Spectra of certain classes of operators in Hilbert space

The structure of certain classes of operators of the form AB, where A and B are self-adjoint, is investigated, and the spectra of certain J-semiunitary operators are described.Translated from Matematicheskie Zametki, Vol. 9, No. 3, pp. 303–310, March, 1971.The present work was written under the direction of I. S. Iokhvidov, whom the author wishes to thank sincerely.     

Factoring operators through Hilbert space

We give an example of a pair of Banach spacesE andF so that neitherE′ norF has cotype 2, but every bounded linear operator fromE intoF factors through Hilbert space.     

Irregular vectors of Hilbert space operators

A vector x in a Hilbert space H is called irregular for an operator provided that supn‖Tnx‖=∞ and infn‖Tnx‖=0. We establish some basic properties of operators having irregular vectors and present examples that highlight the relationship, or lack thereof, between irregularity and hypercyclicity.     

Hilbert space structure and positive operators

Let X be a real Banach space. We prove that the existence of an injective, positive, symmetric and not strictly singular operator from X into its dual implies that either X admits an equivalent Hilbertian norm or it contains a nontrivially complemented subspace which is isomorphic to a Hilbert space. We also treat the nonsymmetric case.     

Commutator inequalities for Hilbert space operators

We prove singular value inequalities for positive operators. Some of these inequalities generalize recent results for commutators due to Bhatia-Kitttaneh, Kittaneh, and Wang-Du. Applications of our results are given.     

Bohr's inequalities for Hilbert space operators

The classical Bohr's inequality states that

²|z+w|?p²|z|+q²|w|     

Generalized random linear operators on a Hilbert space

In this paper, we are concerned with generalized random linear operators on a separable Hilbert space. Generalized random linear bounded operators, generalized random linear normal operators and generalized random linear self-adjoint operators are defined and investigated. The spectral theorems for generalized random linear normal operators and generalized random linear self-adjoint operators are obtained.     

On typical properties of Hilbert space operators

We study the typical behavior of bounded linear operators on infinite-dimensional complex separable Hilbert spaces in the norm, strong-star, strong, weak polynomial and weak topologies. In particular, we investigate typical spectral properties, the problem of unitary equivalence of typical operators, and their embeddability into C ₀-semigroups. Our results provide information on the applicability of Baire category methods in the theory of Hilbert space operators.