A Cosine Functional Equation in Hilbert Space

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Abstract
Throughout this paper R denotes the set of all real numbers, m(K) the Lebesgue measure of K ⊆ R, H a Hilbert space, L(H) the set of all linear continuous mappings of H into H, endowed with the usual structure of a Banach space.We consider the mapping F of the set R into L(H) such that holds for all x, y ∈ R. In (2) we have solved this equation under the assumption that H is of finite dimension. In this paper we prove that a weak measurability of F implies its weak continuity in the case of separable Hilbert space. In Theorem 2 we prove that every weakly continuous solution of (1) in the set of normal transformations has the form F(x) = cos (xN), where the normal transformation N does not depend on x.

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