Journal of Operator Theory

Volume 50, Issue 2, Fall 2003 pp. 297-310.

On measurable operator valued indefinite functions with a finite number of negative squares

Authors: Ramon Bruzual (1) and Marisela Dominguez (2)
Author institution: (1) Escuela de Matem\'atica, Fac. Ciencias, Universidad Central de Venezuela, Apartado Postal 47686, Caracas 1041-A, Venezuela
(2) Escuela de Matem\'atica, Fac. Ciencias, Universidad Central de Venezuela, Apartado Postal 47159, Caracas 1041-A, Venezuela

Summary: Let $f$ be a $\kappa$-indefinite function defined on a locally compact group with values in the space of the continuous linear operators of a Kre\u{\i}n space. We prove that if $f$ is weakly measurable then $f = f^{\rm c} + f^0$, where $f^{\rm c}$ is a $\kappa$-indefinite and weakly continuous function and $f^0$ is a positive definite function which is zero locally almost everywhere. We also prove that if $f$ is weakly continuous then $f$ is strongly continuous. As an application we obtain that a weakly measurable group of unitary operators, on a separable Pontryagin space and with parameter on a locally compact group, is strongly continuous.

Keywords: Weakly measurable, operators on Kre\u{\i}n spaces, groups of linear operators, strong continuity

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