Compactness of Localization Operators on LP- Spaces

We study vector measure. Integration and spaces of p- integrable functions with respect to a vector measure are considered .We establish the spaces of integrable and vector functions with respect to vector measures of convex range and factorization of operators from - spaces. We show some integral identities and inequalities for entire functions and their applications to the coherent state transform, with positive multilinear operators acting on weighted - spaces. We construct the structures of the modulation spaces and multilinear pseudo-differential operators. We show the localization operators and the time-frequency analysis of Sjostrand's class. We study the necessary conditions for Schatten class localization operators and tensor product representations of - space of vector measure duality. We investigate the vector measure Maurey - Rosenthal - type factorizations and sums of - spaces. We show the structure of the short - time Fourier transform analysis of localization operators and study the compactness criteria in function spaces and time-frequency localization operators on Hilbert spaces.