^{1}Department of Mathematics, Payame Noor University, P. O. Box: 19359-3697, Tehran, Iran.

^{2}Department of Mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, Iran.

Abstract

We first show that a bounded linear operator $ T $ on a real Banach space $ E $ is quasicompact (Riesz, respectively) if and only if $T': E_{\mathbb{C}}\longrightarrow E_{\mathbb{C}}$ is quasicompact (Riesz, respectively), where the complex Banach space $E_{\mathbb{C}}$ is a suitable complexification of $E$ and $T'$ 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.

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