Joint seminormality and Dirac operators

In this article, an approach to joint seminormality based on the theory of Dirac and Laplace operators on Dirac vector bundles is presented. To eachn-tuple of bounded linear operators on a complex Hilbert space we first associate a Dirac bundle furnished with a metric-preserving linear connection defined in terms of thatn-tuple. Employing standard spin geometry techniques we next get a Bochner type and two Bochner-Kodaira type identities in multivariable operator theory. Further, four different classes of jointly seminormal tuples are introduced by imposing semidefiniteness conditions on the remainders in the corresponding Bochner-Kodaira identities. Thus we create a setting in which the classical Bochner's method can be put into action. In effect, we derive some vanishing theorems regarding various spectral sets associated with commuting tuples. In the last part of this article we investigate a rather general concept of seminormality for self-adjoint tuples with an even or odd number of entries.     

Commutant lifting theorem and interpolation in discrete nest algebras

Ball in [Ba] showed that the commutant lifting theorem for the nest algebras due to Paulsen and Power gives a unified approach to a wide class of interpolation problems for nest algebras. By restricting our attention to the case when nest algebras associated with the problems are discrete we derive a variant of the commutant lifting theorem which avoids language of representation theory and which is sufficient to treat an analog of the generalized Schur-Nevannlinna-Pick (SNP) problem in the setting of upper triangular operators.     

Minimal and optimal linear discrete time-invariant dissipative scattering systems

For an operator-valued function in the Schur class a new geometric proof, using state space considerations only, of the construction of a minimal and optimal realization is given. A minimal and optimal realization also appears as a restricted shift realization where the state space is the completion of the range of the associated Hankel operator in the de Branges-Rovnyak norm associated with . It is also shown that minimal and optimal, and minimal and star-optimal realizations of a rational matrix function in the Schur class are intimately connected to the extremal positive solutions of the associated Kalman-Yakubobich-Popov operator inequality.The first author thanks the International Association for the promotion of co-operation with scientists from the New Independent States of the former Soviet Union for its support (under project INTAS 93-249), and the Vrije Universiteit, Amsterdam, for its hospitality     

Linear operator pencilsA-λB with discrete spectrum

We study the spectral properties of non-self-adjoint linear pencilsA-B of bounded operators with discrete spectrum whereB is not necessarily bijective. The main results concern the minimality, completeness and basis properties of the corresponding eigenvectors and associated vectors.     

Some inequalities for the Euclidean operator radius of two operators in Hilbert spaces

Some sharp bounds for the Euclidean operator radius of two bounded linear operators in Hilbert spaces are given. Their connection with Kittaneh’s recent results which provide sharp upper and lower bounds for the numerical radius of linear operators are also established.     

Compact linear operators on functional spaces with two norms

Some principles of the operator theory in a linear space with two norms are established in this paper. The well-known Hilbert-Schmidt theorem on the eigenfunction expansion of sourcewise represented functions, Mercer's theorem and other results can be consider as special cases of the statements presented. The general approach proposed is used to construct the theory of symmetrizable operators and to investigate the asymptotic behaviour of eigenvalues of compact operators.This paper was translated by M. Gorbuchuk and V. GorbachukThis paper was translated by M. Gorbuchuk and V. Gorbachuk.     

Dual analytic models of seminormal operators

In [6] (after Clancey's work [2]), Martin and Putinar introduced their two-dimensional functional model of a hyponormal operator, which reduces it to the multiplication by the independent variable in a space of distributions. Here we define another model which does (almost) the same for the adjoint operator. We also explain a close relation between these two models and dual bundle shift models of linear operators introduced in [13]. As application, an estimate of the effectual rational multiplicity of hyponormal operators is given.The research described in this publication was made possible in part by Grant No. NW8000 from the International Science Foundation     

Bounded characteristic functions and models for noncontractive sequences of operators

The Sz.-Nagy-FoiaŞ functional model for completely non-unitary contractions is extended to completely non-coisometric sequences of bounded operatorsT = (T₁,...,T _d) (d finite or infinite) on a Hilbert space, with bounded characteristic functions. For this class of sequences, it is shown that the characteristic function θ_T is a complete unitary invariant. We obtain, as the main result, necessary and sufficient conditions for a bounded multi-analytic operator on Fock spaces to coincide with the characteristic function associated with a completely non-coisometric sequence of bounded operators on a Hilbert space. Research supported in part by a COBASE grant from the National Research Council. The first author was partially supported by a grant from Ministerul Educaţiei Şi Cercetarii. The second author was partially supported by a National Science Foundation grant.     

