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1
Content available Jan Stochel, a stellar mathematician
Chavan Sameer, Curto Raúl, Jabłoński Zenon Jan, Jung Il Bong, Putinar Mihai
Opuscula Mathematica
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2024
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Vol. 44, no. 3
303--321
EN
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.
2
Content available The composition operators between Morrey type spaces
Xu Jiajia
Science, Technology and Innovation
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2022
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Vol. 16, no. 3-4
50--56
EN
In this paper, we characterize the boundedness of composition operator Cφ from Morrey space H²K₁ to H²K₂ on the unit complex disk.
3
Content available remote Local spectral theory of endomorphisms of the disk algebra
Trivedi S., Chandra H.
Demonstratio Mathematica
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2016
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Vol. 49, nr 1
95--106
EN
Let A(D) denote the disk algebra. Every endomorphism of A(D) is induced by some φ (…) A(D) with ║φ║ ≤ 1. In this paper, it is shown that if φ is not an automorphism of D and φ has a fixed point in the open unit disk then the endomorphism induced by φ is decomposable if and only if the fixed set of φ is singleton. Further, we determine the local spectra of the endomorphism induced by φ in the cases when the fixed set of φ either includes unit circle or is a singleton.
4
Content available remote On the Nemytskii operator in the space of functions of bounded (p, 2, α)-variation with respect to the weight function
Aziz W.
Demonstratio Mathematica
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2014
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Vol. 47, nr 4
933--948
EN
In this paper, we consider the Nemytskii operator (H f) (t) = h(t, f(t)), generated by a given function h. It is shown that if H is globally Lipschitzian and maps the space of functions of bounded (p, 2, α)-variation (with respect to a weight function α) into the space of functions of bounded (q, 2, α)-variation (with respect to α) 1 < q < p, then H is of the form (H f) (t) = A(t)f(t) + B(t). On the other hand, if 1 < p < q then H is constant. It generalize several earlier results of this type due to Matkowski–Merentes and Merentes. Also, we will prove that if a uniformly continuous Nemytskii operator maps a space of bounded variation with weight function in the sense of Merentes into another space of the same type, its generator function is an affine function.
5
Content available remote Functions of bounded variations on compact subsets of C
Gimenez J., Merentes N., Vivas M.
Commentationes Mathematicae
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2014
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Vol 54, No. 1
3--19
EN
In this paper we introduce the concept of bounded variation for functions defined on compact subsets of the complex plane C, based on the notion of variation along a curve as defined by Ashton and Doust; We describe in detail the space so generated and show that it can be equipped, in a natural way, with the structure of a Banach algebra. We also present a necessary condition for a composition operator Cφ to act between two such spaces.
6
Content available Remarks on uniformly bounded composition operator acting between Banach spaces of functions of two variables of bounded schramm Φ-variation
Ereú T., Merentes N., Wróbel M.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
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2013
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Vol. 18
19--27
EN
In this paper we prove that if the composition operator H of generator h : Ib a × C → Y (X is a real normed space, Y is a real Banach space, C is a convex cone in X and Ib a ⸦ R2) maps Φ1 BV (Ib a, C) into Φ2 BV (Ib a, Y) and is uniformly bounded, then the left-left regularization h* of h is an affine function in the third variable.
7
Content available remote On Nemytskii operator in the space of set-valued functions of bounded p-variation in the sense of Riesz with respect to the weight function
Aziz W., Guerrero J. A., Merentes N.
Fasciculi Mathematici
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2013
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Nr 50
17--32
EN
In this paper we consider the Nemytskii operator (Hf) (t) = h(t, f (t)), generated by a given set-valued function h is considered. It is shown that if H is globally Lipschitzian and maps the space of functions of bounded p-variation (with respect to a weight function α) into the space of set-valued functions of bounded q-variation (with respect to α) ) 1 < q < p, then H is of the form (Hϕ)(t) = A(t)ϕ(t) + B(t). On the other hand, if 1 < p < q, then H is constant. It generalizes many earlier results of this type due to Chistyakov, Matkowski, Merentes-Nikodem, Merentes-Rivas, Smajdor-Smajdor and Zawadzka.
