Some Results on (A; (m, n))-Isosymmetric Operators on a Hilbert Space

In this paper, we introduce the class of (A; (m, n))-isosymmetric operators and we study some of their properties, for a positive semi-definite operator A and m, n ∈ N, which extend, by changing the initial inner product with the semi-inner product induced by A, the well-known class of (m, n)-isosymmetric operators introduced by Stankus (Isosymmetric linear transformations on complex Hilbert space. University of California, San Diego, Thesis, 1993, Integral Equ Oper Theory 75(3):301–321, 2013). In particular, we characterize a family of A-isosymmetric (2 × 2) upper triangular operator matrices. Moreover, we show that if T is (A; (m, n))-isosymmetric and if Q is a nilpotent operator of order r doubly commuting with T , then T p is (A; (m, n))-isosymmetric symmetric for any p ∈ N and (T + Q) is (A; (m + 2r − 2, n + 2r −1))-isosymmetric. Some properties of the spectrum are also investigated.