Spokes,mirrors and alternating projections

We give an elementary iterative construction of an example, similar to that of Hundal, of a cone and a half-space in the Hilbert space _?2${?}_2$, which intersect at the origin, but such that the corresponding sequence of alternating nearest point mappings (although weakly convergent) does not converge in norm.     

A generalized contraction proximal point algorithm with two monotone operators

Abstract

In this work, we introduce a generalized contraction proximal point algorithm and use it to approximate common zeros of maximal monotone operators A and B in a real Hilbert space setting. The algorithm is a two step procedure that alternates the resolvents of these operators and uses general assumptions on the parameters involved. For particular cases, these relaxed parameters improve the convergence rate of the algorithm. A strong convergence result associated with the algorithm is proved under mild conditions on the parameters. Our main result improves and extends several results in the literature.     

A family of projective splitting methods for the sum of two maximal monotone operators

A splitting method for two monotone operators A and B is an algorithm that attempts to converge to a zero of the sum A + B by solving a sequence of subproblems, each of which involves only the operator A, or only the operator B. Prior algorithms of this type can all in essence be categorized into three main classes, the Douglas/Peaceman-Rachford class, the forward-backward class, and the little-used double-backward class. Through a certain “extended” solution set in a product space, we construct a fundamentally new class of splitting methods for pairs of general maximal monotone operators in Hilbert space. Our algorithms are essentially standard projection methods, using splitting decomposition to construct separators. We prove convergence through Fejér monotonicity techniques, but showing Fejér convergence of a different sequence to a different set than in earlier splitting methods. Our projective algorithms converge under more general conditions than prior splitting methods, allowing the proximal parameter to vary from iteration to iteration, and even from operator to operator, while retaining convergence for essentially arbitrary pairs of operators. The new projective splitting class also contains noteworthy preexisting methods either as conventional special cases or excluded boundary cases. Dedicated to Clovis Gonzaga on the occassion of his 60th birthday.     

Degree of convergence of modified averaged iterations for fixed points problems and operator equations

Let H be a real Hilbert space. We propose a modification for averaged mappings to approximate the unique fixed point of a mapping T:H→H such that T is boundedly Lipschitzian and −T is monotone. We not only prove strong convergence theorems, but also determine the degree of convergence. Using this result, an iteration process is given for finding the unique solution of the equation Ax=f, where A:H→H is strongly monotone and boundedly Lipschitzian.     

Krasnosel’skii type fixed point theorems under weak topology features

In this paper we prove the following Krasnosel’skii type fixed point theorem: Let M be a nonempty bounded closed convex subset of a Banach space X. Suppose that A:M→X and B:X→X are two weakly sequentially continuous mappings satisfying:

(i)

AM is relatively weakly compact;

(ii)

B is a strict contraction;

(iii)

.

Then A+B has at least one fixed point in M.This result is then used to obtain some new fixed point theorems for the sum of a weakly compact and a nonexpansive mapping. The results presented in this paper encompass several earlier ones in the literature.     

A contraction proximal point algorithm with two monotone operators

It is a known fact that the method of alternating projections introduced long ago by von Neumann fails to converge strongly for two arbitrary nonempty, closed and convex subsets of a real Hilbert space. In this paper, a new iterative process for finding common zeros of two maximal monotone operators is introduced and strong convergence results associated with it are proved. If the two operators are subdifferentials of indicator functions, this new algorithm coincides with the old method of alternating projections. Several other important algorithms, such as the contraction proximal point algorithm, occur as special cases of our algorithm. Hence our main results generalize and unify many results that occur in the literature.     

Existence and uniqueness of best proximity points in geodesic metric spaces

A mapping T:A∪B→A∪B such that T(A)⊆B and T(B)⊆A is called a cyclic mapping. A best proximity point x for such a mapping T is a point such that d(x,Tx)= dist(A,B). In this work we provide different existence and uniqueness results of best proximity points in both Banach and geodesic metric spaces. We improve and extend some results on this recent theory and give an affirmative partial answer to a recently posed question by Eldred and Veeramani in [A.A. Eldred, P. Veeramani Existence and convergence of best proximity points, J. Math. Anal. Appl. 323 (2) (2006) 1001-1006].     

