Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2024
Volume
Vol. 44
Number
No. 3
Description
The issue is dedicated to Professor Jan Stochel on the occasion of his 70th birthday
Journal Volume
Opuscula Mathematica
Vol. 44 (2024)
Projects
Pages
Articles
Jan Stochel, a stellar mathematician
(Wydawnictwa AGH, 2024) Chavan, Sameer; Curto, Raúl; Jabłoński, Zenon Jan; Jung, Il Bong; Putinar, Mihai
The occasion for this survey article was the 70th birthday of Jan Stochel, professor at Jagiellonian University, former head of the Chair of Functional Analysis and a prominent member of the Kraków school of operator theory. In the course of his mathematical career, he has dealt, among other things, with various aspects of functional analysis, single and multivariable operator theory, the theory of moments, the theory of orthogonal polynomials, the theory of reproducing kernel Hilbert spaces, and mathematical aspects of quantum mechanics.
Finitely additive functions in measure theory and applications
(Wydawnictwa AGH, 2024) Alpay, Daniel; Jørgensen, Palle E.T.
In this paper, we consider, and make precise, a certain extension of the Radon-Nikodym derivative operator, to functions which are additive, but not necessarily sigma-additive, on a subset of a given sigma-algebra. We give applications to probability theory, in particular, to the study of $\mu$-Brownian motion, to stochastic calculus via generalized Itô-integrals, and their adjoints (in the form of generalized stochastic derivatives), to systems of transition probability operators indexed by families of measures $\mu$ , and to adjoints of composition operators.
Shifted model spaces and their orthogonal decompositions
(Wydawnictwa AGH, 2024) Câmara, M. Cristina; Kliś-Garlicka, Kamila; Ptak, Marek
We generalize certain well known orthogonal decompositions of model spaces and obtain similar decompositions for the wider class of shifted model spaces, allowing us to establish conditions for near invariance of the latter with respect to certain operators which include, as a particular case, the backward shift $S^{*}$. In doing so, we illustrate the usefulness of obtaining appropriate decompositions and, in connection with this, we prove some results on model spaces which are of independent interest. We show moreover how the invariance properties of the kernel of an operator $T$, with respect to another operator, follow from certain commutation relations between the two operators involved.
A note on the general moment problem
(Wydawnictwa AGH, 2024) El Azhar, Hamza; Hanine, Abdelouahab; Zerouali, El Hassan
In this note we show that given an indeterminate Hamburger moment sequence, it is possible to perturb the first moment in such way that the obtained sequence remains an indeterminate Hamburger moment sequence. As a consequence we prove that every sequence of real numbers is a moment sequence for a signed discrete measure supported in $\mathbb{R}_{+}$.
Cesàro summability of Taylor series in higher order weighted Dirichlet-type spaces
(Wydawnictwa AGH, 2024) Ghara, Soumitra; Gupta, Rajeev; Reza, Md. Ramiz
For a positive integer $m$ and a finite non-negative Borel measure $\mu$ on the unit circle, we study the Hadamard multipliers of higher order weighted Dirichlet-type spaces $\mathcal H_{\mu, m}$. We show that if $\alpha\gt\frac{1}{2}$, then for any $f$ in $\mathcal H_{\mu, m}$ the sequence of generalized Cesàro sums $\{\sigma_n^{\alpha}[f]\}$ converges to $f$. We further show that if $\alpha=\frac{1}{2}$ then for the Dirac delta measure supported at any point on the unit circle, the previous statement breaks down for every positive integer $m$.