A variant of Weyl theorem for a class of quasi-class A acting on an infinite complex Hilbert space were discussed. If the adjoint of T is a quasi-class A operator, then the generalized a-Weyl holds for f(T) , for every function that analytic on the spectrum of T. The generalized Weyl theorem holds for a quasi-class A was proved. Also, a characterization of the Hilbert space as a direct sum of range and kernel of a quasi-class A was given. Among other things, if the operator is a quasi-class A, then the B-Weyl spectrum satisfies the spectral theorem was characterized
In this paper, we prove the following assertions:(1) If the pair of operators $ (A,B^*) $ satisfiesthe Fuglede-Putnam Property and $ S\in \ker(\delta_{A,B}) $, where $ S\in \bh $, then we have$$ \|\delta_{A,B}X+S\|\geq\|S\|.$$(2) Suppose the pair of operators $ (A,B^*) $ satisfies the Fuglede-Putnam Property. If $ A^{2}X=XB^{2} $ and $ A^{3}X=XB^{3} $, then $ AX=XB $.(3) Let $ A,B\in \bh $ be such that $ A,B^* $ are $ p $-hyponormal. Then for any $ X\in\c_{2} $, $ AX-XB\in \mathcal{C}_{2} $ implies $ A^*X-XB^*\in \mathcal{C}_{2} $.(4) Let $ T,S\in \bh $ be such that $ T $ and$ S^* $ are quasihyponormal operators. If $ X\in\bh $ and $ TX=XS$ ,then $T^*X=XS^* $.
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