### Quasinormality and Fuglede-Putnam Theorem for w-Hyponormal Operators

#### Abstract

We investigate several properties of Aluthge transform $\tT=|T|^{\frac{1}{2}}U|T|^{\frac{1}{2}}$ of an operator $T=U|T|$ . We prove (i) if $T$ is a $w$-hyponormal operatorand $\tT$ is quasi-normal (resp., normal), then $T$ is quasi-normal (resp.,normal), (ii) if $T$ is a contraction with ker $T=$ ker $T^{2}$ and $\tT$ is a partialisometry, then $T$ is a quasinormal partial isometry, and (iii) we show that if either(a) $T$ is a $w$-hyponormal operator such that $\n(T)\subset\n(T^*)$ and $S^*$ is $w$-hyponormal operator such that $\n(S^*)\subset \n(S)$or (b) $T$ is an invertible $w$-hyponormal operator and $S^*$ is $w$-hyponormal operator or (c) $T$ is a $w$-hyponormal such that $\n(T)\subset\n(T^*)$ and $S^*$ is a class $\Y$, then the pair $(T,S)$ satisfy Fuglede-Putnam property.

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