Operator roots of polynomials: Iso-symmetric operators

B P Duggal ; I H Kim

Filomat, Tome 36 (2022) no. 13, p. 4539

Résumé

Given Hilbert space operators A i , B i , i = 1, 2, and X such that A 1 commutes with A 2 and B 1 commutes with B 2 , and integers m, n ≥ 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)) ∈ left − (X, (m, n)) − symmetric, if m j=0 n k=0 (−1) j+k m j n k B m− j 1 B n−k 2 XA n−k 2 A j 1 = 0. An important class of left-(X, (m, n))−symmetric operators is obtained upon choosing 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 [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.

Détail

DOI : 10.2298/FIL2213539D

Classification : 47A05, 47A55, 47A11, 47B47
Keywords: Hilbert space, Left/right multiplication operator, m-left invertible, m-isometric and m-selfadjoint operators, product of operators, perturbation by nilpotents, commuting operators