Extension of the Notion of P-symmetric Operators Using The Aluthge Transform I

Abstract

The class of $\tilde{P}$-symmetric operators is introduced. Certain properties of this class of operators are obtained.
Among other things, it is shown that $(1)$ This class includes the quasinormal operators, isometries, co-isometries, the cyclic subnormal operators and all P-symmetric
operators $(2)$ If $A$ is an $i w$-hyponormal operator then $A$ is $P$-symmetric if and only if $\tilde{A}$ is $P$-symmetric. Conditions under which a $\tilde{P}$-symmetric
operator becomes normal are given.