Banach Space Lp Research Articles

Analytic Invariants of Semidirect Products of Symmetric Groups on Banach Spaces

We consider algebras of polynomials and analytic functions that are invariant with respect to semidirect products of groups of bounded operators on Banach spaces with symmetric bases. In particular, we consider algebras of so-called block-symmetric and double-symmetric analytic functions on Banach spaces ℓp(Cn) and the homomorphisms of these algebras. In addition, we describe an algebraic basis in the algebra of double-symmetric polynomials and discuss a structure of the spectrum of the algebra of double-symmetric analytic functions on ℓp(Cn).

Stability results for backward heat equations with time-dependent coefficient in the Banach space [formula omitted

In this paper, we investigate the problem of backward heat equations with time-dependent coefficient in the Banach space Lp(R), (1<p<∞). For this problem, we first prove the stability estimates of Hölder type. After that the Tikhonov-type regularization is applied to solve the problem. A priori and a posteriori parameter choice rules are investigated, which yield error estimates of Hölder type. Numerical implementations are presented to show the validity of the proposed scheme.

Stability of some essential B-spectra of pencil operators and application

In this paper, we give some results on the essential B-spectra of a linear operator pencil, which are used to determine the essential B-spectra of an integro-differential operator with abstract boundary conditions in the Banach space Lp([−a, a] × [−1, 1]), p ≥ 1 and a &gt; 0.

Supermixing and hypermixing operators

As stronger versions of the mixing property, supermixing and hypermixing notions are introduced and investigated. We show that most of the newly known strongly hypercyclic operators satisfy the stronger condition of being hypermixing. For Banach space operators, we see that some suitable scalar multiples of any non-invertible strongly hypercyclic operator are hypermixing. We prove that on Banach spaces ℓp and c0, a weighted backward shift is supermixing if and only if it is mixing, but, a supermixing weighted backward shift is not necessarily hypermixing.

Про моделювання гауссового процесу із точністю та надійністю в просторі $L_p([0,T])$

Стаття присвячена моделюванню випадкового процесу із наперд заданоюВ останнi часи теорiя стохастичних процесiв та полiв широко використовується в рiзних галузях науки i не тiльки в природничих сферах, а саме її використання є важливим у фiзицi, радiофiзицi, iнформатицi, програмнiй iнженерiї, соцiологiї, бiологiї, океаналогiї, метеорологiї, фiнансовiй математицi, теорiї прийняття рiшень, системах масового обслуговування тощо. Тому актуальною проблемою для ймовiрносникiв є побудова математичної моделi випадкового процесу або поля та вивчення її аналiтичних властивостей. Проблеми чисельного моделювання стають особливо важливими завдяки потужним можливостям комп’ютерних технологiй, що дозволяють створювати програмнi засоби для моделювання та для передбачення поведiнки випадкового процесу. Пiд статистичним моделюванням ми розумiємо комп’ютерну реалiзацiю спочатку випадкової величини, а потiм вже випадкового процесу або поля при заданих характеристиках даних об’єктiв моделювання. Стаття присвячена моделюванню випадкового процесу iз наперед заданою точнiстю та надiйнiстю в банаховому просторi Lp([0, T]). Припускається, що випадковий процес є стацiонарним гауссовим iз вiдомою скiнченною коварiацiйною функцiєю. Якщо випадковий процес подано як збiжний у середньому квадратичному ряд iз випадковими доданками, то, зазвичай, у якостi моделi можна розглядати скiнченну суми перших доданкiв, тобто зрiзку ряду. Тому, перша проблема, яка виникає у статтi, як розкласти випадковий процес у ряд при вiдомiй коварiацiйнiй функцiї. Для цього у статтi використовується Теорема Карунена-Лоєва i для побудови моделi застосовуємо розклад Карунена-Лоєва випадкового процесу. У данiй роботi особливу увагу придiлено точностi та надiйностi побудованої моделi. Це означає, що спочатку ми будуємо модель, а потiм її перевiряємо за допомогою певних тестiв на адекватнiсть iз заданими вхiдними параметрами. Отже, знаючи наперед точнiсть та надiйнiсть та з використанням доведених у статтi результатiв для перевiрки адекватностi, можна стверджувати, що побудова модель буде гарно описувати початковий випадковий процес.

