scattering matrix is ap×q matrix valued function in the Wiener class (and the associated pairs are homogeneous). These formulas are then further specialized to the rational case. Whenp=q, these formulas are connected to the earlier results of Alpay-Gohberg and Gohberg-Kaashoek-Sakhnovich, who studied inverse problems for a related system of differential equations.This research was partially supported by a Minerva Foundation grant that is acknowledged with thanks.