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)

Let E be a real Banach space with a uniformly Gteaux differentiable norm and which possesses uniform normal structure, K a nonempty bounded closed convex subset of E,{T i} i= 1 N a finite family of asymptotically nonexpansive self-mappings on K with common sequence {k n} n= 1[1,),{t n},{s n} be two sequences in (0, 1) such that s n+ t n= 1 (n 1) and f be a contraction on K. Under suitable conditions on the sequences {s n},{t n}, we show the existence of a sequence {x n} satisfying the relation x n=(1 1 k n) x n+ s n k n f (x n)+ t n k n T r n n x n where n= l n N+ r n for some unique integers l n 0 and 1 r n N. Further we prove that {x n} converges strongly to a common fixed point of {T i} i= 1 N, which solves some variational inequality, provided x n T i x n 0 as n for i= 1, 2,, N. As an application, we prove that the iterative process defined by z 0 K, z n+ 1=(1 1 k n) z n+ s n k n f (z n)+ t n k n

In this paper, a degree theory for finite dimensional generalized variational inequalities is built and employed to prove some results on solution existence and solution stability.

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 A be an operator ideal on LCSs. A continuous seminorm p of a LCS X is said to be Acontinuous if Qp Ainj (X, Xp), where Xp is the completion of the normed space Xp= X/p 1 (0) and Qp is the canonical map. p is said to be a Groth (A)seminorm if there is a continuous seminorm q of X such that p q and the canonical map Qpq: Xq Xp belongs to A (Xq, Xp). It is well-known that when A is the ideal of absolutely summing (resp. precompact, weakly compact) operators, a LCS X is a nuclear (resp. Schwartz, infraSchwartz) space if and only if every continuous seminorm p of X is Acontinuous if and only if every continuous seminorm p of X is a Groth (A)seminorm. In this paper, we extend this equivalence to arbitrary operator ideals A and discuss several aspects of these constructions which are initiated by A. Grothendieck and D. Randkte, respectively. A bornological version of the theory is obtained, too.