On class $\mathcal{A}(k^{*})$ operators

Abstract

This paper deals with some classes of bounded linear operators on Hilbert spaces. The main emphasis is put onto the classes $\mathcal{A}(k^{*})$ and $\mathcal{A}_{(k^{*})}P,\, k>0$. Some additional results are given for other classes, like $P\mathcal{A}(k^{*})$, $M-\mathcal{A}(k^{*})$ and spectral properties of operators belonging to $\mathcal{A}(k^{*})$ are considered. We also describe under what conditions a matrix-operator $T_{A,B}$ belongs to $\mathcal{A}(k^{*})$, $\mathcal{A}_{(k^{*})}P$ or $P\mathcal{A}(k^{*})$.