Properties of absolute-*-k-paranormal operator and contractions for *-A(k) operators

Abstract

First, we see if $T$ is absolute-$*$-$k$-paranormal for $k\geq 1$, then $T$ is a normaloid operator. We also see some properties of absolute-$*$-$k$-paranormal operator and $*$-$\mathcal{A}(k)$ operator.  Then, we will prove the spectrum continuity of the class $*$-$\mathcal{A}(k)$ operator for $k>0$. Moreover, it is proved that if $T$ is a contraction of the class $*$-$\mathcal{A}(k)$ for $k>0$, then either $T$ has a nontrivial invariant subspace or $T$ is a proper contraction, and the nonnegative operator $$D=\left(T^{*}|T|^{2k}T\right)^{\frac{1}{k+1}}-|T^{*}|^{2}$$ is a strongly stable contraction. Finally if $T\in *$-$\mathcal{A}(k)$ is a contraction for $k>0,$ then $T$ is the direct sum of a unitary and $C_{\cdot 0}$ (c.n.u) contraction.