Direct Sums Of Banach Spaces Research Articles

Mixed Tikhonov regularization in Banach spaces based on domain decomposition

We analyze Tikhonov regularization where the forward operator is defined on a direct sum U of several Banach spaces Ui,i=1,…,m. The regularization term on U is a sum of different regularizations on each Ui. The theoretical framework, known for the case m=1, can be easily reformulated to the case m>1. Under certain assumptions on weak topologies and on the forward operator it is possible to extend the results on the well-posedness of the minimization process, on its stability with respect to the data, and on the corresponding existence and convergence results, to the general case m⩾1.We consider two particular regularizations, the total or the so-called bounded variation (BV) regularization and the smooth (H1) regularization. Assume that the domain Ω is decomposed into two non-overlapping subdomains Ω1,Ω2. The direct sum of Banach spaces for m=2 will be U=H1(Ω1)⊕BV(Ω2) and the underlying regularization will be mixed H1–BV regularization. We define proper weak topology on U and prove that the assumptions for the above mentioned framework are fulfilled. In such a way we get the theoretical results for a broad range of H1–BV regularizations including the image denoising for the identity forward operator, the deconvolution problems for the convolution forward operator, or any other inverse problem for the continuous nonlinear forward operators.We demonstrate how the mixed regularization can be applied in practice. We suggest a denoising algorithm and compare its behavior with the BV regularization and the H1 regularization. We discuss the results of the denoising on several real world images justifying the usefulness of the mixed H1–BV regularization.

Polyhedral direct sums of Banach spaces, and generalized centers of finite sets

A Banach space X is said to satisfy (GC) if the set Ef(a) of minimizers of the function X∋x↦f(‖x−a1‖,…,‖x−an‖) is nonempty for each integer n⩾1, each a∈Xn and each continuous nondecreasing coercive real-valued function f on R+n. We study stability of certain polyhedrality properties under making direct sums, in order to be able to use results from a paper by Fonf, Lindenstrauss and the author to show that if X satisfies (GC) and an appropriate polyhedrality property then the function space Cb(T,X) satisfies (GC) for every topological space T. This generalizes the authorʼs result from 1997, proved for finite-dimensional polyhedral spaces X. Moreover, under more restrictive conditions on X and f, the mappings Ef(⋅) on C(K,X)n (n⩾1) are continuous in the Hausdorff metric for each compact K.

On the fixed points of nonexpansive mappings in direct sums of Banach spaces

We show that if a Banach space X has the weak fixed point property for nonexpansive mappings and Y has the generalized Gossez-Lami Dozo property or is uniformly convex in every direction, then the direct sum of X and Y with a strictly monotone norm has the weak fixed point property. The result is new even if Y is finite-dimensional.

Daugavet centers and direct sums of Banach spaces

Abstract A linear continuous nonzero operator G: X → Y is a Daugavet center if every rank-1 operator T: X → Y satisfies ||G + T|| = ||G|| + ||T||. We study the case when either X or Y is a sum X 1⊕F X 2 of two Banach spaces X 1 and X 2 by some two-dimensional Banach space F. We completely describe the class of those F such that for some spaces X 1 and X 2 there exists a Daugavet center acting from X 1⊕F X 2, and the class of those F such that for some pair of spaces X 1 and X 2 there is a Daugavet center acting into X 1⊕F X 2. We also present several examples of such Daugavet centers.

On Semi‐Uniform Kadec‐Klee Banach Spaces

Inspired by the concept of U‐spaces introduced by Lau, (1978), we introduced the class of semi‐uniform Kadec‐Klee spaces, which is a uniform version of semi‐Kadec‐Klee spaces studied by Vlasov, (1972). This class of spaces is a wider subclass of spaces with weak normal structure and hence generalizes many known results in the literature. We give a characterization for a certain direct sum of Banach spaces to be semi‐uniform Kadec‐Klee and use this result to construct a semi‐uniform Kadec‐Klee space which is not uniform Kadec‐Klee. At the end of the paper, we give a remark concerning the uniformly alternative convexity or smoothness introduced by Kadets et al., (1997).

Uniform nonsquareness of direct sums of Banach spaces

An inequality between James constants of Banach spaces $X_s$ and the James constant of their direct sum is obtained. This gives a characterization of uniform nonsquareness of sums of Banach spaces.