Analytic equivalence and similarity of operators

The analytic equivalence of two operators is a generalization of similarity. We prove that under some conditions the analytic equivalence between two Hilbert space operatorsT andR implies the similarity of their restrictions on generalized ranges. We also prove that, in certain cases, the similarity ofT to a contraction implies that ofR. An improvement of a well-known criterion of similarity to an isometry due to Sz.-Nagy is given and an extension of a result of Apostol is obtained.     

Some Selfadjoint 2$\times$2 Operator Matrices Associated with Closed Operators

We study a 22 operator matrix associated with a closed densely defined operator. Among others, the selfadjointness of a closed symmetric operator and the strong commutativity of two (unbounded) self-adjoint operators are characterized in terms of the related operator matrices. We propose a definition of strong commutativity for closed symmetric operators. Submitted: November 8, 2001     

Core–EP pre-order of Hilbert space operators

A new binary relation associated with the core–EP inverse is presented and studied on the corresponding subset of all generalized Drazin invertible bounded linear Hilbert space operators. Using the (dual) core partial order between core parts of operators and the minus partial order between quasinilpotent parts of operators, new pre-orders and partial orders are also introduced and characterized.     

Intertwining liftings and Schur contractions

In their paper [5], C. Foias and A. Frazho have found explicit formulas for the Schur contraction associated with a given contractive intertwining lifting. In this Note we give a more direct and probably the simplest way to find the Schur contraction for a given contractive intertwining lifting.     

On the relation of closed forms and Trotter's product formula

The aim of this paper is to give a characterization in Hilbert spaces of the generators of C₀-semigroups associated with closed, sectorial forms in terms of the convergence of a generalized Trotter's product formula. In the course of the proof of the main result we also present a similarity result which can be of independent interest: for any unbounded generator A of a C₀-semigroup e^tA it is possible to introduce an equivalent scalar product on the space, such that e^tA becomes non-quasi-contractive with respect to the new scalar product.     

Factorization of staircase class operators in Hilbert space

A class of linear bounded staircase operators (H, G spaces) defined by (1) with two infinite sequences of orthogonal decompositions ofH and chain property (2) is considered. Necessary and sufficient conditions for the factorizationZ=XY are obtained, whereX, Y are block-diagonal, bounded, andY has a bounded inverse. All the pairs (X, Y) are explicitly constructed. These conditions are specialized for finite and infinite dimensions of the blocks ofX, Y and for differentX, Y. A direct application to bitriangular and biquasitriangular operators is indicated.     

Zero measure Cantor spectra for continuum one-dimensional quasicrystals

We study Schrödinger operators on R$\mathbb{R}$ with measures as potentials. Choosing a suitable subset of measures we can work with a dynamical system consisting of measures. We then relate properties of this dynamical system with spectral properties of the associated operators. The constant spectrum in the strictly ergodic case coincides with the union of the zeros of the Lyapunov exponent and the set of non-uniformities of the transfer matrices. This result enables us to prove Cantor spectra of zero Lebesgue measure for a large class of operator families, including many operator families generated by aperiodic subshifts.     

On the filling in holes problem for operator matrices

We consider upper-triangular 2-by-2 operator matrices and are interested in the set that has to be added to certain spectra of the matrix in order to get the union of the corresponding spectra of the two diagonal operators. We show that in the cases of the Browder essential approximate point spectrum, the upper semi-Fredholm spectrum, or the lower semi-Fredholm spectrum the set in question need not to be an open set but may be just a singleton. In addition, we modify and extend known results on Hilbert space operators to operators on Banach spaces.     

Mosaic and Principal Functions of log-hyponormal Operators

The purpose of this paper is to introduce mosaic and principal functions of log-hyponormal operators associated with Aluthge transformation and discuss determining sets.     

On explicit factorization and applications

This paper is devoted to two topics connected with factorization of triangular 2 by 2 matrix functions. The first application is concerned with explicit factorization of a class of matrices of Daniel-Khrapkov type and the second is related to inversion of finite Toeplitz matrices. In the first section we present the scheme of factorization of triangular 2 by 2 matrix functions.     

Factorization of matrix functions and characteristic properties of the circle

In the present paper we discuss two problems on factorizations of matrix-valued functions with respect to a simple closed rectifiable curve . These two problems are related and we show that in both of them circular contours play a remarkable role.