arXiv subjects

Marcin Moszyński

Publications and source records attributed to Marcin Moszyński.

3 recordsLinked to original sources

arXiv2025

Block Jacobi matrices and Titchmarsh-Weyl function

We collect some results and notions concerning generalizations for block Jacobi matrices of several concepts, which have been important for spectral studies of the simpler and better known scalar Jacobi case. We focus here on some issues related to the matrix Titchmarsh-Weyl function, but we also consider generalizations of some other tools used by subordinacy theory, including the matrix orthogonal polynomials, the notion of finite cyclicity, a variant of a notion of nonsubordinacy, as well as Jitomirskaya-Last type semi-norms. The article brings together some issues already known, our new concepts, and also improvements and strengthening of some results already existing. We give simpler proofs of some known facts or we add details usually omitted in the existing literature. The introduction contains a separate part devoted to a brief review of the main spectral analysis methods used so far for block Jacobi operators.

math.SP↗

arXiv2025

Barrier nonsubordinacy and absolutely continuous spectrum of block Jacobi matrices

We explore to what extent the relation between the absolute continuous spectrum and non-existence of subordinate generalized eigenvectors, known for scalar Jacobi operators, can be formulated also for block Jacobi operators with $d$-dimensional blocks. The main object here allowing to make some progress in that direction is the new notion of the barrier nonsubordinacy. We prove that the barrier nonsubordinacy implies the absolute continuity for block Jacobi operators. Finally, we extend some well-known $d=1$ conditions guaranteeing the absolute continuity to $d \geq 1$ and we give applications of our results to some concrete classes of block Jacobi matrices.

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arXiv2022

Spectral Theory of Self-adjoint Finitely Cyclic Operators and Introduction to Matrix Measure $L^2$-spaces

We study finitely cyclic self-adjoint operators in a Hilbert space, i.e. self-adjoint operators that posses such a finite subset in the domain that the orbits of all its elements with respect to the operator are linearly dense in the space. One of the main goals here is to obtain the representation theorem for such operators in a form analogous to the one well-known in the cyclic self-adjoint operators case. To do this, we present here a detailed introduction to matrix measures, to the matrix measure $L^2$ spaces, and to the multiplication by scalar functions operators in such spaces. This allows us to formulate and prove in all the details the less known representation result, saying that the finitely cyclic self-adjoint operator is unitary equivalent to the multiplication by the identity function on $\mathbb{R}$ in the appropriate matrix measure $L^2$ space. We study also some detailed spectral problems for finitely cyclic self-adjoint operators, like the absolute continuity.

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