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A Cosine Functional Equation in Hilbert Space

Published online by Cambridge University Press:  20 November 2018

Svetozar Kurepa
Svetozar Kurepa*
Affiliation:
Department of Mathematics, Zagreb
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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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 12 , 1960 , pp. 45 - 50
Copyright
Copyright © Canadian Mathematical Society 1960

References

1. Phillips, R. and Hille, E., Functional analysis and semigroups, Amer. Math. Sci. Coll. Pub. (1957).Google Scholar
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3. Kurepa, S., On the (C)-property of functions, Glasnik mat. fiz. i astr., 13 (1958), 33–38.Google Scholar
4. Nagy, B. Sz., Spektraldarstellung Linearer Transformationen des Hilberschen Raumes (Berlin, 1942).Google Scholar