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Tomasz Kiwerski

Publications and source records attributed to Tomasz Kiwerski.

9 recordsLinked to original sources
arXiv2025

A few last words on pointwise multipliers of Calderón--Lozanovskiĭ spaces

We will provide a complete description of the space $M(X_F,X_G)$ of pointwise multipliers between two Calderón--Lozanovskiĭ spaces $X_F$ and $X_G$ built upon a rearrangement invariant space $X$ and two Young functions $F$ and $G$. Meeting natural expectations, the space $M(X_F,X_G)$ turns out to be another Calderón--Lozanovskiĭ space $X_{G \ominus F}$ with $G \ominus F$ being the appropriately understood generalized Young conjugate of $G$ with respect to $F$. Nevertheless, our argument is not a mere transplantation of existing techniques and requires a rather delicate analysis of the interplay between the space $X$ and functions $F$ and $G$. Furthermore, as an example to illustrate applications, we will solve the factorization problem for Calderón--Lozanovskiĭ spaces. All this not only complements and improves earlier results (basically giving them the final touch), but also confirms the conjecture formulated by Kolwicz, Leśnik and Maligranda in [Pointwise multipliers of Calderón--Lozanovskiĭ spaces, Math. Nachr. 286 (2012), no. 8-9, 876--907]. We will close this work by formulating a number of open questions that outline a promising panorama for future research.

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arXiv2024

Direct sums and abstract Kadets--Klee properties

Let $\mathcal{X} = \{ X_γ \}_{γ\in Γ}$ be a family of Banach spaces and let $\mathcal{E}$ be a Banach sequence space defined on $Γ$. The main aim of this work is to investigate the abstract Kadets--Klee properties, that is, the Kadets--Klee type properties in which the weak convergence of sequences is replaced by the convergence with respect to some linear Hausdorff topology, for the direct sum construction $(\bigoplus_{γ\in Γ} X_γ)_{\mathcal{E}}$. As we will show, and this seems to be quite atypical behavior when compared to some other geometric properties, to lift the Kadets--Klee properties from the components to whole direct sum it is not enough to assume that all involved spaces have the appropriate Kadets--Klee property. Actually, to complete the picture one must add a dichotomy in the form of the Schur type properties for $X_γ$'s supplemented by the variant of strict monotonicity for $\mathcal{E}$. Back down to earth, this general machinery naturally provides a blue print for other topologies like, for example, the weak topology or the topology of local convergence in measure, that are perhaps more commonly associated with this type of considerations. Furthermore, by limiting ourselves to direct sums in which the family $\mathcal{X}$ is constant, that is, $X_γ = X$ for all $γ\in Γ$ and some Banach space $X$, we return to the well-explored ground of K{ö}the--Bochner sequence spaces $\mathcal{E}(X)$. Doing all this, we will reproduce, but sometimes also improve, essentially all existing results about the classical Kadets--Klee properties in K{ö}the--Bochner sequence spaces.

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arXiv2024

Essential norms of pointwise multipliers in the non-algebraic setting

Motivated by some recent results, but also referring to recognized classics, we compute the essential norm and the weak essential norm of multiplication operators acting between two distinct K{\" o}the spaces both defined over the same $σ$-finite measure space. A by-product of the technology we have developed here are some applications to Banach sequence spaces related to decreasing functions and to Banach spaces of analytic functions on the unit disc, in particular, Hardy spaces. We will close our work with some specific examples illustrating the previously obtained results including Musielak--Orlicz sequence spaces (in particular, Nakano sequence spaces) as well as Lorentz and Marcinkiewicz sequence spaces.

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arXiv2024

Arithmetic, interpolation and factorization of amalgams

Building upon Bennett's and Grosse-Erdmann's ideas falling under the conceptual umbrella of factorization of inequalities, we propose a unified approach towards the structure of certain Banach ideal spaces defined in terms of the least decreasing majorant. The key to our results, and, it seems, the main novelty in general, is the synthesis of discretization process, usually called the blocking technique, along with some tools from the interpolation theory. This blend allows us to obtain an abstract versions of several remarkable results proposed by Bennett and to show certain phenomena in new, somehow more complete perspective. Furthermore, with the help of technology we have developed, we re-prove and sometimes also improve many more recent results belonging to this circle of ideas.

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arXiv2021

Quotients, $\ell_\infty$ and abstract Cesàro spaces

Investigating some re-arrangement properties of the norm in the quotient spaces $X/X_a$ we determine the properties of the spaces $X$ guaranteeing the existence of a lattice isometric copy of $\ell_\infty$ in the abstract Cesàro spaces $CX$.

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arXiv2019

Isomorphic and isometric structure of the optimal domains for Hardy-type operators

We investigate structure of the optimal domains for the Hardy-type operators including, for example, the classical Cesàro, Copson and Volterra operators as well as for some of their generalizations. We prove that, in some sense, the abstract Cesàro and Copson function spaces are closely related to the space $L^1$, namely, they contain "in the middle" a complemented copy of $L^1[0,1]$, asymptotically isometric copy of $\ell^1$ and also can be renormed to contain an isometric copy of $L^1[0,1]$. Moreover, the generalized Tandori function spaces are quite similar to $L^\infty$ because they contain an isometric copy of $\ell^\infty$ and can be renormed to contain an isometric copy of $L^\infty[0,1]$. Several applications to the metric fixed point theory will be given. Next, we prove that the Cesàro construction $X \mapsto CX$ does not commutate with the truncation operation of the measure space support. We also study whether a given property transfers between a Banach function space $X$ and the space $TX$, where $T$ is the Cesàro or the Copson operator. In particular, we find a large class of properties which do not lift from $TX$ into $X$ and prove that the abstract Cesàro and Copson function spaces are never reflexive, are not isomorphic to a dual space and do not have the Radon--Nikodym property in general.

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arXiv2017

Local approach to order continuity in Cesàro function spaces

The goal of this paper is to present a complete characterisation of points of order continuity in abstract Cesàro function spaces $CX$ for $X$ being a symmetric function space. Under some additional assumptions mentioned result takes the form $(CX)_a = C(X_a)$. We also find simple equivalent condition for this equality which in the case of $I=[0,1]$ comes to $X\neq L^\infty$. Furthermore, we prove that $X$ is order continuous if and only if $CX$ is, under assumption that the Cesàro operator is bounded on $X$. This result is applied to particular spaces, namely: Cesàro-Orlicz function spaces, Cesàro-Lorentz function spaces and Cesàro-Marcinkiewicz function spaces to get criteria for OC-points.

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arXiv2016

Isomorphic copies of $l^\infty$ in Cesàro-Orlicz function spaces

We characterize Cesàro-Orlicz function spaces $Ces_φ$ containing isomorphic copy of $l^\infty$. We also describe the subspaces $(Ces_φ)_a$ of all order continuous elements of $Ces_φ$. Finally, we study the monotonicity structure of the spaces $Ces_φ$ and $(Ces_φ)_a$.

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