8
Content available remote Injective composition operators on Lorentz-Bochner spaces
Datt G., Arora S. C.
Demonstratio Mathematica
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2013
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Vol. 46, nr 4
741--756
EN
In this paper, we extend the notion of essential range to vector-valued functions and present various equivalent conditions for the injectiveness of the composition operators alongwith a characterisation for measurable transformations inducing composition operators between Lorentz-Bochner spaces. Some aspects of the weighted composition operators on Lorentz-Bochner spaces, induced by a measurable transformation and an operator valued map, are also discussed.
9
Content available Uniformly continuous composition operators in the space of bounded Φ-variation functions in the Schramm sense
Ereu T., Merentes N., Sanchez J. L., Wróbel M.
Opuscula Mathematica
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2012
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Vol. 32, no. 2
239-247
EN
We prove that any uniformly continuous Nemytskii composition operator in the space of functions of bounded generalized Φ-variation in the Schramm sense is affine. A composition operator is locally defined. We show that every locally defined operator mapping the space of continuous functions of bounded (in the sense of Jordan) variation into the space of continous monotonic functions is constant.
10
Content available remote Generalized integration operators from Bloch type spaces to weighted BMOA spaces
Sharma S. D., Sharmat A.
Demonstratio Mathematica
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2011
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Vol. 44, nr 2
373--390
EN
In this paper, we consider a generalized integration operator (…) induced by holomorphic maps g and φ of the open unit disk D, where φ(D) ⊂ D and n is a positive integer. We characterize boundedness and compactness of (…) from Bloch type spaces to weighted BMOA spaces by using logarithmic Carleson measure characterization of the weighted BMOA spaces.
11
Content available remote On composition operator in the algebra of functions of two variables with bounded total Φ-variation in Schramm sense
Ereu T., Merentes N., Rzepka B., Sanchez J. L.
Journal of Mathematics and Applications
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2010
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Vol. 33
35-50
EN
In this paper we consider the so called composition operator being a self-mapping of the Banach algebra of the function of two variables with bounded total Φ-variation in the Schramm sense. The main result of the paper characterizes the composition operator mentioned above which has a generating function being Lipschitzian with respect to the second variable. The basic tool used in our considerations is the concept of the left-left regularization.
12
Content available remote Ascent and descent of composition operators on lp spaces
Chandra H., Kumar P.
Demonstratio Mathematica
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2010
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Vol. 43, nr 1
161-165
EN
Let lp (1 < p
13
Content available remote Characterization of globally Lipschitz Nemytskii operators between spaces of set-valued functions of bounded [phi]- variation in the sense of Riesz
Merentes N., Hernandez S.
Bulletin of the Polish Academy of Sciences. Mathematics
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2004
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Vol. 52, nr 4
417-430
EN
Let (X, || . ||) and [Y, || . ||] be two normed spaces and K be a convex cone in X. Let CC(Y) be the family of all non-empty convex compact subsets of Y. We consider the Nemytskii operators, i.e. the composition operators defined by [Nu)(t) = H(t,u[t)), where H is a given set-valued function. It is shown that if the operator N maps the space RV[phi]1 ([a, b]; K) into RW[phi]2([a, b]; CC[Y)) (both are spaces of functions of bounded [phi]- variation in the sense of Riesz), and if it is globally Lipschitz, then it has to be of the form H(t,u[t)) = A(t]u(t)+B(t), where A(t) is a linear continuous set-valued function and B is a set-valued function of bounded [phi]2-variation in the sense of Riesz. This generalizes results of G. Zawadzka [12], A. Smajdor and W. Smajdor [II], N. Merentes and K. Nikodem [5], and N. Merentes and S. Rivas [7].
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