Matrix Convolution Operators on Groups

✍ Scribed by Cho-Ho Chu (auth.)

• Publisher: Springer-Verlag Berlin Heidelberg
• Year: 2008
• Tongue: English
• Leaves: 118
• Series: Lecture Notes in Mathematics 1956
• Edition: 1
• Category: Library

✦ Synopsis

In the last decade, convolution operators of matrix functions have received unusual attention due to their diverse applications. This monograph presents some new developments in the spectral theory of these operators. The setting is the L^p spaces of matrix-valued functions on locally compact groups. The focus is on the spectra and eigenspaces of convolution operators on these spaces, defined by matrix-valued measures. Among various spectral results, the L²-spectrum of such an operator is completely determined and as an application, the spectrum of a discrete Laplacian on a homogeneous graph is computed using this result. The contractivity properties of matrix convolution semigroups are studied and applications to harmonic functions on Lie groups and Riemannian symmetric spaces are discussed. An interesting feature is the presence of Jordan algebraic structures in matrix-harmonic functions.

✦ Table of Contents

Front Matter....Pages i-ix
Introduction....Pages 1-4
Lebesgue Spaces of Matrix Functions....Pages 5-19
Matrix Convolution Operators....Pages 21-85
Convolution Semigroups....Pages 87-100
Back Matter....Pages 101-108

✦ Subjects

Operator Theory; Abstract Harmonic Analysis; Non-associative Rings and Algebras; Potential Theory; Differential Geometry