# Lebesgue Spaces of Matrix Functions

Chapter

In this Chapter, we introduce the notations and define the spaces *L*^{ p }(*G*,*M*_{ n }) ofmatrix *L*^{ p } functions on locally compact groups *G* as a setting for later developments. We recall some basic definitions and derive some results for convolution operators in the scalar case. We discuss differentiability of the norm in *L*^{ p }(*G*,*M*^{ p } which is needed later, and compute the Gateaux derivative of the norm when the matrix space *M*^{ p } is equipped with the Hilbert-Schmidt norm.

## Keywords

Banach Space Matrix Function Compact Group Banach Algebra Lebesgue Space
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## Copyright information

© Springer-Verlag Berlin Heidelberg 2008