  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.
