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We investigate uniform, strong, weak and almost weak stability ofmultiplication semigroups on Banach space valued $L^p$-spaces. We show that,under certain conditions, these properties can be characterized by analogousones of the pointwise semigroups. Using techniques from selector theory, weprove a spectral mapping theorem for the point spectra of the pointwise andglobal semigroups and apply this as a major tool for determining almost weakstability.

Eigenvalues and Stability Properties of Multiplication Operators and Multiplication Semigroups