Representation and normality of $\ast$-paranormal absolutely norm attaining operators

Abstract

In this article, we give a representation of $\ast$-paranormal absolutely norm attaining operator. Explicitly saying, every $\ast$-paranormal absolutely norm attaining ($\mathcal{A}\mathcal{N}$ in short) $T$ can be decomposed as $U\oplus D$, where $U$ is a direct sum of scalar multiple of unitary operators and $D$ is a $2\times 2$ upper diagonal operator matrix. By the representation it is clear that the class of $\ast$-paranormal $\mathcal{A}\mathcal{N}$-operators is bigger than the class of normal $\mathcal{A}\mathcal{N}$-operators but here we observe that a $\ast$-paranormal $\mathcal{A}\mathcal{N}$-operator is normal if either it is invertible or dimension of its null space is same as dimension of null space of its adjoint.