Demonstratio Mathematica (Oct 2020)
Structure of n-quasi left m-invertible and related classes of operators
Abstract
Given Hilbert space operators T,S∈B(ℋ)T,S\in B( {\mathcal H} ), let Δ\text{Δ} and δ∈B(B(ℋ))\delta \in B(B( {\mathcal H} )) denote the elementary operators ΔT,S(X)=(LTRS−I)(X)=TXS−X{\text{Δ}}_{T,S}(X)=({L}_{T}{R}_{S}-I)(X)=TXS-X and δT,S(X)=(LT−RS)(X)=TX−XS{\delta }_{T,S}(X)=({L}_{T}-{R}_{S})(X)=TX-XS. Let d=Δd=\text{Δ} or δ\delta . Assuming T commutes with S∗{S}^{\ast }, and choosing X to be the positive operator S∗nSn{S}^{\ast n}{S}^{n} for some positive integer n, this paper exploits properties of elementary operators to study the structure of n-quasi [m,d]{[}m,d]-operators dT,Sm(X)=0{d}_{T,S}^{m}(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)(m,C) symmetric operators (for some conjugation C of ℋ {\mathcal H} ). It is proved that Sn{S}^{n} is the perturbation by a nilpotent of the direct sum of an operator S1n=S|Sn(ℋ)¯n{S}_{1}^{n}={\left(S{|}_{\overline{{S}^{n}( {\mathcal H} )}}\right)}^{n} satisfying dT1,S1m(I1)=0{d}_{{T}_{1},{S}_{1}}^{m}({I}_{1})=0, T1=TSn(ℋ)¯{{T}_{1}=T}_{\overline{{S}^{n}( {\mathcal H} )}}, with the 0 operator; if S is also left invertible, then Sn{S}^{n} is similar to an operator B such that dB∗,Bm(I)=0{d}_{{B}^{\ast },B}^{m}(I)=0. For power bounded S and T such that ST∗−T∗S=0S{T}^{\ast }-{T}^{\ast }S=0 and ΔT,S(S∗nSn)=0{\text{Δ}}_{T,S}({S}^{\ast n}{S}^{n})=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,ST,S satisfying dT,Sm(I)=0{d}_{T,S}^{m}(I)=0, given certain commutativity properties, transfers to operators satisfying S∗ndT,Sm(I)Sn=0{S}^{\ast n}{d}_{T,S}^{m}(I){S}^{n}=0.
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