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We show basic characterizations of Hilbert spaces and the vector -valued Littlewood-Paley and Fourier multipliers theorems. We also show the operator-valued Fourier multiplier theorem and maximal L_p-regularity. We study the behavior of inner functions and summability kernels for the Lebesgue space multipliers. We investigate the multipliers in Hardy-Sobolev spaces and show remarks on vector-valued BMOA and vector-valued multipliers. We develop a very general operator-valued functional calculus for operators with an H^ -calculus. We study the operator -valued Fourier multiplier theorem on the Lebesgue space and geometry of Banach spaces. We give the Gaussian estimates and the L^p-boundedness of Riesz means. The Plancerel-type estimates and sharp spectral multipliers are considered.