Lebesgue Spaces of Matrix Functions

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.