Author/Authors :
- - نويسنده Department of Mathematics, Payame Noor University, P. O. Box: 19359-3697, Tehran, Iran. Mayghani Maliheh , - - نويسنده Department of Mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, Iran. Alimohammadi Davood
Abstract :
We first show that a bounded linear operator $ T $ on a real Banach space $ E $ is quasicompact (Riesz, respectively) if and only if $Tʹ: E_{mathbb{C}}longrightarrow E_{mathbb{C}}$ is quasicompact (Riesz, respectively), where the complex Banach space $E_{mathbb{C}}$ is a suitable complexification of $E$ and $Tʹ$ is the complex linear operator on $E_{mathbb{C}}$ associated with $T$. Next, we prove that every unital endomorphism of real Lipschitz algebras of complex-valued functions on compact metric spaces with Lipschitz involutions is a composition operator. Finally, we study some properties of quasicompact and Riesz unital endomorphisms of these algebras.