Operator roots of polynomials:iso-symmetric operators

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 $m, n\geq 1$, we say that the pairs of operators $(B_1,A_1)$ and $(B_2,A_2)$ are left-$(X, (m,n))$-symmetric, denoted $((B_1,A_1),(B_2,A_2))\in {\rm left}-(X,(m,n))-{\rm symmetric}$ if $$\sum_{j=0}^m\sum_{k=0}^n (-1)^{j+k}\left(\begin{array}{clcr}m\\j\end{array}\right) \left(\begin{array}{clcr}n\\k\end{array}\right) B_1^{m-j}B_2^{n-k} X A_2^{n-k}A_1^{j}=0.$$An important class of left-$(X,(m,n))-$symmetric operators is obtained uponchoosing $B_1=B_2=A^*_1=A^*_2=A^*$ 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.