Harmonic analysis in spaces of matrices and operators

Résumé

This thesis, which extends into five chapters, deals with three areas of mathematical analysis: measurability theory, harmonic analysis and matricial analysis. The study is interested in an operator-valued approach. More precisely, we want to extract the concepts, elements and some of the classical results of the aforementioned theories from the scalar setting, and extend or generalize them to a vectorial context. In the first chapter, devoted to measurability, the vectorial approach consists of considering functions taking values in the space of bounded and linear operators between two Banach spaces. In this operator-valued setting, the three main topologies (namely the norm topology, the strong operator topology and the weak operator topology) shall play an important role. We study measurable spaces without an underlying measure, and introduce a terminology that allows to differentiate all the possible notions in the operator-valued setting and unravel their relations. With such purpose, the notions of measurability are organized in two types: measurability with respect to a basis of the corresponding topology and measurability in terms of approximability by a sequence of countably-valued functions. The most important result is a version of Pettis’s measurability Theorem for the strong operator topology. The subject of the next three chapters is the matricial harmonic analysis. This time, the operator-valued philosophy leads us to consider matrices whose entries belong to the space of bounded and linear operators from a Hilbert space to itself. The Hadamard product (also known as the Schur product), which is the element-wise product of matrices, is an important device in the study. The concept of Schur multiplier is defined and studied in our operator setting, with an emphasis on Toeplitz matrices. The relations that connect the worlds of functions/measures and matrices are explored in detail, and the results obtained include generalizations of classical theorems such as Toeplitz’s Theorem of Bennett’s Theorem in our setting. Also, we explore some particular types of matrices connected with Schur multipliers: the space of “continuous matrices" and the space of “integrable matrices”, which consist of matrices that are limit in the operator norm (respectively multiplier norm) of “polynomial matrices” (matrices that can be written as a finite sum of diagonal matrices). The last chapter, which takes a rather algebraic tone, presents new alternative versions of Schur and Kronecker products for block matrices and explores some of their properties providing examples and some applications. A proof of Schur’s Theorem for block matrices equipped with this new Schur product is provided, and the chapter also investigates the trace operator in conjunction with those two matrix products, extending trace equalities and inequalities from the scalar case to the block matrix setting.