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This paper gives the concepts of finite dimensional irreducible operators ((FDI) operators) and infinite dimensional irreducible operators ((IDI) operators). Discusses the relationships of (FDI) operators, (IDI) operators and strongly irreducible operators ((SI) operators) and illustrates some properties of the three classes of operators. Some sufficient conditions for the finite-dimensional irreducibility of operators which have the forms of upper triangular operator matrices are given. This paper proves that every operator with a singleton spectrum is a small compact perturbation of an (FDI) operator on separable Banach spaces and shows that every bounded linear operator T can be approximated by operators in (ΣFDI)( X) with respect to the strong-operator topology and every compact operator K can be approximated by operators in (ΣFDI)( X) with respect to the norm topology on a Banach space X with a Schauder basis, where (ΣFDI)( X):= { T ∈ B( X): T = Σ ⊕ T, T ∈ (FDI), k ∈ ℕ}. [ABSTRACT FROM AUTHOR]

A non-Archimedean Banach space has the orthogonal finite-dimensional decomposition property (OFDDP) if it is the orthogonal direct sum of a sequence of finite-dimensional subspaces. This property has an influence in the non-Archimedean Grothendieck's approximation theory, where an open problem is the following: Let E be a non-Archimedean Banach space of countable type with the OFDDP and let D be a closed subspace of E. Does D have the OFDDP? In this paper we give a negative answer to this question; we construct a Banach space of countable type with the OFDDP having a one-codimensional subspace without the OFDDP. Next we prove that, however, for certain classes of Banach spaces of countable type, the OFDDP is preserved by taking finite-codimensional subspaces. [ABSTRACT FROM AUTHOR]

In this paper, we establish a general solution and the generalized Hyers-Ulam-Rassias stability of the following general mixed additive-cubic functional equation in the quasi-Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we give some equivalent conditions on a Banach frame for a Banach space by using the pseudoinverse operator. We also consider the stability of a Banach frame for a Banach space X with respect to X or an X-frame for a Banach space X under perturbation. These results generalize and improve the related works of Balan, Casazza, Christensen, Stoeva and Jian et al. [ABSTRACT FROM AUTHOR]

This paper is concerned with the problem of best weighted simultaneous approximations to totally bounded sequences in Banach spaces. Characterization results from convex sets in Banach spaces are established under the assumption that the Banach space is uniformly smooth. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to provide an error analysis for the multicategory support vector machine (MSVM) classificaton problems. We establish the uniform convergency approach for MSVMs and estimate the misclassification error. The main difficulty we overcome here is to bound the offset vector. As a result, we confirm that the MSVM classification algorithm with polynomial kernels is always efficient when the degree of the kernel polynomial is large enough. Finally the rate of convergence and examples are given to demonstrate the main results. [ABSTRACT FROM AUTHOR]

In this paper we study the connection between the metric projection operator P K : B → K, where B is a reflexive Banach space with dual space B* and K is a non–empty closed convex subset of B, and the generalized projection operators ∏ K : B → K and π K : B* → K. We also present some results in non–reflexive Banach spaces. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we will prove that, Ky Fan's Theorem (Math. Z. 112(1909), 234-240) is true for 1-set-contractive maps defined on a bounded closed convex subset K in a Banach space with int K is not equal to 0. This class of 1-set-contractive maps includes condensing maps, nonexpansive maps, semicontractive maps, LANE maps and others. As applications of our theorems, some fixed point theorems of non-self-maps are proved under various well-known boundary conditions. Our results are generalizations and improvements of the recent results obtained by many authors. [ABSTRACT FROM AUTHOR]

The authors discuss the dual relation of nearly very convexity and property WS. By two kinds of near convexities and two kinds of near smoothness, the authors prove a series of characterizations such that every half-space in Banach space X and every weak* half-space in the dual space X* are approximatively weakly compact and approximatively compact. They show a sufficient condition such that a Banach space X is a Asplund space. Using upper semi-continuity of duality mapping, the authors also give two characterizations of property WS and property S. [ABSTRACT FROM AUTHOR]

In this study, the estimates of approximation numbers of embedding operators in weighted spaces have been analyzed. These estimates depend on orders of differential operators, dimensions of function spaces and weighted functions. This fact implies that the associated embedding operators belong to Schatten class of compact operators. By using these estimates, the discreetness of spectrum and completion of root elements relating to principal nonselfedjoint degenerate differential operators is obtained. [ABSTRACT FROM AUTHOR]

We prove strong convergence of the viscosity approximation process for nonexpansive nonself multimaps. Furthermore, an explicit iteration process which converges strongly to a fixed point of multimap T is constructed. It is worth mentioning that, unlike other authors, we do not impose the condition “ Tz = “ z”” on the map T. [ABSTRACT FROM AUTHOR]

The BCQ and the Abadie CQ for infinite systems of convex inequalities in Banach spaces are characterized in terms of the upper semi–continuity of the convex cones generated by the subdifferentials of active convex functions. Some relationships with other constraint qualifications such as the CPLV and the Slate condition are also studied. Applications in best approximation theory are provided. [ABSTRACT FROM AUTHOR]

Kernel theorems are established for Banach space–valued multilinear mappings. A moment characterization theorem for Banach space–valued generalized functionals of white noise is proved by using the above kernel theorems. A necessary and sufficient condition in terms of moments is given for sequences of Banach space–valued generalized functionals of white noise to converge strongly. The integration is also discussed of functions valued in the space of Banach space–valued generalized functionals. [ABSTRACT FROM AUTHOR]

The concept of anRS-set in a complex Banach space is introduced and the problem of best approximation from anRS-set in a complex space is investigated. Results consisting of characterizations, uniqueness and strong uniqueness are established. [ABSTRACT FROM AUTHOR]