Additivity of Jordan maps on standard Jordan operator algebras

Let A be a standard Jordan operator algebra on a Hilbert space of dimension >1 and B be an arbitrary Jordan algebra. In this note, we prove that if a bijection ?:A→B satisfies

     

All-derivable points of operator algebras

Let A be an operator subalgebra in B(H), where H is a Hilbert space. We say that an element Z∈A is an all-derivable point of A for the norm-topology (strongly operator topology, etc.) if, every norm-topology (strongly operator topology, etc.) continuous derivable linear mapping φ at Z (i.e. φ(ST)=φ(S)T+Sφ(T) for any S,T∈A with ST=Z) is a derivation. In this paper, we show that every invertible operator in the nest algebra is an all-derivable point of the nest algebra for the strongly operator topology. We also prove that every nonzero element of the algebra of all 2×2 upper triangular matrixes is an all-derivable point of the algebra.     

Duality and normal parts of operator modules

For an operator bimodule X over von Neumann algebras A⊆B(H) and B⊆B(K), the space of all completely bounded A,B-bimodule maps from X into B(K,H), is the bimodule dual of X. Basic duality theory is developed with a particular attention to the Haagerup tensor product over von Neumann algebras. To X a normal operator bimodule X_n is associated so that completely bounded A,B-bimodule maps from X into normal operator bimodules factorize uniquely through X_n. A construction of X_n in terms of biduals of X, A and B is presented. Various operator bimodule structures are considered on a Banach bimodule admitting a normal such structure.     

Algorithms with strong convergence for a system of nonlinear variational inequalities in Banach spaces

In this paper, a general system of nonlinear variational inequality problem in Banach spaces was considered, which includes some existing problems as special cases. For solving this nonlinear variational inequality problem, we construct two methods which were inspired and motivated by Korpelevich’s extragradient method. Furthermore, we prove that the suggested algorithms converge strongly to some solutions of the studied variational inequality.     

Convergence theorems, best approximation and best proximity for set-valued dynamic systems of relatively quasi-asymptotic contractions in cone uniform spaces

In cone uniform spaces X, using the concept of the D-family of cone pseudodistances, the distance between two not necessarily convex or compact sets A and B in X is defined, the concepts of cyclic and noncyclic set-valued dynamic systems of D-relatively quasi-asymptotic contractions T:A∪B→2^A∪B are introduced and the best approximation and best proximity point theorems for such contractions are proved. Also conditions are given which guarantee that for each starting point each generalized sequence of iterations of these contractions (in particular, each dynamic process) converges and the limit is a best proximity point. Moreover, D-families are constructed, characterized and compared. The results are new for set-valued and single-valued dynamic systems in cone uniform, cone locally convex and cone metric spaces. Various examples illustrating ideas, methods, definitions and results are constructed.     

Additive results for the Wg-Drazin inverse

In this paper we prove the formula for the expression (A+B)^d,W in terms of A,B,W,A^d,W,B^d,W, assuming some conditions for A,B and W. Here S^d,W denotes the generalized W-weighted Drazin inverse of a linear bounded operator S on a Banach space.     

On operator order and chaotic operator order for two operators

In this paper we characterize operator order A?B?O and chaotic operator order log A?logB for positive and invertible operators A and B in terms of operator inequalities via the Furuta inequality and operator equalities due to the Douglas’s majorization and factorization. Related results are obtained, which include generalizations and characterizations of some well-known results.     

Annihilator-preserving maps, multipliers, and derivations

For a commutative subspace lattice L in a von Neumann algebra N and a bounded linear map f:N∩algL→B(H), we show that if Af(B)C=0 for all A,B,C∈N∩algL satisfying AB=BC=0, then f is a generalized derivation. For a unital C^∗-algebra A, a unital Banach A-bimodule M, and a bounded linear map f:A→M, we prove that if f(A)B=0 for all A,B∈A with AB=0, then f is a left multiplier; as a consequence, every bounded local derivation from a C^∗-algebra to a Banach A-bimodule is a derivation. We also show that every local derivation on a semisimple free semigroupoid algebra is a derivation and every local multiplier on a free semigroupoid algebra is a multiplier.     