Analogs of the Lebesgue Measure in Spaces of Sequences and Classes of Functions Integrable with Respect to These Measures

We examine translation-invariant measures on Banach spaces lp, where p ∈ [1,∞]. We construct analogs of the Lebesgue measure on Borel σ-algebras generated by the topology of pointwise convergence (σ-additive, invariant under shifts by arbitrary vectors, regular measures). We show that these measures are not σ-finite. We also study spaces of functions integrable with respect to measures constructed and prove that these spaces are not separable. We consider various dense subspaces in spaces of functions that are integrable with respect to a translation-invariant measure. We specify spaces of continuous functions, which are dense in the functional spaces considered. We discuss Borel σ-algebras corresponding to various topologies in the spaces lp, where p ∈ [1,∞]. For p ∈ [1,∞), we prove the coincidence of Borel σ-algebras corresponding to certain natural topologies in the given spaces of sequences and the Borel σ-algebra corresponding to the topology of pointwise convergence. We also verify that the space l∞ does not possess similar properties.

Local Uniform Kadec-Klee Property (LUKK) and Modulus of (LUKK)

A new geometry property and two new moduli are introduced in Banach space. First, the concept of local uniform Kadec-Klee property (LUKK) is introduced and the implication relationships between LUKK and local near uniform convexity LNUC, uniformly Kadec-Klee (UKK), (H) are investigated in Banach space. Furthermore, the modulus PXLε of (LUKK) and the modulus ΔXLε of LNUC are introduced and the relationship of size between PXLε and ΔXLε is also investigated in Banach space. Finally, several formulas for PXLε are calculated in classical Banach space lp.

A new regular infinite matrix defined by Jordan totient function and its matrix domain in ℓp

In this paper, we first define a new regular matrix by using the arithmetic function called Jordan totient function and study the matrix domain of this newly introduced matrix in the Banach space ℓp. After computing the dual spaces of this new space, we characterize certain matrix mappings related to this space.

Solvability for infinite systems of fractional differential equations in Banach sequence spaces lp and c0

This paper is devoted to an infinite system of nonlinear fractional differential equations in the Banach spaces c0 and lp with p ? 1. Existence results are obtained, by using the theory of measure of noncompactness and a new generalization of Darbo?s fixed point theorem. Some examples are also included to show the efficiency of our results.

Estimate for the Entropy Numbers of the Weighted Hardy Operators that Act from Banach Space to q-Banach Space

We study weighted Hardy operators from the Banach space Lp to the q-Banach space Lq and obtain estimates for the entropy numbers of one- and two-weighted Hardy operators with nonnegative measurable weight functions.

A generalization of Cesàro sequence spaces in the Orlicz space

In this paper, we generalize the Cesàro sequence spaces in the classic Banach space Lp to the generalized Orlicz space Lφ. We construct the space by replacing the norm in Lp with modular ρφ in Lφ. This generalization has lead to the use of the Luxemburg norm to discuss some topological properties of the spaces. We prove results regarding to modular and norm convergence. We also describe some properties of the spaces and a closed subspaces of the space.

The β-dual of the Cesàro sequence spaces defined on a generalized Orlicz space

In this paper we characterized the β-dual of the Cesàro sequence space with terms in a generalized Orlicz space. Further, we find that the dual is a generalization of the dual of the Cesàro space in the classical Banach space Lp for1 < p < ∞.

