Product Of Banach Spaces Research Articles

Gustafson, Jeribi, Weidmann and Wolf essential spectra and generator properties for one-sided coupled operator matrices

This paper is concerned with the closedness (closability), essential spectra and generator properties for a one-sided coupled operator matrix A 0 acting in a product of Banach spaces X and Y. By combining the Browder Resolvent with the Schur factorization, some basic properties related to the operator entries of A 0 are collected, including perturbations, closedness and spectra. Utilizing these results, we present a necessary and sufficient condition for A 0 to be closed (closable), and the essential spectra as well as generator properties of A 0 are further described.

MIONet: Learning Multiple-Input Operators via Tensor Product

As an emerging paradigm in scientific machine learning, neural operators aim to learn operators, via neural networks, that map between infinite-dimensional function spaces. Several neural operators have been recently developed. However, all the existing neural operators are only designed to learn operators defined on a single Banach space; i.e., the input of the operator is a single function. Here, for the first time, we study the operator regression via neural networks for multiple-input operators defined on the product of Banach spaces. We first prove a universal approximation theorem of continuous multiple-input operators. We also provide a detailed theoretical analysis including the approximation error, which provides guidance for the design of the network architecture. Based on our theory and a low-rank approximation, we propose a novel neural operator, MIONet, to learn multiple-input operators. MIONet consists of several branch nets for encoding the input functions and a trunk net for encoding the domain of the output function. We demonstrate that MIONet can learn solution operators involving systems governed by ordinary and partial differential equations. In our computational examples, we also show that we can endow MIONet with prior knowledge of the underlying system, such as linearity and periodicity, to further improve accuracy.

Dominated multilinear operators defined on tensor products of Banach spaces

Dominated multilinear operators defined on tensor products of Banach spaces

The dual of the space of bounded operators on a Banach space

Abstract Given Banach spaces X and Y, we ask about the dual space of the 𝒧(X, Y). This paper surveys results on tensor products of Banach spaces with the main objective of describing the dual of spaces of bounded operators. In several cases and under a variety of assumptions on X and Y, the answer can best be given as the projective tensor product of X ** and Y *.

Positive periodic solutions to an indefinite Minkowski-curvature equation

We investigate the existence, non-existence, multiplicity of positive periodic solutions, both harmonic (i.e., T-periodic) and subharmonic (i.e., kT-periodic for some integer k≥2) to the equation(u′1−(u′)2)′+λa(t)g(u)=0, where λ>0 is a parameter, a(t) is a T-periodic sign-changing weight function and g:[0,+∞[→[0,+∞[ is a continuous function having superlinear growth at zero. In particular, we prove that for both g(u)=up, with p>1, and g(u)=up/(1+up−q), with 0≤q≤1<p, the equation has no positive T-periodic solutions for λ close to zero and two positive T-periodic solutions (a “small” one and a “large” one) for λ large enough. Moreover, in both cases the “small” T-periodic solution is surrounded by a family of positive subharmonic solutions with arbitrarily large minimal period. The proof of the existence of T-periodic solutions relies on a recent extension of Mawhin's coincidence degree theory for locally compact operators in product of Banach spaces, while subharmonic solutions are found by an application of the Poincaré–Birkhoff fixed point theorem, after a careful asymptotic analysis of the T-periodic solutions for λ→+∞.

Factorization of multilinear operators defined on products of function spaces

This paper deals with multilinear operators acting in products of Banach spaces that factor through a canonical mapping. We prove some factorization theorems and characterizations by means of norm inequalities for multilinear operators defined on the n-fold Cartesian product of the space of bounded Borel measurable functions, respectively, products of Banach function spaces. These factorizations allow us to obtain integral dominations and lattice geometric properties and to present integral representations for abstract classes of multilinear operators. Finally, bringing together these ideas, some applications are shown, regarding for example summability properties and representation of multilinear maps as orthogonally additive n-homogeneous polynomials.

Perturbation theory, M-essential spectra of operator matrices

In this paper, we will establish some results on perturbation theory of block operator matrices acting on Xn, where X is a Banach space. These results are exploited to investigate the M-essential spectra of a general class of operators defined by a 3x3 block operator matrix acting on a product of Banach spaces X3.

Stability of essential B-spectra of unbounded linear operators and applications

In this paper, we study the stability of some essential B-spectra of closed linear operators on a Banach space X, under polynomially finite rank operators and we give the characterization of some essential B-spectra of a $$2\times 2$$ of unbounded matrix operator acting in the product of Banach spaces $$X\times Y$$ . Then, using the functional calculus, we prove that a spectral mapping type theorem holds for these essential B-spectra. As an application, we study the effect of the functional calculus on the class of meromorphic operators, and on the class of isoloid operators with sable sign index, satisfying generalized Weyl theorem.

