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Structure of n-quasi left m-invertible and related classes of operators

Duggal Bhagwati Prashad, Kim In Hyun

Demonstratio Mathematica

2020

Vol. 53, nr 1

249--268

Given Hilbert space operators T, S ∈ B(H), let Δ and δ ∈ B(B (H)) denote the elementary operators ΔT,S(X) = (LT RS − I) (X) = TXS - X and δT,S(X) = (LT – RS)(X) = TX - XS. Let d = Δ or δ. Assuming T commutes with S∗, and choosing X to be the positive operator S∗nSn for some positive integer n, this paper exploits properties of elementary operators to study the structure of n-quasi [m, d]-operators dm T,S (X) = 0 to bring together, and improve upon, extant results for a number of classes of operators, such as n-quasi left m-invertible operators, n-quasi m-isometric operators, n-quasi m-self-adjoint operators and n-quasi (m, C) symmetric operators (for some conjugation C of H). It is proved that Sn is the perturbation by a nilpotent of the direct sum of an operator Sn1 = (…)n satisfying dmT1S1(I1) = 0 , T1 = (…) , with the 0 operator; if S is also left invertible, then Sn is similar to an operator B such that dmB∗,B(I) = 0. For power bounded S and T such that ST∗ - T∗S = 0 and ΔTS(S∗nSn) = 0, S is polaroid (i.e., isolated points of the spectrum are poles). The product property, and the perturbation by a commuting nilpotent property, of operators T, S satisfying dmT,S (I) = 0, given certain commutativity properties, transfers to operators satisfying S∗ndmT,S (I)Sn = 0.