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Dilability of sesquilinear form-valued kernels. (English) Zbl 0655.47007

Following the classical work of Sz.-Nagy on the dilation theory of Hilbert space operator-valued functions on unital *-semigroups, the author studies the dilatability of sesquilinear form-valued kernels on noninvolutory and nonunital semigroups, that is, functions C:S\(\times S\to F(X)\), where S is a semigroup and F(X) is the space of all sesquilinear forms over the vector space X with values in the underlying field F (\({\mathbb{R}}\) or \({\mathbb{C}}).\)
The main theorem of this paper gives a necessay and sufficient condition for positive-definite kernels to be dilatable; it is some kind of a boundedness condition involving a new kernel \(\tilde C\) from (S\(\times S)\times (S\times S)\) to F(X) associated with C. From this main theorem follow other *-dilatability results on *-semigrops. The paper concludes with an example of a kernel which has a dilation but not a minimal one.
Reviewer: Wu Pei Yuan

MSC:

47A20 Dilations, extensions, compressions of linear operators
47B38 Linear operators on function spaces (general)
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