Weyl and Weyl type theorems for class Ak* and quasi class Ak* operators

Recently many mathematicians are working on finding operators which have Bishop's property, SVEP and which satisfy Weyl's theorem, where mostly these operators are normaloids. In our previous papers, we have defined quasi class $A_k$ and m-quasi k-paranormal operators and showed that they have these properties but are not normaloids. In this paper, we define quasi class $A_k^{*}$ operators for positive integers $k$ and show that restriction of quasi class $A_k^{*}$ operators to an invariant subspace are class $A_k^{*}$, they have Bishop' property, (H) property and satisfy Weyl and other Weyl type theorems but are not normaloids. We also define algebraically quasi class $A_k^{*}$ operators and prove that spectral mapping theorem for Weyl spectrum and for essential approximate point spectrum hold. Also if $T$ is an algebraically quasi class $A_k^{*}$ operator, then $T$ is polaroid, Generalised Weyl's theorem holds for $T$ and other Weyl type theorems are discussed. We further prove that tensor product of two quasi class $A_k^{*}$ operators is closed.