Opuscula Mathematica

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.

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Finitely additive functions in measure theory and applications

(Wydawnictwa AGH, 2024) Alpay, Daniel; Jorgensen, Palle

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.

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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.

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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 $\mathbb{R}_{+}$.

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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 Ces?ro 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 $m$.

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On the Möbius invariant principal functions of Pincus

(Wydawnictwa AGH, 2024) Ghosh, Sagar; Misra, Gadadhar

In this semi-expository short note, we prove that the only homogeneous pure hyponormal operator $T$ with $\operatorname{rank} (T^*T-TT^*) =1$, modulo unitary equivalence, is the unilateral shift.

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Positive solutions for nonparametric anisotropic singular solutions

(Wydawnictwa AGH, 2024) Papageorgiou, Nikolaos S.; Rǎdulescu, Vicenţiu; Sun, Xueying

We consider an elliptic equation driven by a nonlinear, nonhomogeneous differential operator with nonstandard growth. The reaction has the combined effects of a singular term and of a "superlinear" perturbation. There is no parameter in the problem. Using variational tools and truncation and comparison techniques, we show the existence of at least two positive smooth solutions.

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Reduction of positive self-adjoint extensions

(Wydawnictwa AGH, 2024) Tarcsay, Zsigmond; Sebestyén, Zoltán

We revise Krein's extension theory of semi-bounded Hermitian operators by reducing the problem to finding all positive and contractive extensions of the "resolvent operator" $(I+T)^{-1}$ of $T$. Our treatment is somewhat simpler and more natural than Krein's original method which was based on the Krein transform $(I-T)(I+T)^{-1}$. Apart from being positive and symmetric, we do not impose any further constraints on the operator $T$: neither its closedness nor the density of its domain is assumed. Moreover, our arguments remain valid in both real or complex Hilbert spaces.

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