The James–Schreier spaces Vp, where 1 6 p < ∞, were recently introduced by Bird and Laustsen [5] as an amalgamation of James’ quasi-reflexive Banach space on the one hand and Schreier’s Banach space giving a counterexample to the Banach–Saks property on the other. The purpose of this note is to answer some questions left open in [5]. Specifically, we prove that (i) the standard Schauder basis f...

let a be a bounded linear operator on a banach space x. we investigate the conditions of existing rank-one operator b such that i+f(a)b is invertible for every analytic function f on sigma(a). also we compare the invariant subspaces of f(a)b and b. this work is motivated by an operator method on the banach space ell^2 for solving some pdes which is extended to general operator space under some ...

In this paper we present inequalities for integrals of functions that are the composition of nonnegative convex functions on an open convex set of a vector space R and vectorvalued functions in a weakly compact subset of a Banach vector space generated by m Lμ-spaces for 1 ≤ p < +∞ and inequalities when these vector-valued functions are in a weakly* compact subset of a Banach vector space gener...

In 1972, the late B. E. Johnson introduced the notion of an amenable Banach algebra and asked whether the Banach algebra B(E) of all bounded linear operators on a Banach space E could ever be amenable if dimE = ∞. Somewhat surprisingly, this question was answered positively only very recently as a by-product of the Argyros– Haydon result that solves the “scalar plus compact problem”: there is a...

In this paper we introduce new modified implicit and explicit algorithms and prove strong convergence of the two algorithms to a common fixed point of a family of uniformly asymptotically regular asymptotically nonexpansive mappings in a real reflexive Banach space  with a uniformly G$hat{a}$teaux differentiable norm. Our result is applicable in $L_{p}(ell_{p})$ spaces, $1 < p

For a Banach space X we shall denote the set of all closed subspaces of X by G(X). In some kinds of problems it turned out to be useful to endow G(X) with a topology. The main purpose of the present paper is to survey results on two the most common topologies on G(X). The organization of this paper is as follows. In section 2 we introduce some definitions and notation. In sections 3 and 4 we in...

a normed space $mathfrak{x}$ is said to have the fixed point property, if for each nonexpansive mapping $t : e longrightarrow e $ on a nonempty bounded closed convex subset $ e $ of $ mathfrak{x} $ has a fixed point. in this paper, we first show that if $ x $ is a locally compact hausdorff space then the following are equivalent: (i) $x$ is infinite set, (ii) $c_0(x)$ is infinite dimensional, (...

We prove that the Banach spaces (⊕n=1`p )`q , which are isomorphic to the Besov spaces on [0, 1], have greedy bases, whenever 1 ≤ p ≤ ∞ and 1 < q < ∞. Furthermore, the Banach spaces (⊕n=1`p )`1 , with 1 < p ≤ ∞, and (⊕n=1`p )c0 , with 1 ≤ p < ∞ do not have a greedy bases. We prove as well that the space (⊕n=1`p )`q has a 1-greedy basis if and only if 1 ≤ p = q ≤ ∞.

We show that, if E is a Banach space with a basis satisfying a certain condition, then the Banach algebra l∞(K(l2 ⊕ E)) is not amenable; in particular, this is true for E = l with p ∈ (1,∞). As a consequence, l∞(K(E)) is not amenable for any infinite-dimensional Lp-space. This, in turn, entails the non-amenability of B(lp(E)) for any Lp-space E, so that, in particular, B(lp) and B(Lp[0, 1]) are...

Very few Banach spaces E are known for which the lattice of closed ideals in the Banach algebra B(E) of all (bounded, linear) operators on E is fully understood. Indeed, up to now the only such Banach spaces are, up to isomorphism, Hilbert spaces and the sequence spaces c0 and `p for 1 6 p < ∞. We add a new member to this family by showing that there are exactly four closed ideals in B(E) for t...