Weak supermajorization and families as doubly superstochastic operators on ℓp(I)

We present necessary and sufficient conditions that a family A={aij:i,j∈I} of real numbers may be considered as a bounded linear operator on Banach spaces ℓ1(I) and ℓ∞(I), where I is an arbitrary non-empty set. Moreover, we get that these conditions are sufficient for a family to be a bounded linear operator on ℓp(I), for each p∈[1,∞]. Within this class of operators, the notion of doubly superstochastic operator is introduced as an extension of the doubly superstochastic matrix, and some of its essentially properties are presented. In the second part, we extend the notion of weak supermajorization relation on the Banach space ℓp(I) using doubly superstochastic operators, and present close relationship between this relation and superstochastic operators as generalisation well-known results in the theory of majorization. Among others, for two functions f,g∈ℓ1(I)+ we show that relations f≺wsg and g≺wsf hold if and only if there exist a permutation P such that g=Pf.

On certain uniformly open multilinear mappings

We obtain two results stating the uniform openness of bilinear operators and multilinear functionals. The first result deals with Banach spaces Lp:=LKp (over K∈{R,C}) and pointwise multiplication from Lp×Lq to Lr (where 1/p+1/q=1/r). The second result is concerned with the nontrivial n-linear functionals from the product X1×⋯×Xn of normed spaces (over K∈{R,C}) to the field K.

On strongly separately continuous functions on sequence spaces

We study strongly separately continuous real-valued function defined on the Banach spaces ℓp. Determining sets for the class of strongly separately continuous functions on ℓp are characterized. We prove that for every 2≤α<ω1 there exists a strongly separately continuous function which belongs to the α'th Baire class and does not belong to the β'th Baire class on ℓp for β<α. We show that any open set in ℓp is the set of discontinuities of a strongly separately continuous real-valued function.

Isomorphic classification of Lp,q-spaces

We prove that for all 1≤p,q<∞, p≠q, p≠1, p≠2 and p=2, q>2 the Banach space lp,q does not isomorphically embed into the space Lp,q(0,1). We also prove under the same conditions that the space Lp,q(0,∞) does not isomorphically embed into the space Lp,q(0,1)⊕lp,q.

Weak majorization and doubly substochastic operators on [formula omitted

We extend the notion of weak majorization on the Banach space ℓp(I), where I is a non-empty set and p∈[1,∞), using doubly substochastic operators on ℓp(I). The close relationship between weak majorization and these operators are presented. Among others, we show that f≺wg and g≺wf, iff there is a partial permutation P such that g=Pf. Using the specific type of positive increasing convex functions, a necessary and sufficient condition for f≺wg and g≺wf is given.

Extended eigenvalues for Cesàro operators

A complex scalar λ is said to be an extended eigenvalue of a bounded linear operator T on a complex Banach space if there is a non-zero operator X such that TX=λXT. Such an operator X is called an extended eigenoperator of T corresponding to the extended eigenvalue λ. The purpose of this paper is to give a description of the extended eigenvalues for the discrete Cesàro operator C0, the finite continuous Cesàro operator C1 and the infinite continuous Cesàro operator C∞ defined on the complex Banach spaces ℓp, Lp[0,1] and Lp[0,∞) for 1<p<∞ by the expressions(C0f)(n):=1n+1∑k=0nf(k),(C1f)(x):=1x∫0xf(t)dt,(C∞f)(x):=1x∫0xf(t)dt. It is shown that the set of extended eigenvalues for C0 is the interval [1,∞), for C1 it is the interval (0,1], and for C∞ it reduces to the singleton {1}.

Spectrum localizations for matrix operators on [formula omitted] spaces

We investigate localization of spectra of infinite matrices that can act as linear operators from the Banach space lp,1≤p≤∞, to itself. To that end, we propose a method that is based on generalization of a strict diagonal dominance of infinite matrices that is adapted to lp and lq norms. Results are illustrated with several examples, and some interesting applications are discussed.

Extension of isometries between the unit spheres of complex lp(Γ)(p>1)spaces

In this paper, we study the extension of isometries between the unit spheres of complex Banach spaces lp(Γ) and lp(Δ)(p > 1). We first derive the representation of isometries between the unit spheres of complex Banach spaces lp(Γ) and lp(Δ). Then we arrive at a conclusion that any surjective isometry between the unit spheres of complex Banach spaces lp(Γ)and lp(Δ) can be extended to be a linear isometry on the whole space.