Some results on higher order isosymmetries in Semi-Hilbertian Spaces

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 Mark Stankus (\cite{mark1}, \cite{mark}). In particular, we characterize a family of $A$-isosymmetric $(2\times2)$ upper triangular operator matrices. Moreover, we show that 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 $\big(A,(m+2r -2, n+2r -1)\big)$-isosymmetric. Some properties of the spectrum are also investigated.