Algorithms for a system of generalized mixed equilibrium problems and a countable family of some nonlinear multi-valued nonexpansive-type maps

In this paper, relaxed iterative algorithms of Krasnoselskii-type and Halperntype that approximate a solution of a system of a generalized mixed equilibrium problem and a common fixed point of a countable family of totally quasi-φ-asymptotically nonexpansive multi-valued maps are constructed. Strong convergence of the sequence generated by these algorithms is proved in uniformly smooth and strictly convex real Banach spaces with KadecKlee property. Furthermore, several applications of our theorems are also presented. Finally,our theorems are significant improvements on several important recent results for this class of nonlinear problems.     

The countable Telescope Conjecture for module categories

By the Telescope Conjecture for Module Categories, we mean the following claim: “Let R be any ring and (A,B) be a hereditary cotorsion pair in Mod-R with A and B closed under direct limits. Then (A,B) is of finite type.”We prove a modification of this conjecture with the word ‘finite’ replaced by ‘countable.’ We show that a hereditary cotorsion pair (A,B) of modules over an arbitrary ring R is generated by a set of strongly countably presented modules provided that B is closed under unions of well-ordered chains. We also characterize the modules in B and the countably presented modules in A in terms of morphisms between finitely presented modules, and show that (A,B) is cogenerated by a single pure-injective module provided that A is closed under direct limits. Then we move our attention to strong analogies between cotorsion pairs in module categories and localizing pairs in compactly generated triangulated categories.     

On orthogonal systems of matrix algebras

In this work it is shown that certain interesting types of orthogonal system of subalgebras (whose existence cannot be ruled out by the trivial necessary conditions) cannot exist. In particular, it is proved that there is no orthogonal decomposition of M_n(C)⊗M_n(C)≡M_n²(C) into a number of maximal abelian subalgebras and factors isomorphic to M_n(C) in which the number of factors would be 1 or 3.In addition, some new tools are introduced, too: for example, a quantity c(A,B), which measures “how close” the subalgebras A,B⊂M_n(C) are to being orthogonal. It is shown that in the main cases of interest, c(A^′,B^′) - where A^′ and B^′ are the commutants of A and B, respectively - can be determined by c(A,B) and the dimensions of A and B. The corresponding formula is used to find some further obstructions regarding orthogonal systems.     

Noisy colored point set matching

We propose a process for determining approximated matches, in terms of the bottleneck distance, under color preserving rigid motions, between two colored point sets A,B∈R², |A|≤|B|. We solve the matching problem by generating all representative motions that bring A close to a subset B^′ of set B and then using a graph matching algorithm. We also present an approximate matching algorithm with improved computational time. In order to get better running times for both algorithms we present a lossless filtering preprocessing step. By using it, we determine some candidate zones which are regions that contain a subset S of B such that A may match one or more subsets B^′ of S. Then, we solve the matching problem between A and every candidate zone. Experimental results using both synthetic and real data are reported to prove the effectiveness of the proposed approach.     

Additive maps on rings behaving like derivations at idempotent-product elements

For every ring R with the unit I containing a nontrivial idempotent P, we describe the additive maps δ from R into itself which behave like derivations, and show that derivations on such kinds of rings can be determined by the action on the elements A,B∈R with AB=0, AB=P and AB=I respectively. Those results of An and Hou [R. An, J. Hou, Characterizations of derivations on triangular rings: additive maps derivable at idempotents, Linear Algebra Appl. 431 (2009) 1070-1080], Bres?ar [M. Bres?ar, Characterizing homomorphisms, multipliers and derivations in rings with idempotents, Proc. Roy. Soc. Edinburgh. Sect. A. 137 (2007) 9-21] and Chebotar et al. [M.A. Chebotar, W.-F. Ke, P.-H. Lee, Maps characterized by action on zero products, Pacific J. Math. 216 (2) 2004 217-228] are improved.