Putnam-Fuglede type theorem for class $ \mathcal{A}_k $ operators

Abstract

We will call $U\in B(X)$ as an operator of class $\mathcal{A}_k$ if for some integer $k$, the following inequality is satisfied: $$\vert U^{k+1}\vert^{\frac{2}{k+1}}\geq \vert U\vert^{2}.$$ In the present article, some basic spectral properties of this class are given, also the asymmetric Putnam-Fuglede theorem and the range kernel orthogonality for class $\mathcal{A}_k$ operators are proved.

Keywords

• putnam-fuglede theorem
• hyponormal operator
• class $\mathcal{a}_k$ operator