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Zbigniew Burdak

Publications and source records attributed to Zbigniew Burdak.

6 recordsLinked to original sources
arXiv2026

On the diversity of twisted commuting operators

Operators are called twisted commuting if the deformation from commutativity is determined by a unitary operator, including multiplication by a unimodular constant. The aim of the paper is to investigate constraints on the diversity of twisted commuting pairs depending on the deforming unitary called the twist. It turns out that some operators, like projections, are never twisted commuting. For many twists, twisted commutativity is possible only if at least one of the operators has a nontrivial kernel. In particular, bounded below operators, like isometries, are never twisted commuting by such twists. Twisted commuting unitaries are modeled by pairs of unitarily equivalent and commuting unitaries.

math.FA↗
arXiv2024

The Taylor spectrum of pairs of isometries

In the paper we fully describe Taylor spectrum of pairs of isometries given by diagrams. In most cases both isometries in such pairs have non-trivial shift part and its Taylor spectrum is a proper subset (of Lebesgue measure in $(0,\pi^2)$) of the closed bidisc.

math.SP↗
arXiv2024

An operator theory approach to the evanescent part of a two-parametric weak-stationary stochastic process

A new approach to the evanescent part of a two-dimensional weak-stationary stochastic process with the past given by a half-plane is proceed. The classical result due to Helson and Lowdenslager divides a two-parametric weak-stationary stochastic process into three parts. In this paper we describe the most untouchable one - the evanescent part. Moreover, we point out how this part depends on the shape of the past.

math.PR↗
arXiv2019

The decomposition theorems in Baer $*$-rings

We show a general decomposition theorem in Baer *-rings. As a consequence the vast majority of decompositions known in the algebra of bounded Hilbert space operators are generalized to Baer *-rings. There are also results which are new in the algebra of bounded Hilbert space operators. The model of summands in Wold-Słociński decomposition in Baer *-rings is given.

math.RA↗
arXiv2012

On the unitary part of isometries in commuting, completely non doubly commuting pairs

There are considered isometries on a Hilbert space. By the Wold theorem any isometry can be decomposed into a unitary operator and a unilateral shift. For a pair of isometries, even commuting, a maximal subspace reducing one isometry to a unitary operator might not reduce the other isometry. In the paper are considered pairs of commuting isometries which are completely non doubly commuting. For such pairs there are no nontrivial subspaces reducing both isometries and one of them to a unitary operator. The results describe a unitary part of an isometry in such a pair.

math.FA↗