Positive Hankel operators, positive definite kernels and related topics

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 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 V(A) of all linear isometries V from H into H such that AV is positive as well is investigated. In particular, operators A such that 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.