Mappings preserving the ascent or descent of triple skew products of operators

Abstract

Let $\mathcal{B}(\mathcal{H})$ be the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert
space $\mathcal{H}$. Let $asc(A)$ and $desc(A)$ be, respectively, the ascent and descent of an operator $A$ in $\mathcal{B}(\mathcal{H})$. In this paper, we determine the explicit form of all maps  $ \phi $ from $\mathcal{B}(\mathcal{H})$ into itself that preserve the ascent or descent of triple skew product of operators.