Symmetric Banach Space Research Articles

A Green’s Function Based Iterative Approach for Solutions of BVPs in Symmetric Spaces

We consider the Banach space C[0,1], which is a symmetric Banach space, and prove the existence and approximation of numerical solutions for a broad class of third-order BVPs. Our approach is based on an integral operator that is constructed using Green’s function. The Banach contraction principle (BCP) is applied to guarantee a unique solution to our problem. Moreover, in order to find the value of the numerical solution, this new operator is embedded within the three-step Noor iterative scheme; we named this new iterative scheme the Noor–Green iterative scheme. We provide a convergence theorem for the proposed scheme by employing suitable restrictions on the parameters involved in the problem and in the scheme. The results of the stability of our scheme are also reported. It is worth mentioning that unlike the concept of stability in the classical sense, our result for stability is based on the concept of weak w2 stability. In order to support our findings, we carried out various numerical experiments using different third-order BVPs. Finally, we report on the application of our iterative scheme to solve a class of fractional BVPs in the same symmetric Banach space. Our results are essentially new in the present literature and extend several of the results found in the current literature.

2-co-lacunary sequences in noncommutative symmetric Banach spaces

We characterize noncommutative symmetric Banach spaces for which every bounded sequence admits either a convergent subsequence, or a 2 2 -co-lacunary subsequence. This extends the classical characterization, due to Räbiger.

An extension of Feller’s strong law of large numbers

This paper presents a general result that allows for establishing a link between the Kolmogorov–Marcinkiewicz– Zygmund strong law of large numbers and Feller’s strong law of large numbers in a Banach space setting. Let {X,Xn;n≥1} be a sequence of independent and identically distributed Banach space valued random variables and set Sn=∑i=1nXi,n≥1. Let {an;n≥1} and {bn;n≥1} be increasing sequences of positive real numbers such that limn→∞an=∞ and bn∕an;n≥1 is a nondecreasing sequence. We show that Sn−nEXI{‖X‖≤bn}bn→0almost surelyfor every Banach space valued random variable X with ∑n=1∞P(‖X‖>bn)<∞ if Sn∕an→0 almost surely for every symmetric Banach space valued random variable X with ∑n=1∞P(‖X‖>an)<∞. To establish this result, we invoke two tools (obtained recently by Li, Liang, and Rosalsky): a symmetrization procedure for the strong law of large numbers and a probability inequality for sums of independent Banach space valued random variables.

On p-Convexity and q-Concavity in Non-Commutative Symmetric Spaces

It is shown that if a symmetric Banach space E on the positive semi-axis is p-convex (q-concave) then so is the corresponding non-commutative symmetric space E(τ) of τ-measurable operators affiliated with some semifinite von Neumann algebra \({({\mathcal{M}}, \tau)}\) , with preservation of the convexity (concavity) constants in the case that \({{\mathcal{M}}}\) is non-atomic. Similar statements hold in the case that E satisfies an upper (lower) p-estimate and extend to the more general semifinite setting earlier results due to Arazy and Lin for unitary matrix spaces.

Self-normalized Wittmann's laws of iterated logarithm in Banach space

For a sequence of independent symmetric Banach space valued random variables { X n , n ⩾ 1 } , we obtain the self-normalized Wittmann's law of iterated logarithm (LIL) and give the upper bound for the non-random constant.

On the self-normalized bounded laws of iterated logarithm in Banach space

For a sequence of independent symmetric Banach space valued random variables { X n , n⩾1}, we obtain the self-normalized law of iterated logarithm and give the upper bound for the non-random constant.

An improvement of Hoffmann-Jørgensen’s inequality

Let B be a Banach space and F any family of bounded linear functionals on B of norm at most one. For x ∈ B set || x || = supΛ∈F Λ (x) (||· || is at least a seminorm on B). We give probability estimates for the tail probability of S* n = max1≤ k≤ n ||Σk j=1 Xj || where {Xi}n i=1 are independent symmetric Banach space valued random elements. Our method is based on approximating the probability that S* n exceeds a threshold defined in terms of Σk j=1 Y(j), where Y(r) denotes the rth largest term of {|| Xi ||}n i=1. Using these tail estimates, essentially all the known results concerning the order of magnitude or finiteness of quantities such as EΦ(|| Sn ||) and EΦ(S* n) follow (for any fixed 1 ≤ n ≤ ∞). Included in this paper are uniform Lp bounds of S* n which are within a factor of 4 for all p ≥ 1 and within a factor of 2 in the limit as p → ∞.

Asymptotic Estimates for Approximation Quantities of Tensor Product Identities

LetEnandFmbe two symmetric Banach spaces, andαa reasonable norm on their tensor product. We give asymptotically best possible estimates for the approximation and Gelfand numbers of the natural embedding from thenm-dimensional Hilbert spacelnm2intoEn⊗αFm, and its inverse. Our results are used in order to compute some related characteristics of such tensor products (e.g., type and cotype constants).

SOME ASPECTS OF THE THEORY OF SYMMETRIC OPERATOR SPACES

Abstract We discuss several aspects of the theory of symmetric Banach spaces of measurable operators, including their construction and certain topological and geometric properties. Particular emphasis is given to the role played by rearrangement inequalities.

Noncommutative Köthe duality

Using techniques drawn from the classical theory of rearrangement invariant Banach function spaces we develop a duality theory in the sense of Köthe for symmetric Banach spaces of measurable operators affiliated with a semifinite von Neumann algebra equipped with a distinguished trace. A principal result of the paper is the identification of the Köthe dual of a given Banach space of measurable operators in terms of normality.

On Rosenthal-Shields' problem

In this paper it will be shown that the spectrum of every unicellular unilateral weighted shift operator on a symmetric Banach space is the singleton set {0}. From this, we give an affirmative answer to Rosenthal-Shields' problem.

Weakly compact subsets of symmetric operator spaces

AbstractUnder natural conditions it is shown that the rearrangement invariant hull of a weakly compact subset of a properly symmetric Banach space of measurable operators affiliated with a semi-finite von Neumann algebra is again relatively weakly compact.

Entropy numbers of diagonal operators between symmetric Banach spaces

We give the exact order of the dyadic entropy numbers of the identities from l n p to l n r where p < r. Weaker estimates can be found in [3, 4]. The crucial lemma is a combinatorial result from [5]. Then we consider (dyadic) entropy numbers of identities between finite-dimensional symmetric Banach spaces. We obtain a simple expression that gives the exact order up to some logarithmic factor. This allows us to generalize a theorem due to B. Carl [1] about diagonal operators. It turns out that the result still holds under much weaker assumptions on the spaces. More precisely, the assumptions are not so much concerned with the spaces themselves but (what seems to be intuitively clear) with the relation between the spaces.

On the Banach-Mazur distance of finite-dimensional symmetric Banach spaces and the hypergeometric distribution

On the Banach-Mazur distance of finite-dimensional symmetric Banach spaces and the hypergeometric distribution

Isometries of diagonally symmetric Banach spaces

We describe all compact symmetric subgroups of the orthogonal groupO n which contain the permutation groupS n. There are seven such groups, each one can be realised as the group of all isometries on somen-dimensional Banach space.