We study disjointness preserving (quasi-) n-shift operators on C 0 (X), where X is locally compact and Hausdorff. When C 0 (X) admits a quasi-n-shift T, there is a countable subset of X= X{} equipped with a tree-like structure, called -tree, with exactly n joints such that the action of T on C 0 (X) can be implemented as a shift on the -tree. If T is an n-shift, then the -tree is dense in X and thus X is separable. By analyzing the structure of the -tree, we show that every (quasi-) n-shift on c 0 can always be written as a product of n (quasi-) 1-shifts. Although it is not the case for general C 0 (X) as shown by our counter examples, we can do so after dilation.

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.

It is well known that a vector variational inequality can be a very efficient model for use in studying vector optimization problems. By using the Ky Fan fixed point theorem and the scalarization method we will prove some existence theorems for strong solutions for generalized vector variational inequalities where discontinuous and star-pseudomonotone operators are involved. Our results can be applied to the study of the existence of solutions of vector optimal problems. Some examples are given and analyzed.

Inspired by the recent work of Bekka, we study two reasonable analogues of property (T) for not necessarily unital C-algebras. The stronger one of the two is called property (T) and the weaker one is called property (T e). It is shown that all non-unital C-algebras do not have property (T)(neither do their unitalizations). Moreover, all non-unital -unital C-algebras do not have property (T e).

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.