P-summable Sequences Research Articles

Representation of sequence classes by operator ideals

It is well known that weakly p-summable sequences in a Banach space E are associated to bounded operators from ℓp⁎ to E, and unconditionally p-summable sequences in E are associated to compact operators from ℓp⁎ to E. Generalizing these results to a quite wide environment, we characterize the classes of Banach spaces-valued sequences that are associated to (or represented by) some Banach operator ideal. Using these characterizations, we decide, among all sequence classes that usually appear in the literature, which are represented by some Banach operator ideal and which are not. Moreover, to each class that is represented by some Banach operator ideal, we show an ideal that represents it. Illustrative examples and additional applications are provided.

A New Semi-Inner Product and pn-Angle in the Space of p-Summable Sequences

In this paper, we propose a definition for a semi-inner product in the space of p-summable sequences equipped with an n-norm. Using this definition, we introduce the concepts of pn-orthogonality and the pn-angle between two vectors in the space of p-summable sequences . For the special case n = 1, these concepts are identical to previous studies. We also introduce the notion of the pn-angle between one-dimensional subspaces and arbitrary-dimensional subspaces. The authors believe that the results obtained in this paper are very significant, especially in the theory of n-normed space in functional analysis.

A Class of Sets in a Banach Space Coarser Than Limited Sets

A wide new class of subsets of a Banach space X named coarse p-limited sets (\( 1\le p < \infty \)) is introduced by considering weak* p-summable sequences in \(X'\) instead of weak* null sequences. We study its basic properties and compare it with the class of compact and weakly compact sets. Results concerning the relationship of coarse p-limited sets with operators are obtained.

Measure of noncompactness for an infinite system of fractional Langevin equation in a sequence space

A new sequence space related to the space ell _{p}, 1leq p<infty (the space of all absolutely p-summable sequences) is established in the present paper. It turns out that it is Banach and a BK space with Schauder basis. The Hausdorff measure of noncompactness of this space is presented and proven. This formula with the aid the Darbo’s fixed point theorem is used to investigate the existence results for an infinite system of Langevin equations involving generalized derivative of two distinct fractional orders with three-point boundary condition.

A study on Copson operator and its associated sequence space II

In this paper, we investigate some properties of the domains c(C^{n}), c_{0}(C^{n}), and ell _{p}(C^{n})(0< p<1) of the Copson matrix of order n, where c, c_{0}, and ell _{p} are the spaces of all convergent, convergent to zero, and p-summable real sequences, respectively. Moreover, we compute the Köthe duals of these spaces and the lower bound of well-known operators on these sequence spaces. The domain ell _{p}(C^{n}) of Copson matrix C^{n} of order n in the sequence space ell _{p}, the norm of operators on this space, and the norm of Copson operator on several matrix domains have been investigated recently in (Roopaei in J. Inequal. Appl. 2020:120, 2020), and the present study is a complement of our previous research.

Product factorable multilinear operators defined on sequence spaces

We prove a factorization theorem for multilinear operators acting in topological products of spaces of (scalar) p-summable sequences through a product. It is shown that this class of multilinear operators called product factorable maps coincides with the well-known class of the zero product preserving operators. Due to the factorization, we obtain compactness and summability properties by using classical functional analysis tools. Besides, we give some isomorphisms between spaces of linear and multilinear operators, and representations of some classes of multilinear maps as n-homogeneous orthogonally additive polynomials.

