ON REPRESENTATION OF ONE CLASS OF SCHMIDT OPERATORS

Abstract

In this paper, unitary symmetrizers are considered. It is well known that using Newton operatoralgorithm, similar to the usual Newton algorithm, for extracting the square root, one can provethat for every Hermitian operator T 0, there exists a unique Hermitian operator S 0 suchthat T = S2. Moreover, S commutes with every bounded operator R with which commutes T. Theoperator S is called a square root of the operator T and is denoted by T1=2. The existence of thesquare root allows one to determine the absolute value jTj = (TT)1=2 of the bounded operator T.For every bounded linear operator T : H ! H there exists a unique partially isometric operatorU : H ! H such that T = UjTj, KerU = KerT. Such an equality is called a polar expansionof the operator T. The Schmidt operator is understood as the unitary multiplier of the polarexpansion of a compact inverse operator, with the help of which E. Schmidt was the rst to obtainthe expansion of a compact and not-self-adjoint operator and introduced so-called s-numbers.This paper shows that the unitary symmetrizer of an operator diers only in sign from the adjointSchmidt operator. The main result of the paper: if A is an invertible and compact operator, andS is a unitary operator such that the operator SA is self-adjoint, then the operator AS is alsoself-adjoint and the formula S = U holds, where U is the Schmidt operator.