Operator roots of polynomials: Iso-symmetric operators

Abstract

Given Hilbert space operators Ai, Bi, i = 1, 2, and X such that A1 commutes with A2 and B1 commutes with B2, and integers m, n ? 1, we say that the pairs of operators (B1,A1) and (B2,A2) are left-(X, (m, n))-symmetric, denoted ((B1,A1), (B2,A2)) ? left ? (X, (m, n)) ? symmetric, if ?m j=0 ?n k=0 (?1)j+k (mj)(nk) Bm?j 1 Bn?k 2 XAn?k 2 Aj 1 = 0. An important class of left-(X, (m, n))?symmetric operators is obtained upon choosing B1 = B2 = 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 [23]. Using what are essentially algebraic arguments involving elementary operators, we prove results on 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.