Let X be a locally compact Hausdorff space, and T be a disjointness preserving bounded linear operator on C0 (X). We give a sufficient and necessary condition of T being power compact. Moreover, a description of eigenvalues and eigenfunctions of T is given.

Let <i>T</i> be a compact disjointness preserving linear operator from <i>C</i>₀(<i>X</i>) into <i>C</i>₀(<i>Y</i>), where <i>X</i> and <i>Y</i> are locally compact Hausdorff spaces. We show that <i>T</i> can be represented as a norm convergent countable sum of disjoint rank one operators. More precisely, <i>T</i> = _<i>n</i> <i></i> <i>h_n</i> for a (possibly finite) sequence {<i>x_n</i> }_<i>n</i> of distinct points in <i>X</i> and a norm null sequence {<i>h_n</i> }_<i>n</i> of mutually disjoint functions in <i>C</i>₀(<i>Y</i>). Moreover, we develop a graph theoretic method to describe the spectrum of such an operator ( 2009 WILEYVCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Degree theory has been developed as a tool for checking the solution existence of nonlinear equations. In his classic paper published in 1983, Browder developed a degree theory for mappings of monotone type f+ T, where f is a mapping of class (S)+ from a bounded open set in a reflexive Banach space X into its dual X, and T is a maximal monotone mapping from X into X. This breakthrough paved the way for many applications of degree theoretic techniques to several large classes of nonlinear partial differential equations. In this paper we continue to develop the results of Browder on the degree theory for mappings of monotone type f+ T. By enlarging the class of maximal monotone mappings and pseudo-monotone homotopies we obtain some new results of the degree theory for such mappings.