Reflexivity of Spaces of Polynomials on Direct Sums of Banach Spaces

Let P( n E, F) be the space of the continuous n-homogeneous polynomials from E into F and Hb(E, F) be the space of the holomorphic mappings from E into F that are bounded in the bounded subsets of E, both spaces endowed with the topology τb of uniform convergence on the bounded subsets of E. The reflexivity of P( n E, F) is studied in connection with the density of the space of the finite type n-homogeneous polynomials in P( n E, F) and in connection with the equality [P( n E, F) ,τ b] � =[ P( n E, F) ,τ 0] � in case E is a reflexive countable direct sum of complex Banach spaces and F is a reflexive complex Banach space. The reflexivity of Hb(E) is also considered.

On the fixed point property in direct sums of Banach spaces with strictly monotone norms

It is shown that if a Banach space X has the weak Banach-Saks property and the weak fixed point property for nonexpansive mappings and Y satisfies property asymptotic (P) (which is weaker than the condition WCS(Y)>1), then the direct sum of X and Y endowed with a strictly monotone norm enjoys the weak fixed point property. The same conclusion is valid if X admits a 1-unconditional basis.

Extremal structure of the unit ball of direct sums of Banach spaces

We characterize the extreme points and smooth points of the unit ball of certain direct sums of Banach spaces. We use these results to characterize noncreasiness and uniform noncreasiness of direct sums, thereby extending results of the second author [S. Saejung, Extreme points, smooth points and noncreasiness of ψ -direct sum of Banach spaces, Nonlinear Anal. 67 (2007) 1649–1653].

Extreme points, smooth points and noncreasiness of [formula omitted]-direct sum of Banach spaces

Characterizations for noncreasiness and uniform noncreasiness of the ψ -direct sum of Banach spaces are given. As a consequence, the geometry of the infinite ℓ p -direct sum ℓ p ( X n ) and the Lebesgue–Bochner space L p ( μ , X ) with respect to these notions is investigated where 1 < p < ∞ .

Dense subspaces of quasinormable spaces

AbstractDense linear subspaces of quasinormable Fréchet spaces need not be quasinormable, as an example due to J. Bonet and S. Dierolf proved. A characterization of the quasinormability of dense linear subspaces of quasinormable locally convex spaces and several consequences are given. Moreover, an example of a dense linear subspace of a countable direct sum of Banach spaces, which is not quasinormable, is provided. (© 2006 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Complex strict convexity of absolute norms on [formula omitted] and direct sums of Banach spaces

We show that an absolute normalized norm on C 2 is complex strictly convex if and only if the associated convex function ψ on [ 0 , 1 ] satisfies ψ ( t ) > max { 1 − t , t } for all 0 < t < 1 . Using this result, we characterize when the direct sum X ⊕ ψ Y of the complex Banach spaces X and Y is complex strictly convex and complex uniformly convex. A characterization of complex strict convexity of C n for n > 2 , equipped with an absolute normalized norm, is also given.

Proximinality in generalized direct sums

We consider proximinality and transitivity of proximinality for subspaces of finite codimension in generalized direct sums of Banach spaces. We give several examples of Banach spaces where proximinality is transitive among subspaces of finite codimension.

On Ψ direct sums of Banach spaces and convexity

AbstractLet X1, X2, …, XN be Banach spaces and ψ a continuous convex function with some appropriate conditions on a certain convex set in RN−1. Let (X1⊕X2⊕…⊕XN)Ψ be the direct sum of X1, X2, …, XN equipped with the norm associated with Ψ. We characterize the strict, uniform, and locally uniform convexity of (X1 ⊕ X2 ⊕ … ⊕ XN)Ψ; by means of the convex function Ψ. As an application these convexities are characterized for the ℓp, q-sum (X1 ⊕ X2 ⊕ … ⊕ XN)p, q (1 &lt; q ≤ p ≤ ∈, q &lt; ∞), which includes the well-known facts for the ℓp-sum (X1 ⊕ X2 ⊕ … ⊕ XN)p in the case p = q.

Strict convexity of absolute norms on and direct sums of banach spaces

Strict convexity of absolute norms on and direct sums of banach spaces

The relations between volume ratios and new concepts of GL constants

In this paper we investigate a property named GL(p,q) which is closely related to the Gordon-Lewis property. Our results on GL(p,q) are then used to estimate volume ratios relative to lp, 1<p≤∞, of unconditional direct sums of Banach spaces.

On projections and unconditional bases in direct sums of Banach spaces II

On projections and unconditional bases in direct sums of Banach spaces II

On projections and unconditional bases in direct sums of Banach spaces

On projections and unconditional bases in direct sums of Banach spaces

The direct sum of Banach spaces with respect to a basis

The direct sum of Banach spaces with respect to a basis