Spectral Properties of M-class $A_{k}^{*}$ Operator
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Keywords:
Class $A_{k}^{*}$, Quasi Class $A_{k}^{*}$, Weyl's theoremAbstract
The Banach algebra on a non-zero complex Hilbert space H of all bounded linear operators are denoted by $B(H)$. An operator T is defined as an element in $B(H)$. If T belongs to $B(H)$, then $T^{*}$ means the adjoint of T in $B(H)$. An operator T is called class $A(k)$ if $\left|T\right|^{2} \le \left(T^*\left|T\right|^{2k} T\right)^{\frac{1}{k+1} } $ for $k > 0$. An operator T is called class $A_{k}$ if $\left|T\right|^{2} \le \left(\left|T^{k+1} \right|^{\frac{2}{k+1} } \right)$ for some positive integer k. S. Panayappan \cite{11} introduced class $A_{k}^{* }$ operator as ``an operator T is called class $A_{k}^{*}$ if $\left|T^{k} \right|^{\frac{2}{k} } \ge \left|T^*\right|^{2} $ where k is a positive integer'' and studied Weyl and Weyl type theorems for the operator \cite{9}. In this paper we introduced extended class $A_{k}^{* }$ operator and studied some of its spectral properties. We also show that extended class $A_{k}^{*}$ operators are closed under tensor product.
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