Journal of the
Korean Mathematical Society JKMS

An operator $T\in L(H),$ is said to belong to $k$-quasi class $\mathcal{A}^{*}_{n}$ operator if $$T^{*k}\left(|T^{n+1}|^{\frac{2}{n+1}}-|T^{*}|^{2}\right)T^{k}\geq O$$ for some positive integer $n$ and some positive integer $k$. First, we will see some properties of this class of operators and prove Weyl's theorem for algebraically $k$-quasi class $\mathcal{A}^{*}_{n}.$ Second, we consider the tensor product for $k$-quasi class $\mathcal{A}^{*}_{n}$, giving a necessary and sufficient condition for $T\otimes S$ to be a $k$-quasi class $\mathcal{A}^{*}_{n}$, when $T$ and $S$ are both non-zero operators. Then, the existence of a nontrivial hyperinvariant subspace of $k$-quasi class $\mathcal{A}^{*}_{n}$ operator will be shown, and it will also be shown that if $X$ is a Hilbert-Schmidt operator, $A$ and $(B^{*})^{-1}$ are $k$-quasi class $\mathcal{A}^{*}_{n}$ operators such that $AX=XB$, then $A^{\ast}X=XB^{\ast}$. Finally, we will prove the spectrum continuity of this class of operators.

Keywords: $k$-quasi class $\mathcal{A}^{*}_{n}$ operators, Weyl's theorem, $a$-Weyl's theorem, polaroid operators, tensor products, Fuglede-Putnam theorem, hyperinvariant, continuity spectrum

MSC numbers: 47B20