EXTREME AND NICE OPERATORS ON CERTAIN FUNCTION SPACES

Let X be a Banach space, and L(X) be the space of bounded linear operators from X into X. B1(X) denotes the closed unit ball of X, and S1(X )i s unit sphere of X. An element T ∈ S1(L(X)) is called extreme operator if there is no A ∈ L(X) such thatT ± A �≤ 1. The set of extreme points of S1(X) will be denoted by ext(S1(X)) .T ∈ S1(L(X)) is called nice if T ∗ (ext(S1(L(X ∗ ))) ⊆ ext(S1(L(X ∗ ))). The object of this paper is to give simpler proofs of old results and present new results on extreme operators of S1(L(� p )). We introduce the concept of k-extreme points. Further, we characterize the nice operators on most of the classical function and sequence spaces. Nice compact operators onp −spaces are characterized. I. Introduction. Let X be a Banach space. The closed unit ball of X will be denoted by B1(X) , and the unit sphere by S1(X). An element x ∈ S1(X) is called an extreme point of B1(X) if whenever x = 1 (y + z), with y and z in S1(X), then x = y = z. The space of bounded linear operators on X will be denoted by L(X), and the compact ones by K(X). Extreme elements of S1(L(X)) are called extreme operators. An operator T ∈ L(X )i s called nice if the set of extreme points of B1(X ∗ ) is an invariant set for T ∗ . L p (I), 1 ≤ p< ∞, denotes the Banach space of p-Bochner integrable functions (equiv- alence classes) defined on the unit interval, with the usual classical norm. Similarly, � p denotes the Banach space of p-summable sequences, with the usual classical norm. The space of continuous functions on a compact set Ω, with the uniform norm is denoted by C(Ω).

Generalized Statistical Convergence of Double Sequences in Paranormed Spaces

We introduce the notion of (λ, µ)-statistical convergence of double sequences in a setting of paranormed space and prove that every convergent sequence is (λ, µ)-statistically convergent but not conversely by supporting an illustrative example. We also define the notions of (λ, µ)-statistical Cauchy and strongly (λ, µ)p-summable double sequences in a paranormed space and obtain their relationship with (λ, µ)- statistical convergence.

Polynomial control on stability, inversion and powers of matrices on simple graphs

Spatially distributed networks of large size arise in a variety of science and engineering problems, such as wireless sensor networks and smart power grids. Most of their features can be described by properties of their state-space matrices whose entries have indices in the vertex set of a graph. In this paper, we introduce novel algebras of Beurling type that contain matrices on a connected simple graph having polynomial off-diagonal decay, and we show that they are Banach subalgebras of B(ℓp),1≤p≤∞, the space of all bounded operators on the space ℓp of all p-summable sequences. The ℓp-stability of state-space matrices is an essential hypothesis for the robustness of spatially distributed networks. In this paper, we establish the equivalence among ℓp-stabilities of matrices in Beurling algebras for different exponents 1≤p≤∞, with quantitative analysis for the lower stability bounds. Admission of norm-control inversion plays a crucial role in some engineering practice. In this paper, we prove that matrices in Beurling subalgebras of B(ℓ2) have norm-controlled inversion and we find a norm-controlled polynomial with close to optimal degree. Polynomial estimate to powers of matrices is important for numerical implementation of spatially distributed networks. In this paper, we apply our results on norm-controlled inversion to obtain a polynomial estimate to powers of matrices in Beurling algebras. The polynomial estimate is a noncommutative extension about convolution powers of a complex function and is applicable to estimate the probability of hopping from one agent to another agent in a stationary Markov chain on a spatially distributed network.

Matrix Transformations on Köthe Spaces

Let $$\mu $$ denotes any of the classical spaces $$c_0, c$$ or $$\ell _{p}$$ of null, convergent or absolutely p-summable sequences and $$1\le p\le \infty $$ . In this paper, we characterize the classes $$(\lambda (P)\,{:}\,\mu )$$ of matrix transformations.

Domain of Euler Mean in the Space of Absolutely $p$-Summable Double Sequences with $0&lt; p&lt;1$

Domain of Euler Mean in the Space of Absolutely $p$-Summable Double Sequences with $0&lt; p&lt;1$

A NOTE ON THE SOME GEOMETRIC PROPERTIES OF THE SEQUENCE SPACES DEFINED BY TAYLOR METHOD

In this paper, it was obtained the new matrix domain with the well known classical sequence spaces and an infinite matrix. The Taylor method which known then as the circle method of order r (0 < r < 1), as an infinite matrix for the matrix domain is used. Newly constructed space is isomorphic copy of the spaces of all absolutely p-summable sequences. It is well known that Hilbert space have the nicest geometric properties. Then, it is proved that the new space is a Hilbert space for p = 2. Further, it was computed dual spaces and characterized some matrix classes of the new Taylor space in the table form. Section 3 is devoted some geometric properties of Taylor space.

