On L. Schwartz's boundedness condition for kernels

TL;DR

The paper develops a unified, constructive framework for L. Schwartz's boundedness condition on hermitian kernels by linking Kolmogorov decompositions, Krein space theory, and semigroup invariance. It shows that -L <= K <= L is equivalent to the existence of a Kolmogorov decomposition, and it situates this within invariant decompositions, Hankel-type constraints, and Stinespring dilation theory, establishing a correspondence between minimal dilations and invariant decompositions. It extends these ideas to decomposability via Haagerup and Paulsen results, and demonstrates that holomorphic kernels in multiple variables possess holomorphic Kolmogorov decompositions, hence Schwartz boundedness automatically holds for holomorphic kernels; non-hermitian holomorphic kernels are shown to be decomposable as K(ξ,η)=V(ξ)^* U V(η). Overall, the work unifies moment problems on free semigroups, dilation theory, and multi-variable holomorphy under Schwartz’s boundedness paradigm, providing concrete criteria and constructive representations.

Abstract

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.

Theorems & Definitions (21)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• proof
• Theorem 3.1
• proof
• Theorem 3.2
• Proposition 4.1
• proof