2018/12/01 by Duggal, B. P.
#47A05 #47A10 #47A55 #47A80 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1812.00221
A Hilbert space operator S∈\B is n-quasi left m-invertible (resp., left m-invertible) by T∈\B, m,n ≥ 1 some integers, if S*np(S,T)Sn=0 (resp., p(S,T)=0), where p(S,T)=∑j=0m(-1)m-j(m
j)TjSj. Left m-invertible and n-quasi left m-invertible operators share a number of properties. Thus, if S is n-quasi left m-invertible, then Sn is the perturbation by a nilpotent of the direct sum of a left m-invertible with the 0 operator. In particular, if T=S^* (so that S is n-quasi m-isomertric) and |(S|_Sn(\H))n| is not the identity operator, then Sn is similar to an m-isometry. For a power bounded n-quasi left m-invertible operator S such that T is (also) power bounded. and ST^*-T^*S=0, S is polaroid (i.e., isolated points of the spectrum are poles); the product of an n-quasi left m1-invertible operator with a left m2-invertible operator, given certain commutativity properties, is n-quasi left (m1+m2-1)-invertible; again, if ST^*-T^*S=0 and N is an n1-nilpotent which commutes with S, then T is an (n+n1-1)-quasi left (m+n1-1)-inverse of S+N1. These results have applications to n-quasi m-isometries \citeAS, [m,C]-isometries \citeCKL, and (left invertible) m-symmetric \citeCLM and m-selfadjoint \citeL operators.