Matrix multiplication operators on Banach function spaces

We analyze a definition of product of Banach spaces that is naturally associated by duality with an abstract notion of space of multiplication operators. This dual relation allows to understand several constructions coming from different fields of the functional analysis, that can be seen as instances of the abstract one when a particular product is considered. Some relevant examples and applications are shown.

In this paper we introduce and study semigroups of operators on spaces of fuzzy-number-valued functions, and various applications to fuzzy differential equations are presented. Starting from the space of fuzzy numbers, many new spaces sharing the same properties are introduced. We derive basic operator theory results on these spaces and new results in the theory of semigroups of linear operators on fuzzy-number kind spaces. The theory we develop is used to solve classical fuz...

For a scalar sequence {(\theta_n)}_{n \in \mathbb{N}}, let C be the matrix defined by c_n^k = \theta_{n-k+1} if n > k, c_n^k = 0 if n < k. The map between K\"{o}the spaces \lambda(A) and \lambda(B) is called a Cauchy Product map if it is determined by the triangular matrix C. In this note we introduced some necessary and sufficient conditions for a Cauchy Product map on a nuclear K\"{o}the space \lambda(A) to nuclear G_1-space \lambda(B) to be linear and continuous. Its trans...

We create a new, functional calculus, approach to approximation of C_0-semigroups on Banach spaces. As an application of this approach, we obtain optimal convergence rates in classical approximation formulas for C_0-semigroups. In fact, our methods allow one to derive a number of similar formulas and equip them with sharp convergence rates. As a byproduct, we prove a new interpolation principle leading to efficient norm estimates in the Banach algebra of Laplace transforms of...

Notes from a course taught by Palle Jorgensen in the fall semester of 2009. The course covered central themes in functional analysis and operator theory, with an emphasis on topics of special relevance to such applications as representation theory, harmonic analysis, mathematical physics, and stochastic integration.

In this survey, we discuss the definition of a (quasi-)Banach function space. We advertise the original definition by Zaanen and Luxemburg, which does not have various issues introduced by other, subsequent definitions. Moreover, we prove versions of well-known basic properties of Banach function spaces in the setting of quasi-Banach function spaces.

We investigate some types of composition operators, linear and not, and conditions for some spaces to be mapped into themselves and for the operators to satisfy some good properties.

One-parameter strongly continuous semigroups of linear bounded operators in Banach spaces (also known as $C_0$-semigroups) is a fundamental operator-theoretic tool used in the study of linear and non-linear evolution PDEs arising in physics, probability, control theory and other areas of science and technology, including quantum theory, transportation problems and finance. Since 2021 the worldwide community of researchers who work in the field of One-parameter semigroups of o...

In this article we investigate some general properties of the multiplier algebras of normed spaces of continuous functions (NSCF). In particular, we prove that the multiplier algebra inherits some of the properties of the NSCF. We show that it is often possible to construct NSCF's which only admit constant multipliers. In order to do that, using a method from [23], we prove that any separable Banach space can be realized as a NSCF over any separable metrizable space. On the o...

We study BMO spaces associated with semigroup of operators and apply the results to boundedness of Fourier multipliers. We prove a universal interpolation theorem for BMO spaces and prove the boundedness of a class of Fourier multipliers on noncommutative Lp spaces for all 1 < p < \infty, with optimal constants in p.