Isometric shifts on [formula omitted

The Mazur–Ulam property in ℓ∞-sum and c0-sum of strictly convex Banach spaces

Tingley's problems on uniform algebras

Let l < p < ∞, p ≠ 2 and α > 0. In what follows sp(α) will denote the space of all real or complex sequences for which(1.1)In this paper we show that the spaces sp(α) are Banach spaces under the natural norm and in fact share many properties that the usual lp spaces have. Our main results give characterizations of the surjective isometries of sp(α). These turn out to be quite different than the results for lp. For example, we show that for α ≠ 1, an operator T is a surjective isometry if and only if T is a modulus one multiple of the identity. The methods used are valid for both real and complex scalars. They involve the use of a disjoint support condition together with a property of semi inner products. In the complex case the information on isometries allows us to give complete descriptions of the Hermitian operators as well as the adjoint abelian operators.

A projection P0 on a complex Banach space is generalized 3- circular if its linear combination with two projections P1 and P2 having coefficients λ1 and λ2, respectively, is a surjective isometry, where λ1 and λ2 are distinct unit modulus complex numbers different from 1 and P0⊕P1⊕P2=I. Such projections are always contractive. In this paper, we prove structure theorems for generalized 3-circular projections acting on the spaces of all n×n symmetric and skew-symmetric matrices over C when these spaces are equipped with unitary congruence invariant norms.

It has previously been shown that separable Banach spaces $V$ with trivial $L$-structure have the ${L^1}$-Banach-Stone property, i.e. every surjective isometry between two Bochner spaces ${L^1}({\mu _i},V)$ induces an isomorphism of the two measure algebras. We remove the separability restriction, employing the topology of the measure algebra’s Stonean space. The result is achieved via a complete description of the $L$-structure of ${L^1}(\mu ,V)$.

The work of Banach [1],2 Mazur [2], and others has emphasized the fundamental role played by convex functions and sets in the study of Banach spaces and (continuous) linear transformations between them. It is natural to suspect that similar considerations underlie the theory of general linear transformations, in which closure rather than the restricted notion of continuity is predominant. In this paper, therefore, we study linear transformations T from a Banach space B1 , to a Banach space B2, where T denotes a single-valued, additive and homogeneous operation with linear domain DT in B1, and linear range RT in B2. Our definition of linearity is purely algebraic and will necessarily imply continuity and total definition (DT = B1) cnly in the case of linear functionals. With these exceptions our terminology conforms to that of Banach. When T is linear the expression K(x) = 11 Tx 11, xe DT is a convex function, and E[K(x) _ a], a > 0, is a convex set, where E[K(x) ? a] denotes the set of points x e DK = DT such that K(x) ? a. An examination of the connections between the closure of T, the closure of E[K(x) < a], and the structure of convex K's yields a number of theorems with interesting applications, including as special cases theorems of Gelfand [3] and Bosanquet and Kestelmann [4] of the Banach-Steinhaus type (cf. Zygmund [5], pp. 97-100). Upon specializing B1 and B2 to Hilbert space we obtain a characterization of the domain of a selfadjoint transformation. A by-product of the analysis is a surprisingly simple proof of Wintner's theorem [6, ?108]: every definite symmetric transformation possesses a self-adjoint extension.

A universal theorem for stability of ε-isometries of Banach spaces

We introduce Mann-type extragradient methods for a general system of variational inequalities with solutions of a multivalued variational inclusion and common fixed points of a countable family of nonexpansive mappings in real smooth Banach spaces. Here the Mann-type extragradient methods are based on Korpelevich’s extragradient method and Mann iteration method. We first consider and analyze a Mann-type extragradient algorithm in the setting of uniformly convex and 2-uniformly smooth Banach space and then another Mann-type extragradient algorithm in a smooth and uniformly convex Banach space. Under suitable assumptions, we derive some weak and strong convergence theorems. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.

The purpose of this paper is to study some iterative algorithms for finding a common element of the set of solutions of systems of variational inequalities for inverse-strongly accretive mappings and the set of fixed points of an asymptotically nonexpansive mapping in uniformly convex and 2-uniformly smooth Banach space or uniformly convex and q-uniformly smooth Banach space. Strong convergence theorems are obtained under suitable conditions. We also give some numerical examples to support our main results. The results obtained in this paper improve and extend the recent ones announced by many others in the literature.

The notion of “exceptional family of elments (EFE)” plays a very important role in solving complementarity problems. It has been applied in finite dimensional spaces and Hilbert spaces by many authors. In this paper, by using the generalized projection defined by Alber, we extend this notion from Hilbert spaces to uniformly smooth and uniformly convex Banach spaces, and apply this extension to the study of nonlinear complementarity problems in Banach spaces.

Hermitian operators and isometries on injective tensor products of uniform algebras and C⁎-algebras

We provide some new type of mappings associated with pseudocontractions by introducing some actual examples in smooth and strictly convex Banach spaces. Moreover, we also find the significant inequality related to the mappings mentioned in the paper and the mappings defined from generalized mixed equilibrium problems on Banach spaces. We propose an iterative shrinking projection method for finding a common solution of generalized mixed equilibrium problems and fixed point problems of closed and ϕ‐quasi‐strict pseudo‐contractions. Our results hold in reflexive, strictly convex, and smooth Banach spaces with the property (K). The results of this paper improve and extend the corresponding results of Zhou and Gao (2010) and many others.

On super fixed point property and super weak compactness of convex subsets in Banach spaces

Convergence of composite iterative methods for finding zeros of accretive operators

We introduce a new iterative process to approximate a common fixed point of a finite family of multivalued maps in a uniformly convex real Banach space and establish strong convergence theorems for the proposed process. Furthermore, strong convergence theorems for finite family of quasi-nonexpansive multivalued maps are obtained. Our results extend important recent results.