Mathematics

A simple construction is given of a class of Euclidean invariant, reflection positive measures on a compactification of the space of distributions. An unusual feature is that the regularizations used are not reflection positive.

We consider a generalization of Hausdorff operators on Lebesgue spaces and under natural conditions prove that such an operator is not a Riesz operator provided it is non-zero. In particular, it cannot be represented as a sum of a quasinilpotent and compact operators.

Coorbit theory is a powerful machinery that constructs a family of Banach spaces, the so-called coorbit spaces, from well-behaved unitary representations of locally compact groups. A core feature of coorbit spaces is that they can be discretized in a way that reflects the geometry of the underlying locally compact group. Many established function spaces such as modulation spaces, Besov spaces, Sobolev-Shubin spaces, and shearlet spaces are examples of coorbit spaces. The goal of this survey is to give an overview of coorbit theory with the aim of presenting the main ideas in an accessible manner. Coorbit theory is generally seen as a complicated theory, filled with both technicalities and conceptual difficulties. Faced with this obstacle, we feel obliged to convince the reader of the theory's elegance. As such, this survey is a showcase of coorbit theory and should be treated as a stepping stone to more complete sources.

We prove the existence of a separable approximately ultra-homogeneous Banach lattice BL that is isometrically universal for separable Banach lattices. This is done by showing that the class of Banach lattices has the Amalgamation Property, and thus finitely generated Banach lattices form a metric Fraïssé class. Some additional results about the structural properties of BL are also proven.

In 1989, G. Godefroy proved that a Banach space contains an isomorphic copy of ??1 if and only if it can be equivalently renormed to be octahedral. It is known that octahedral norms can be characterized by means of covering the unit sphere by a finite number of balls. This observation allows us to connect the theory of octahedral norms with ball-covering properties of Banach spaces introduced by L. Cheng in 2006. Following this idea, we extend G. Godefroy's result to higher cardinalities. We prove that, for an infinite cardinal κ , a Banach space X contains an isomorphic copy of ??1 ( κ + ) if and only if it can be equivalently renormed in such a way that its unit sphere cannot be covered by κ many open balls not containing α B X , where α??0,1) . We also investigate the relation between ball-coverings of the unit sphere and octahedral norms in the setting of higher cardinalities.

In this article, we prove an inner product inequality for Hilbert space operators. This inequality, then, is utilized to present a general numerical radius inequality using convex functions. Applications of the new results include obtaining new forms that generalize and extend some well known results in the literature, with an application to the newly defined generalized numerical radius. We emphasize that the approach followed in this article is different from the approaches used in the literature to obtain the refined versions.

Given a set B of operators between subspaces of L p spaces, we characterize the operators between subspaces of L p spaces that remain bounded on the X -valued L p space for every Banach space on which elements of the original class B are bounded. This is a form of the bipolar theorem for a duality between the class of Banach spaces and the class of operators between subspaces of L p spaces, essentially introduced by Pisier. The methods we introduce allow us to recover also the other direction --characterizing the bipolar of a set of Banach spaces--, which had been obtained by Hernandez in 1983.

A canonical factorization is given for a quadratic pencil of accretive operators in a Hilbert space. Also, we establish some relationships between an m-accretive operator and its Moore-Penorse inverse. As an application, we study a result of existence, uniqueness, and maximal regularity of the strict solution for complete abstract second order differential equation in the non-homogeneous case. The paper is concluded with some questions left open from the preceding discussions.

In this paper we generalize the classical theorems of Brown and Halmos about algebraic properties of Toeplitz operators to the Bergman space over the unit ball in several complex variables. A key result, which is of independent interest, is the characterization of summable functions u on the unit ball whose Berezin transform B(u) can be written as a finite sum ??j f j g ¯ j with all f j , g j being holomorphic. In particular, we show that such a function must be pluriharmonic if it is sufficiently smooth and bounded. We also settle an open question about M -harmonic functions. Our proofs employ techniques and results from function and operator theory as well as partial differential equations.

In this manuscript, a generalized inverse eigenvalue problem is considered that involves a linear pencil (z J [0,n] − H [0,n] ) of matrices arising in the theory of rational interpolation and biorthogonal rational functions. In addition to the reconstruction of the Hermitian matrix H [0,n] with the entries b ′ j s , characterizations of the rational functions that are components of the prescribed eigenvectors are given. A condition concerning the positive-definiteness of J [0,n] and which is often an assumption in the direct problem is also isolated. Further, the reconstruction of H [0,n] is viewed through the inverse of the pencil (z J [0,n] − H [0,n] ) which involves the concept of m -functions.