On Different Definitions of Positive Definiteness
Copyright policy )On Different Definitions of Positive Definiteness
Let the function \(T^*\) be defined by \(T^*(a)=T(a^{-1})^*\) for a function \(T\) of a group \(G\) into a Hilbert space. The authors show that the commutativity of \(G\) is equivalent to each of the conditions below: (1) \(T^*\) is positive definite for every representation \(T\) of \(G\) in Hilbert space. (2) Every weakly positive definite function of \(G\) is positive definite. It answers a question of \textit{S. K. Berberian} [Mich. Math. J. 13, 171--184 (1966; Zbl 0152.13804)].
- Leipzig University Germany
Groups and semigroups of linear operators, weakly positive definite function, Representation theory of groups, representation, Hilbert space, Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones), commutativity of group
Groups and semigroups of linear operators, weakly positive definite function, Representation theory of groups, representation, Hilbert space, Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones), commutativity of group
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