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1
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Interpolation methods of means and orbits

100%
Mastyło M.
Studia Mathematica
|
2005
|
tom 171
|
nr 2
153-175
EN
Banach operator ideal properties of the inclusion maps between Banach sequence spaces are used to study interpolation of orbit spaces. Relationships between those spaces and the method-of-means spaces generated by couples of weighted Banach sequence spaces with the weights determined by concave functions and their Janson sequences are shown. As an application we obtain the description of interpolation orbits in couples of weighted $L_{p}$-spaces when they are not described by the K-method. We also develop a connection between the method of means with a quasi-parameter and the real method of interpolation generated by the Calderón-Lozanovsky space parameters. Applications to interpolation of operators are also discussed.
2
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Factorization and extension of positive homogeneous polynomials

64%
Defant A., Mastyło M.
Studia Mathematica
|
2014
|
tom 221
|
nr 1
87-99
EN
We study the following problem: Given a homogeneous polynomial from a sublattice of a Banach lattice to a Banach lattice, under which additional hypotheses does this polynomial factorize through $L_{p}$-spaces involving multiplication operators? We prove that under some lattice convexity and concavity hypotheses, for polynomials certain vector-valued norm inequalities and weighted norm inequalities are equivalent. We combine these results and prove a factorization theorem for positive homogeneous polynomials which is a variant of a celebrated factorization theorem for linear operators due to Maurey and Rosenthal. Our main application is a Hahn-Banach extension theorem for positive homogeneous polynomials between Banach lattices.
3
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Interpolation of real method spaces via some ideals of operators

64%
Mastyło M., Milman M.
Studia Mathematica
|
1999
|
tom 136
|
nr 1
17-35
EN
Certain operator ideals are used to study interpolation of operators between spaces generated by the real method. Using orbital equivalence a new reiteration formula is proved for certain real interpolation spaces generated by ordered pairs of Banach lattices of the form $(X,L_∞(w))$. As an application we extend Ovchinnikov's interpolation theorem from the context of classical Lions-Peetre spaces to a larger class of real interpolation spaces. A description of certain abstract J-method spaces is also presented.
4
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On the relation between complex and real methods of interpolation

64%
Mastyło M., Ovchinnikov V. I.
Studia Mathematica
|
1997
|
tom 125
|
nr 3
201-218
EN
We study those compatible couples of Banach spaces for which the complex method interpolation spaces are also described by the K-method of interpolation. As an application we present counter-examples to Cwikel's conjecture that all interpolation spaces of a Banach couple are described by the K-method whenever all complex interpolation spaces have this property.
5

Asymptotically isometric and isometric copies of \(\ell_1\) in some Banach function lattices

64%
Kamińska A., ~Mastyło M.
Commentationes Mathematicae
|
2013
|
tom 53
|
nr 2
EN
We identify the class of Caldern-Lozanovskii spaces that do not contain an asymptotically isometric copy of \(\ell_1\), and consequently we obtain the corresponding characterizations in the classes of Orlicz-Lorentz and Orlicz spaces equipped with the Luxemburg norm. We also give a~complete description of order continuous Orlicz-Lorentz spaces which contain (order) isometric copies of \(\ell_1^{(n)}\) for each integer \(n \geq 2\).~As an application we provide necessary and sufficient conditions for order continuous Orlicz-Lorentz spaces to contain an (order) isometric copy of \(\ell_1\).~In particular we give criteria in Orlicz and Lorentz spaces for (order) isometric containment of \(\ell_1^{(n)}\) and \(\ell_1\).~The results are applied to obtain the~description of universal Orlicz-Lorentz spaces for all two-dimensional normed spaces.
6
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Composition of (E,2)-summing operators

64%
Defant A., Mastyło M.
Studia Mathematica
|
2003
|
tom 159
|
nr 1
51-65
EN
The Banach operator ideal of (q,2)-summing operators plays a fundamental role within the theory of s-number and eigenvalue distribution of Riesz operators in Banach spaces. A key result in this context is a composition formula for such operators due to H. König, J. R. Retherford and N. Tomczak-Jaegermann. Based on abstract interpolation theory, we prove a variant of this result for (E,2)-summing operators, E a symmetric Banach sequence space.
7
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Eigenvalues of Hille-Tamarkin operators and geometry of Banach function spaces

64%
Kühn T., Mastyło M.
Studia Mathematica
|
2011
|
tom 207
|
nr 3
275-296
EN
We investigate how the asymptotic eigenvalue behaviour of Hille-Tamarkin operators in Banach function spaces depends on the geometry of the spaces involved. It turns out that the relevant properties are cotype p and p-concavity. We prove some eigenvalue estimates for Hille-Tamarkin operators in general Banach function spaces which extend the classical results in Lebesgue spaces. We specialize our results to Lorentz, Orlicz and Zygmund spaces and give applications to Fourier analysis. We are also able to show the optimality of our eigenvalue estimates in the Lorentz spaces $L_{2,q}$ with 1 ≤ q < 2 and in Zygmund spaces $L_{p}(log L)_a$ with 2 ≤ p < ∞ and a > 0.
8
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Henryk Hudzik – vita et opera

64%
Mastyło M., Musielak J.
Commentationes Mathematicae
|
2015
|
tom 55
|
nr 2
EN
This article contains a short vita of Henryk Hudzik's as well as a non-exhaustive survey of his contribution to various areas of analysis. We focus on the theory of Orlicz−Sobolev spaces and the geometry of Banach spaces. We highlight criteria for some important geometric properties related to the metric fixed point theory in some classes of Banach lattices, including Orlicz and Orlicz−Lorentz spaces, but we do not forget Henryk Hudzik's contribution to nonlinear integral equations and partial differential equations.
9
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The universal right K-property for some interpolation spaces

38%
Mastyło M.
Studia Mathematica
|
1988
|
tom 90
|
nr 2
117-128
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