Related papers: Uniform Property (S)

We study the relationship between the residuality of the set of norm attaining functionals on a Banach space and the residuality and the denseness of the set of norm attaining operators between Banach spaces. Our first main result says that…

A Banach space $X$ is said to have the ball generated property (BGP) if every closed, bounded, convex subset of $X$ can be written as an intersection of finite unions of closed balls. In 2002 S. Basu proved that the BGP is stable under…

It is shown that if the dual of a Banach space satisfies Property($K^*$) then $R(X) \leq 1 + \frac{\displaystyle 1}{\displaystyle 1 + r_{X^*}(1)} < 2$ where $r_{X^*}(c)$ is Opial's modulus for $X^*.$ Thus $X$ has the weak fixed point…

We study $M$-ideals of compact operators by means of the property~$(M)$ introduced in \cite{Kal-M}. Our main result states for a separable Banach space $X$ that the space of compact operators on $X$ is an $M$-ideal in the space of bounded…

Let $B(X)$ be the Banach algebra of all bounded linear operators acting on a Banach space $X$. Are sums and products of commuting decomposable operators on Banach spaces decomposable? This is one of the most important open problems in the…

Let $A$ be a Banach algebra with a bounded left approximate identity $\{e_\lambda\}_{\lambda\in\Lambda}$, let $\pi$ be a continuous representation of $A$ on a Banach space $X$, and let $S$ be a non-empty subset of $X$ such that…

We study some properties of the randomized series and their applications to the geometric structure of Banach spaces. For $n\ge 2$ and $1<p<\infty$, it is shown that $\ell_\infty^n$ is representable in a Banach space $X$ if and only if it…

Another proof that uniformly nonsquare Banach spaces have the fixed point property is presented.

Let $X$ be a completely regular space. For a non-vanishing self-adjoint Banach subalgebra $H$ of $C_B(X)$ which has local units we construct the spectrum $\mathfrak{sp}(H)$ of $H$ as an open subspace of the Stone-Cech compactification of…

A Banach space $X$ with a Schauder basis is defined to have the restricted quotient hereditarily indecomposable (QHI) property if $X/Y$ is hereditarily indecomposable (HI) for any infinite codimensional subspace $Y$ with a successive…

The class of uniformly smooth hyperbolic spaces was recently introduced by the first author as a common generalization of both CAT(0) spaces and uniformly smooth Banach spaces, in a way that Reich's theorem on resolvent convergence could…

We extend a precise renorming result of Godefroy, Kalton, and Lancien regarding asymptotically uniformly smooth norms of separable Banach spaces with Szlenk index $\omega$. For every ordinal $\xi$, we characterize the operators, and…

Property($M$) in separable Banach spaces has played an important role in metric fixed point theory. This paper explores some of the Banach space properties that can be associated with Property($M$) and Property($M^*$).

We present a sufficient condition for smoothness of bounded linear operators on Banach spaces for the first time. Let $T, A \in B(\mathbb{X}, \mathbb{Y}),$ where $\mathbb{X}$ is a real Banach space and $\mathbb{Y}$ is a real normed linear…

We characterize real Banach spaces $Y$ such that the pair $(\ell_\infty ^n, Y)$ has the Bishop-Phelps-Bollob\'as property for operators. To this purpose it is essential the use of an appropriate basis of the domain space $\R^n$. As a…

In this work we construct certain general bundles $<\mathfrak{M},\rho,X>$ and $<\mathfrak{B},\eta,X>$ of Hausdorff locally convex spaces associated with a given Banach bundle $<\mathfrak{E},\pi,X>$. Then we present conditions ensuring the…

We study some generalized metric properties of weak topologies when restricted to the unit sphere of some equivalent norm on a Banach space, and their relationships with other geometrical properties of norms. In case of dual Banach space…

We study Banach spaces with a weak stable unit ball, that is Banach spaces where every convex combination of relatively weakly open subsets in its unit ball is again a relatively weakly open subset in its unit ball. It is proved that the…

This paper investigates strong metric subregularity around a reference point as introduced by H. Gfrerer and J. V. Outrata. In the setting of Banach spaces, we analyse its stability under Lipschitz continuous perturbations and establish its…

Relatively recently it was proved that if $\Gamma$ is an arbitrary set, then any equivalent norm on $c_0(\Gamma)$ can be approximated uniformly on bounded sets by polyhedral norms and $C^\infty$ smooth norms, with arbitrary precision. We…