arXiv subjects
Publications and source records attributed to Michał Buchała.
The aim of this paper is to study $m$-partial isometries on Hilbert spaces, a natural extension of partial isometries and $m$-isometries. We establish structural and spectral results, characterize the $m$-partial isometric weighted shifts, and investigate similarity to $m$-isometries and $m$-partial isometries.
The aim of this paper is to obtain $ m $-isometric dilation of expansive $ m $-concave operator on Hilbert space. The obtained dilation is shown to be minimal. The matrix representation of this dilation is given. It is also proved that in case of 3-concave operators the assumption on expansivity is not necessary. The paper contains an example showing that minimal $ m $-isometric dilations may not be isomorphic.
The aim of this paper is to study $ m $-isometric weighted shifts with operator weights (both unilateral and bilateral). We obtain a characterization of such shifts by polynomials with operator coefficients. The procedure of construction of all $ m $-isometric weighted shifts with positive weights is given. We answer the question when the weights of $ m $-isometric shifts are commuting. The completion problem for $ m $-isometric weighted shifts with operator weights is solved. We characterize $ m $-isometric bilateral shifts the adjoints of which are also $ m $-isometric. Several relevant examples are provided.
In this paper we investigate composition operators on discrete spaces. We establish the classification of underlying graphs of such operators. For one class of such graphs, namely graphs with one cycle, we obtain a characterization of m-isometries. The paper also contains the solution to the Cauchy dual subnormality problem for composition operators on graphs with one cycle.
We study unitarily equivalent bilateral weighted shifts with operator weights. We establish a general characterization of unitary equivalence of such shifts under the assumption that the weights are quasi-invertible. We prove that under certain assumptions unitary equivalence of bilateral weighted shifts with operator weights defined on $ \mathbb{C}^{2} $ can always be given by a unitary operator with at most two non-zero diagonals. We provide examples of unitarily equivalent shifts with weights defined on $ \mathbb{C}^{k} $ such that every unitary operator, which intertwines them has at least $ k $ non-zero diagonals.
For a given directed tree and weights associated with vertices from a subtree the completion problem is to determine if these weights may be completed in a way to obtain a bounded weighted shift on the whole tree, which possibly satisfies also some more restrictive conditions. In this paper we consider subnormal and completely hyper-expansive completion problem for weighted shifts on directed trees with one branching point. We develop new results on backward extensions of truncated moment sequences and, exploiting these results, we obtain a characterization of existence of such a completion