On a class of lambda-hyponormal operators

TL;DR

This paper introduces and studies a new class of lambda-hyponormal operators on infinite-dimensional Hilbert spaces, exploring their properties, hypercyclicity, and specific examples, advancing operator theory understanding.

Contribution

It defines lambda-hyponormal operators, identifies a subclass that cannot be hypercyclic, and analyzes their properties and hypercyclicity in weighted composition operators.

Findings

Certain lambda-hyponormal operators are not hypercyclic.

Closedness of range is studied for weighted composition operators.

Hypercyclicity results are applied to lambda-hyponormal weighted composition operators.

Abstract

In this paper we define $\lambda$-hyponormal operators on an infinite dimensional Hilbert space $\mathcal{H}$ and find a class of $\lambda$-hyponormal operators that can not be hypercyclic. Also, we study closedness of range and $\lambda$-hyponormality of weighted composition operators on the Hilbert space $\mathcal{H}=L^2(\mu)$. Moreover, we apply the hypercyclicity results to $\lambda$-hyponormal weighted composition operators. Finally, we provide some examples to illustrate our main results.