Classes of Operators Related to Polynomially Normal Operators

Abstract

In this paper, we introduce new classes of operators related to the class of polynomially normal operators which are described as follows: (i) $$m$$ -quasi polynomially normal operators includes polynomially normal operators recently studied in [7, 6]. A bounded linear operator $$S$$ on a complex Hilbert space $$\mathcal{H}$$ is said to be $$m$$ -quasi polynomially normal operator if there exists a nontrivial polynomial $$P={\sum }_{0\le k\le n}{b}_{k}{z}^{k}\in {\mathbb{C}}[z]$$ for which $${S}^{*m}(P(S){S}^{*}-{S}^{*}P(S)){S}^{m}=\,0$$ $$(\Leftrightarrow \sum_{0\le k\le n}{b}_{k}{S}^{*m}({S}^{k}{S}^{*}-{S}^{*}{S}^{k}){S}^{m}=\,0),$$ where $$m$$ is a natural number. (ii) Polynomially $$C$$ -normal operators includes $$C$$ -normal operators studied in [13, 23, 25]. An operator $$S$$ is called polynomially $$C$$ -normal operator if there exists a nontrivial polynomial $$P\in {\mathbb{C}}[z]$$ and a conjugation operator $$C$$ on $$\mathcal{H}$$ for which $$CP(S){S}^{*}-{S}^{*}P(S)C=\,0(\Leftrightarrow \sum_{0\le k\le n}{b}_{k}(C{S}^{k}{S}^{*}-{S}^{*}{S}^{k}C)=\,0).$$ A detailed study of certain properties of some members of the first class has been presented. However, an initiation to the study of the second class has been given.