New spaces of Lipschitz type on metric-measure spaces are introduced and they are shown to be just the well-known Besov spaces or Triebel–Lizorkin spaces when the smooth index is less than 1. These theorems also hold in the setting of spaces of homogeneous type, which include Euclidean spaces, Riemannian manifolds and some self-similar fractals. Moreover, the relationships amongst these Lipschitz-type spaces, Hajłasz–Sobolev spaces, Korevaar–Schoen–Sobolev spaces, Newtonian Sobolev space and Cheeger–Sobolev spaces on metric-measure spaces are clarified, showing that they are the same space with equivalence of norms. Furthermore, a Sobolev embedding theorem, namely that the Lipschitz-type spaces with large orders of smoothness can be embedded in Lipschitz spaces, is proved. For metric-measure spaces with heat kernels, a Hardy–Littlewood–Sobolev theorem is establish, and hence it is proved that Lipschitz-type spaces with small orders of smoothness can be embedded in Lebesgue spaces.

For a linear isometry T: C 0 (X) C 0 (Y) of finite corank, there is a cofinite subset Y 1 of Y such that Tf| Y 1= h f is a weighted composition operator and X is homeomorphic to a quotient space of Y 1 modulo a finite subset. When X= Y, such a T is called an isometric quasi-n-shift on C 0 (X). In this case, the action of T can be implemented as a shift on a tree-like structure, called a T-tree, in M (X) with exactly n joints. The T-tree is total in M (X) when T is a shift. With these tools, we can analyze the structure of T.

We state a conjecture for the formulas of the depth 4 low-weight rotational eigenvectors and their corresponding eigenvalues for the 3^G subfactor planar algebras. We prove the conjecture in the case when |G| is odd. To do so, we find an action of G on the reduced subfactor planar algebra at f^{(2)}, which is obtained from shading the planar algebra of the even half. We also show that this reduced subfactor planar algebra is a Yang-Baxter planar algebra.

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).

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

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.

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.

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.