CLASSES OF SEQUENTIALLY LIMITED OPERATORS

AbstractThe purpose of this paper is to present a brief discussion of both the normed space of operator p-summable sequences in a Banach space and the normed space of sequentially p-limited operators. The focus is on proving that the vector space of all operator p-summable sequences in a Banach space is a Banach space itself and that the class of sequentially p-limited operators is a Banach operator ideal with respect to a suitable ideal norm- and to discuss some other properties and multiplication results of related classes of operators. These results are shown to fit into a general discussion of operator [Y,p]-summable sequences and relevant operator ideals.

Two equivalent n-norms on the space of p-summable sequences

We prove the (strong) equivalence between two known n-norms on the space lp of p-summable sequences (of real numbers). The first n-norm is derived from Gahler’s formula [3], while the second is due to Gunawan [7]. The equivalence is proved by using the properties of the volume of n-dimensional parallelepipeds in lp.

Domain of the double sequential band matrix in the classical sequence spaces

Let λ denote any one of the classical spaces , c, and of bounded, convergent, null and absolutely p-summable sequences, respectively, and also be the domain of the double sequential band matrix in the sequence space λ, where and are given convergent sequences of positive real numbers and . The present paper is devoted to studying the sequence space . Furthermore, the β- and γ-duals of the space are determined, the Schauder bases for the spaces , and are given, and some topological properties of the spaces , and are examined. Finally, the classes and of infinite matrices are characterized, where and . MSC:46A45, 40C05.

On some new subfamilies of classical spaces of absolutely p-summable sequences

ABSTRACT In this paper the properties of some new subfamilies of the spaces lp(ℂ) are discussed

ASCENT AND DESCENT OF COMPOSITION OPERATORS ON lp SPACES

Let lp, (1 ≤ p ≤ ∞) be the Banach space of all p-summable sequences (bounded sequences for p = ∞) of complex numbers under the standard p-norm on it and Cφ be a composition operator on l p induced by a function φ on N into itself. In this paper we give a charcterization of composition operators whose ascent and descent are infinite.

Stability of localized operators

Let ℓ p , 1 ⩽ p ⩽ ∞ , be the space of all p-summable sequences and C a be the convolution operator associated with a summable sequence a. It is known that the ℓ p -stability of the convolution operator C a for different 1 ⩽ p ⩽ ∞ are equivalent to each other, i.e., if C a has ℓ p -stability for some 1 ⩽ p ⩽ ∞ then C a has ℓ q -stability for all 1 ⩽ q ⩽ ∞ . In the study of spline approximation, wavelet analysis, time–frequency analysis, and sampling, there are many localized operators of non-convolution type whose stability is one of the basic assumptions. In this paper, we consider the stability of those localized operators including infinite matrices in the Sjöstrand class, synthesis operators with generating functions enveloped by shifts of a function in the Wiener amalgam space, and integral operators with kernels having certain regularity and decay at infinity. We show that the ℓ p -stability (or L p -stability) of those three classes of localized operators are equivalent to each other, and we also prove that the left inverse of those localized operators are well localized.

A Description of the Space of Banach Space-valued Absolutely p-summable Sequences

Conradie and Swart described the sequence space l p strong(X) of Banach space-valued absolutely p-summable sequences as the Banach space of cone absolutely summing operators (1-concave operators) from c 0 to X for p = 1, or l q to X for 1 < p < ∞, where 1/p + 1/q = 1. We give an elementary proof for their result.

Banach space sequences and projective tensor products

In this paper we use results from the theory of tensor products of Banach spaces to establish the isometry of the space of (1, p)-summing sequences (also known as strongly p-summable sequences) in a Banach space X, the space of nuclear X-valued operators on ℓ p and the complete projective tensor product of ℓ p with X. Through similar techniques from the theory of tensor products, the isometry of the sequence space L p 〈 X〉 (recently introduced in a paper by Bu, Quaestiones Math. (2002), to appear), the space of nuclear X-valued operators on L p (0,1) (with a suitable equivalent norm) and the complete projective tensor product of L p (0,1) with X is established. Moreover, we find conditions for the space of ( p, q)-summing multipliers to have the GAK-property (generalized AK-property), use multiplier sequences to characterize Banach space valued bounded linear operators on the vector sequence space of absolutely p-summable sequences in a Banach space and present short proofs for results on p-summing multipliers.