In the decade, convolution operators of matrix capabilities have got strange consciousness because of their varied functions. This monograph offers a few new advancements within the spectral idea of those operators. The atmosphere is the Lp spaces of matrix-valued features on in the community compact teams. the point of interest is at the spectra and eigenspaces of convolution operators on those areas, outlined via matrix-valued measures. between numerous spectral effects, the L2-spectrum of such an operator is totally made up our minds and as an software, the spectrum of a discrete Laplacian on a homogeneous graph is computed utilizing this consequence. The contractivity houses of matrix convolution semigroups are studied and functions to harmonic capabilities on Lie teams and Riemannian symmetric areas are mentioned. an attractive characteristic is the presence of Jordan algebraic constructions in matrix-harmonic functions.
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