Operator roots of polynomials:iso-symmetric operators

Abstract

Given Hilbert space operators $A_i,B_i$, $i=1,2$, and $X$ such that $A_1$ commutes with $A_2$ and $B_{!}$ commutes with $B_2$, and integers $m,n\ge 1$, we say that the pairs of operators $(B_1,A_1)$ and $(B_2,A_2)$ are left-$(X,(m,n))$-symmetric, denoted $((B_1,A_1),(B_2,A_2))\in \mathrm{left}-(X,(m,n))-\mathrm{symmetric}$ if $$\sum _{j=0}^m\sum _{k=0}^n(-1{)}^{j+k}\left(\begin{array}{c}m\\ j\end{array}\right)\left(\begin{array}{c}n\\ k\end{array}\right)B_1^{m-j}{B}_2^{n-k}X{A}_2^{n-k}{A}_1^j=0.$$An important class of left-$(X,(m,n))-$symmetric operators is obtained uponchoosing $B_1=B_2=A_1^{\ast }=A_2^{\ast }=A^{\ast }$ and $X=I$: such operators have been called $(m,n)-$isosymmetric, and a study of the spectral picture and maximal invariant subspaces of $(m,n)-$isosymmetric operators has been carried out by Stankus \cite{St}. The current work considers stability under perturbations by commuting nilpotents, and products of commuting, left-$(X,(m,n))-$symmetric operators. It is seen that $(X,(m,n))-$isosymmetric Drazin invertible operators $A$ have a particularly interesting structure.

References in corpus (1)

• Left m-invertibility by the adjoint of Drazin inverse and m-selfadjointness of Hilbert space operators