On L. Schwartz's boundedness condition for kernels

Details On L. Schwartz's boundedness condition for kernels On L. Schwartz's boundedness condition for kernels

2002-12-16

arxiv.org/abs/math/0212218v1

Mathematics

Functional Analysis

21 pages

Scientific paper

In previous works we analysed conditions for linearization of hermitian kernels. The conditions on the kernel turned out to be of a type considered previously by L. Schwartz in the related matter of characterizing the real space generated by positive definite kernels. The aim of this note is to find more concrete expressions of the Schwartz type conditions: in the Hamburger moment problem for Hankel type kernels on the free semigroup, in dilation theory (Stinespring type dilations and Haagerup decomposability), as well as in multi-variable holomorphy. Among other things, we prove that any hermitian holomorphic kernel has a holomorphic linearization, and hence that holomorphic kernels automatically satisfy L. Schwartz's boundedness condition.

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