On extension of multi-parametric local semigroups of isometric operators and some applications (Q2928683)

The classical result of M. G. Krein stating that every continuous positive definite function defined on an interval of the real line can be extended to a continuous definite function on the whole line, is considered in this article for a Hilbert space. The author addresses the problem of unitary extensions for \(n\)-parametric local semigroups of isometric operators on Hilbert spaces. With the requirement that each uni-parametric local semigroup of isometric operators has a unique unitary extension to a strongly continuous group of unitary operators on the Hilbert space, it is proved that a strongly continuous \(n\)-parametric local semigroup of isometric operators on a Hilbert space can be extended to a strong continuous group of unitary operator on a larger Hilbert space. The result was previously addressed by the first author in [Integral Equations Oper. Theory 10, 80--801 (1987; Zbl 0656.47031)] for uni-parametric local semigroups and in [Integral Equations Oper. Theory 17, No. 3, 301--321 (1993; Zbl 0796.47030)] for bi-parametric local semigroups.