László Kérchy

Harmonic analysis of operators on Hilbert space

Contractions and Their Dilations.- Geometrical and Spectral Properties of Dilations.- Functional Calculus.- Extended Functional Calculus.- Operator-Valued Analytic Functions.- Functional Models.- Regular Factorizations and Invariant Subspaces.- Weak Contractions.- The Structure of C1.-Contractions.- The Structure of Operators of Class C0.

On the hyperinvariant subspace problem for asymptotically nonvanishing contractions

Our aim in the present note is to obtain new information on the structure of contractions of class C 1., and to develop new ways for obtaining hyperinvariant subspaces for these operators.

On the spectra of contractions belonging to special classes

Abstract Complete characterizations of the possible spectra of C10 and C00 contractions and their ∗-residual parts are given.

Recent Advances in Operator Theory and Related Topics

Inverse problems associated to a canonical differential system.- Construction of Schwarz norms.- On the class of extremal extensions of a nonnegative operator.- On Livsic-Brodskii nodes with strongly regular J-inner characteristic matrix functions in the Hardy class.- Scalar perturbations of the Sz.-Nagy-Foias charac-teristic function.- Inequalities for eigenvalues of sums in a von Neu-mann algebra.- Weighted variants of the Three Chains Completion Theorem.- Semigroups in finite von Neumann algebras.- Singly generated algebras containing a coin-pact operator.- Analytic extension of vector valued functions.- On quotient modules.- On the structure of spherical contractions.- Apostols bilateral weighted shifts are hyper-reflexive.- Wielandt type extensions of the Heinz-Kato-Furuta inequality.- Logarithmic order and dual logarithmic order.- On the generalized von Neumann inequality.- Ultraproducts of C*-algebras.- Intertwining extensions and a two-sided corona problem.- On self-polar Hilbertian norms on (indefinite) inner productspaces.- Schur norms and the multivariate von Neumann inequality.- Spectral properties of selfadjoint Jacobi matrices coming from birth and death processes.- On the hyperinvariant subspace problem for asymptotically non-vanishing contractions.- Unstable dynamics on a Markov background and stability in average.- A relation for the spectral shift function of two self-adjoint extensions.- Beppo Levi and Lebesgue type theorems for bundle convergence in noncommutative L2-spaces.- *-semigroup endomorphisms of B(H).- Spectral singularities, Szokefalvi-Nagy-Foias functional model and the spectral analysis of the Boltzmann operator.- Uniqueness of invariant measures for the stochastic Cauchy problem in Banach spaces.- The Friedrichs operator of a planar domain. II.- Localization of the Wielandt-Wintner Theorem.- Order and square roots in hermitian Banach *-algebras.- Unitary dilation of several contractions.- Inequalities for semibounded operators and their applications to log-hyponormal operators.- Operator moment problems in unbounded sets.- The argument principle and boundaries of analytic varieties.- Conference Program.- List of Participants.

Unitary asymptotes of Hilbert space operators

1. Power bounded operators. In operator theory it is a generally used, fruitful method that in order to explore the structure and properties of operators belonging to a large, undetected class one relates these operators to those of a special, well-understood class, and then exploiting this connection obtains theorems on the operators in the large class. Unitary operators, the automorphisms of Hilbert spaces form certainly the most thoroughly investigated and best understood class. It is sufficient to refer to the spectral theorem which is one of the main tools in their study (see e.g. [29], [30] or [6]). On the other hand, power bounded operators, that is, the operators with bounded sequence of iterates form an extensive, broad class. The idea of using Banach limits to relate power bounded operators to unitaries stems from the paper [33] by B. Sz.-Nagy. It was shown in that paper that every invertible power bounded operator with a power bounded inverse is similar to a unitary operator. It was observed in [22] that Sz.-Nagy’s method works for every power bounded operator and that the unitary operator associated with the power bounded operator has a useful property of universality. To be more precise, let us give the exact definitions. Let H be a complex Hilbert space and let B(H) denote the set of all bounded linear operators acting on H. Consider a power bounded operator T ∈ B(H), that is, sup{‖T‖ : n = 0, 1, 2, . . .} is finite. Let L be a Banach limit on the sequence space `∞, that is, let L be a positive linear functional with the properties L(1, 1, 1, . . .) = 1 and L({cn}n=0) = L({cn+1}n=0). The existence of such an L, which is an extension

Criteria of regularity for norm-sequences

General sufficient conditions are given for a regularity property of norm-sequences. This regularity of the norm-sequence {‖Tn‖}n∈N makes possible to associate an isometryV with the operatorT in a similar way as it has been known in the power bounded case.

Contractions being Weakly Similar to Unitaries

In this paper contractions being weakly similar in a certain sense to unitary operators are studied. Extending the investigations of [12], where only cyclic contractions were considered, it is examined when a contraction T, weakly similar to unitary possesses the bicommutant property Alg T = {T}″. Finally, a characterization of cyclic C11 -contractions is given.

On the commutant of asymptotically non-vanishing contractions

The injectivity of the commutant mapping of asymptotically nonvanishing contractions is examined. We show that this mapping can be injective even in the presence of a non-trivial stable subspace. Various characterizations of injectivity are provided.

Unbounded Representations of Discrete Abelian Semigroups

This survey article contains the lecture delivered on the conference; it is based on the papers [5]– [9], where complete proofs can be found. In the first section we present results concerning single operators, and the second section is devoted to the study of representations of general discrete abelian semigroups. The main idea in both sections is to apply an appropriate normalizing gauge function in order to get connection with an isometry, or isometric representation.

On a Conjecture of Teodorescu and Vasyunin

Let T be a contraction acting on the separable Hilbert space H. Let us assume that T is of class C10, which means that for every non-zero vector h e H we have \(\mathop{{\lim }}\limits_{n} \parallel {{T}^{n}}h\parallel \ne 0 = \mathop{{\lim }}\limits_{n} \parallel {{T}^{{*n}}}h\parallel\). As far as we know it is yet open whether T necessarily has a non-trivial invariant subspace. An isometry can be attached to T in a natural way. Indeed, a new scalar product can be introduced in H by \( }_{ \sim }} = \mathop{{\lim }}\limits_{n} ,h, k\varepsilon H\). In the inner product space \((H, }_{ \sim }})\) T acts as an isometry. Let \({\tilde{H}}\) denote the completion of \((H, }_{ \sim }})\) and let \({\tilde{T}}\) stand for the continuous extension of T to \({\tilde{H}}\). The isometry \({\tilde{T}}\) is called the minimal isometric extension of T. The identical mapping X from H to \({\tilde{H}}\) is a quasi-affinity, i. e. it is injective and has dense range. Moreover, X intertwines T and \(\tilde{T}:XT = \tilde{T}X\), hence T is a quasi-affine transform of its minimal isometric extension.