On a class of lambda-hyponormal operators

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.