Let *G* be a locally compact group and 1 ≤ p ≤ ∞. In this monograph, we study the basic structures of the convolution operators *f* ↦*f* * σ on *L*^{p}
spaces of matrixvalued functions on *G*, induced by a matrix-valued measure σ on *G*. This study is motivated by recent works in [9, 10, 12–14, 16] on complex and matrix-valued σ-harmonic functions on *G* which are eigenfunctions of the operator *f* ↦*f* * σ, as well as their applications in [45] and the fact that a system of scalar convolution equations is equivalent to a matrix convolution equation. The ubiquity of matrixvalued functions gives another impetus to our investigation, for example, the matrix convolution *f* * σ of a matrix distribution *f* and a matrix measure σ on ℝ^{n}
has been used in [49] to study partial differential and convolution equations and recently, applications of vector-valued *L*^{2}-convolution operators with matrix-valued kernels have been described in depth in [6], and the Fredholm properties of finite sums of weighted shift operators on ℓ^{p}
spaces of Banach space valued functions on ℤ^{n} have been analysed in detail in [54]. Convolution operators on *L*^{p}
spaces of real and complex functions are well-studied in literature, however, there are at least two new elements in the matrix setting, namely, the non-commutativity of the matrix multiplication and the non-associative structures of the harmonic functions, which add complexity to the subject and often require more delicate treatment. Some of our results for matrix convolution operators are also new in the scalar case.

## Keywords

Harmonic Function Compact Group Convolution Operator Scalar Case Fredholm Property## Preview

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