System of multivariate pseudo-contractive operator equations and the existence of solutions

The purpose of this paper is to study the following system of nonlinear operator equations for N-variable pseudo-contractive mappings $$T_1, T_2,\ldots ,T_N$$ in a Banach space: $$\begin{aligned} {\left\{ \begin{array}{ll} T_1(x_{1},x_{2},\ldots ,x_{N})=x_1,\\ T_2(x_{1},x_{2},\ldots ,x_{N})=x_2,\\ T_3(x_{1},x_{2},\ldots ,x_{N})=x_3,\\ \ \ \ \ \ \ \ \cdots \\ T_N(x_{1},x_{2},\ldots ,x_{N})=x_N.\\ \end{array}\right. } \end{aligned}$$ The existence theorems of solutions are proved using a comprehensive analytical method. To get the expected results, the reflexivity of the product of Banach space and Opial’s condition of the product of Banach spaces are discussed. Meanwhile, the weak topology of the product of Banach spaces and normal structure of the product of Banach spaces are also discussed. The results of this article improve and extend the previous classical results given in the literature.

Coincidence Results for Summing Multilinear Mappings

AbstractIn this paper we prove coincidence results concerning spaces of absolutely summing multilinear mappings between Banach spaces. The nature of these results arises from two distinct approaches: the coincidence of two a priori different classes of summing multilinear mappings, and the summability of all multilinear mappings defined on products of Banach spaces. Optimal generalizations of known results are obtained. We also introduce and explore new techniques in the field: for example, a technique to extend coincidence results for linear, bilinear and even trilinear mappings to general multilinear ones.

Multiresolution analysis for compactly supported interpolating tensor product wavelets

We construct multidimensional interpolating tensor product multiresolution analyses (MRA's) of the function spaces C0(ℝn, K), K = ℝ or K = ℂ, consisting of real or complex valued functions on ℝn vanishing at infinity and the function spaces Cu(ℝn, K) consisting of bounded and uniformly continuous functions on ℝn. We also construct an interpolating dual MRA for both of these spaces. The theory of the tensor products of Banach spaces is used. We generalize the Besov space norm equivalence from the one-dimensional case to our n-dimensional construction.

Note on Arens regularity of symmetric tensor products

We investigate symmetric regularity of sums of symmetric tensor products of Banach spaces and Arens regularity of symmetric tensor products of Banach algebras. An example for the Hilbert space is obtained.

CONVEX-TRANSITIVITY OF BANACH ALGEBRAS VIA IDEALS

We investigate a method for producing concrete convex-transitive Banach spaces. The gist of the method is in getting rid of dissymmetries of a given space by taking a carefully chosen quotient. The spaces of interest here are typically Banach algebras and their ideals. We also investigate the convex-transitivity of ultraproducts and tensor products of Banach spaces.

Shifts on products of banach spaces

It is well-known in the literature that the product of two Banach spaces does not necessarily admit a shift even with the case where each factor has a shift. The purpose of this paper is to explicitly construct a complex shift on the product of Banach spaces, c and c 0. Furthermore, we show that the product spaces c × ℓ p and c 0 × ℓ p , 0 ≤ p ≤ ∞ admit a complex shift.

On some subsets of Schechter's essential spectrum of a matrix operator and application to transport operator

This paper is devoted to the investigation of the essential approximate point spectrum and the essential defect spectrum of a 2 × 2 block operator matrix on a product of Banach spaces. The obtained results are applied to a two-group transport operators with general boundary conditions in the Banach space Lp ([–a, a ] × [–1, 1]) × Lp ([–a, a ] × [–1, 1]), a > 0, p ≥ 1 (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Into isomorphisms in tensor products of Banach spaces

We establish quantitative extensions of two Grothendieck's results on into isomorphisms in projective tensor products. Among others, we prove the following. Let Y be a closed subspace of a Banach space Z and let j : Y → Z denote the identity embedding. If Y is complemented in its bidual Y**, then the injection modulus of the natural inclusion Id ⊗ j : Y*⊗Y → Y*⊗Z satisfies 1/λ loc (Y,Z) ≤ i(Id ⊗ j) ≤ λ(Y,Y**)/λ(Y,Z), where λ(·,·) and λloc(·,·) are, respectively, the projection and the local projection constants.

Projections on tensor products of Banach spaces

We characterize norm hermitian operators on classes of tensor products of Banach spaces and derive results for the particular settings of injective and projective tensor products. We provide a characterization of bi-circular and generalized bi-circular projections on tensor products of Banach spaces supporting only dyadic surjective isometries.

Integrability of subdifferentials of certain bivariate functions

We study integrability properties of bivariate functions defined on the product of Banach spaces.

Some type of rotational scheme

The aim of this paper is to construct a rotational product scheme in the product of Banach spaces with rotational schemes. Besides we give an equivalent definition of the generalized Kolmogorov diameters suggested by Aksoy and Nakamura in 1986, and give the generalized Kolmogorov diameters of the product (∏ i∈n Di) of bounded subsets Di in the product of Banach spaces having schemes in terms of the generalized diameters of Di.

Polynomial properties and symmetric tensor product of Banach spaces

We study some classes of distinguished subsets of a Banach space in terms of polynomials and their relationship. This allows us to develop a systematic approach to study polynomial properties on a Banach space. We apply this approach to obtain several known and new results on the symmetric tensor product of a Banach space in a unified way.