In this Chapter, we study the basic structures of matrix convolution operators Tσ : f ∈ Lp(G, M n ) ↦f * σ∈ Lp(G,M n ). Noncommutativity of the matrix multiplication necessitates the introduction of the left convolution operator Lσ : f ↦σ *ℓ f for a consistent duality theory. We first characterise these operators and show they are translation invariant operators satisfying some continuity condition.We also determine when these operators are weakly compact on L1 and L∞ spaces.
Keywords
- Abelian Group
- Harmonic Function
- Compact Group
- Convolution Operator
- Left Translation
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© 2008 Springer-Verlag Berlin Heidelberg
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(2008). Matrix Convolution Operators. In: Matrix Convolution Operators on Groups. Lecture Notes in Mathematics, vol 1956. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69798-5_3
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DOI: https://doi.org/10.1007/978-3-540-69798-5_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-69797-8
Online ISBN: 978-3-540-69798-5
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