Positive Hankel operators, positive definite kernels and related topics

Piotr Niemiec

doi:10.1007/s43036-020-00058-6

Abstract

It is shown that a positive (bounded linear) operator on a Hilbert space with trivial kernel is unitarily equivalent to a Hankel operator that satisfies the double positivity condition if and only if it is non-invertible and has simple spectrum (that is, if this operator admits a cyclic vector). More generally, for an arbitrary positive (bounded linear) operator A on a Hilbert space H with trivial kernel the collection {mathscr {V}}(A) of all linear isometries V:H rightarrow H such that A V is positive as well is investigated. In particular, operators A such that {mathscr {V}}(A) contains a pure isometry with a given deficiency index are characterized. Some applications to unbounded positive self-adjoint operators as well as to positive definite kernels are presented. In particular, positive definite matrix-type square roots of such kernels are studied and kernels that have a unique such root are characterized. The class of all positive definite kernels that have at least one such a square root is also investigated.

Highlights

In [15] the authors characterized bounded self-adjoint operators that are unitarily equivalent to Hankel
If A is unitarily equivalent to a Hankel operator, the essential supremum of the multiplicity function of A does not exceed 2;
If the essential supremum of the multiplicity function of A does not exceed 1, A is unitarily equivalent to a Hankel operator

Summary

Introduction

In [15] the authors characterized (in the language of the multiplicity theory of separable Hilbert space self-adjoint operators) bounded self-adjoint operators that are unitarily equivalent to Hankel This is a deep result whose proof is difficult and long. In Theorem 5 below we gather conditions on a dense operator range R in a Hilbert space H related to the foregoing statement ðcÞ—that is, conditions equivalent to the existence of a closed linear subspace Z of H such that Z \ R 1⁄4 f0g and dimðZÞ 1⁄4 dimðHÞ. We use basics of the operator theory and of the spectral theory of self-adjoint operators Another topic we deal with in this paper is related to Hilbert space reproducing (that is, positive definite) kernels.

Notation and terminology

Isometries of class VðAÞ

Operator ranges

Unbounded positive operators

Positive definite kernels