LAUSR

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 $$ m,n\in {\mathbb {N}}$$ , 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 $$(2\times 2)$$ 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 $$p\in {\mathbb {N}}$$ and $$(T +Q)$$ is $$(A;(m+2r -2, n+2r -1))$$ -isosymmetric. Some properties of the spectrum are also investigated.