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Let $widetilde{{C}_{varphi}}$ be the Aluthge transform of composition operator on $L^{2}(Sigma)$. The main result of this paper is characterizations of Aluthge transform of composition operators in some operator classes that are weaker than hyponormal, such as hyponormal, quasihyponormal, paranormal, $*$-paranormal on $L^{2}(Sigma)$. Moreover, to explain the results, we provide several useful related examples to show that $widetilde{{C}_{varphi}}$ lie between these classes.

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In this paper, we introduce a new model of a block matrix operator  induced by two sequences and characterize its absolute-(p; r)-
∗-paranormality. Next, we give examples of these operators to show that absolute-(p; r)-∗-paranormal classes are distinct.