PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE (BEOGRAD) (N.S.) Vol. 29(43), pp. 5--13 (1981)

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$M$ -- PARANORMAL OPERATORS

S.C. Arora and Ramesh Kumar

Department of Mathematics, Hans Raj College, University of Delhi, Delhi 11007, India and Department of Mathematics, Khalsa College, University of Delhi, Delhi 11007, India

Abstract: V. Istratescu has recently defined $M$-paranormal operators on a Hilbert space $H$ as: An operator $T$ is called $M$-paranormal if for all $x\in H$ with $\|x\|=1$, $$\|T^2 x\|\geqq\frac1M\|Tx\|^2$$ We prove the following results: \item{1.} $T$ is $M$-paranormal if and only if $M^2T^*2T^2-2\lambda T^*T+\lambda^2 \geq 0$ for all $\lambda > 0$. \item{2.} If a $M$-paranormal operator $T$ double commutes with a hyponormal operator $S$, then the product $TS$ is $M$-paranormal. \item{3.} If a paranormal operator $T$ doble commutes with a $M$-hyponormal operator, then the product $TS$ is $M$-paranormal. \item{4.} If $T$ is invertible $M$-paranormal, then $T^{-1}$ is also $M$-paranormal. \item{5.} If $Re W (T) \leq 0$, where $W (T)$ denotes the numerical range of $T$, then $T$ is $M$-paranormal for $M \geq 8$. \item{6.} If a $M$-paranormal partial isometry $T$ satisfies $\|T\| \leq \frac1M$, then it is subnormal.

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