Suppose \(X\) and \(Y\) are Banach spaces, K is a compact Hausdorff space, Σ is the σ-algebra of Borel subsets of \(K\), \(C(K, X)\) is the Banach space of all continuous \(X\)-valued functions (with the supremum norm), and T : \(C(K, X)\) → \(Y\) is a strongly bounded operator with representing measure m : Σ → \(L(X, Y)\). We show that if T is a strongly bounded operator and is its extension, then T* is pseudo weakly compact (resp. limited completely continuous, limited p-convergent, 1 ≤ p < ∞) if and only if has the same property.
We formulate a quantitative finite-dimensional conjecture about frame multipliers and prove that it is equivalent to Conjecture 1 in [SB2]. We then present solutions to the conjecture for certain classes of frame multipliers. In particular, we prove that for all $C>0$ and $N\in\mathbb{N}$ the following is true: Let $(x_j)_{j=1}^N$ and $(f_j)_{j=1}^N$ be sequences in a finite dimensional Hilbert space which satisfy $\|x_j\|=\|f_j\|$ for all $1\leq j\leq N$ and $$\Big\|\sum_{j=1}^N \varepsilon_j\langle x,f_j\rangle x_j\Big\|\leq C\|x\|, \qquad\textrm{ for all $x\in \ell_2^M$ and $|\varepsilon_j|=1$}. $$ If the frame operator for $(f_j)_{j=1}^N$ has eigenvalues $\lambda_1\geq...\geq\lambda_M$ and $\beta>0$ is such that $\lambda_1\leq \beta M^{-1}\sum_{j=1}^M\lambda_j$ then $(f_j)_{j=1}^N$ has Bessel bound $27 \beta^2 C$. The same holds for $(x_j)_{j=1}^N$.
Let (Ω, Σ, μ) be a σ-finite measure space and consider the Lebesgue function space \(L_1(\mu)\) endowed with its standard norm. We obtain a characterization of weak compactness for closed bounded convex subsets of \(L_1(\mu)\) in terms of the existence of fixed points for certain classes of eventually affine, uniformly Lipschitzian mappings.
The aim of this paper is to study the fixed point property for nonexpansive mappings in the duals of separable Lindenstrauss spaces by means of suitable geometrical properties of the dual ball. First we show that a property concerning the behaviour of a class of -closed subsets of the dual sphere is equivalent to the fixed point property. Then, the main result of our paper shows an equivalence between another, stronger geometrical property of the dual ball and the stable fixed point property. The last geometrical notion was introduced by Fonf and Veselý as a strengthening of the notion of polyhedrality. In the last section we show that also the first geometrical assumption that we have introduced can be related to a polyhedral concept for the predual space. Indeed, we give a hierarchical structure among various polyhedrality notions in the framework of Lindenstrauss spaces. Finally, as a by-product, we obtain an improvement of an old result about the norm-preserving compact extension of compact operators
The main aim of the paper is to study some quantitative aspects of the stability of the \(weak^*\) fixed point property for nonexpansive mappings in \(ℓ_1\) (shortly, \(weak^*\)-fpp). We focus on two complementary approaches to this topic. First, given a predual \(X\) of \(ℓ_1\) such that the \(σ(ℓ_1, X)\)-fpp holds, we precisely establish how far, with respect to the Banach–Mazur distance, we can move from \(X\) without losing the \(weak^*\)-fpp. The interesting point to note here is that our estimate depends only on the smallest radius of the ball in \(ℓ_1\) containing all \(σ(ℓ_1, X)\)-cluster points of the extreme points of the unit ball. Second, we pass to consider the stability of the \(weak^*\)-fpp in the restricted framework of preduals of \(ℓ_1\). Namely, we show that every predual \(X\) of \(ℓ_1\) with a distance from \(c_0\) strictly less than 3, induces a \(weak^*\) topology on \(ℓ_1\) such that the \(σ(ℓ_1, X)\)-fpp holds.
We show that there exists a non-weakly compact, closed, bounded, convex subset W of the Banach space of convergent sequences \((c,‖⋅‖_∞)\), such that every nonexpansive mapping T: W⟶ W has a fixed point. This answers a question left open in the 2003 and 2004 papers of Dowling, Lennard and Turett. This is also the first example of a non-weakly compact, closed, bounded, convex subset W of a Banach space \(X\) isomorphic to \(c_0\), for which W has the fixed point property for nonexpansive mappings. We also prove that the sets W may be perturbed to a large family of non-weakly compact, closed, bounded, convex subsets \(W_q\) of \((c,‖⋅‖_∞)\) with the fixed point property for nonexpansive mappings; and we discuss similarities and differences with work of Goebel and Kuczumow concerning analogous subsets of \(ℓ_1\).
In the first chapter we construct a new example of an affine norm continuous mapping on a closed, convex, non-weakly compact set \(C\) that cannot be extended to a continuous linear map on the entire space \(X\). Although often used in the field of the fixed point theory, most of the examples known in the literature are restrictions of continuous, linear mappings from \(X\) to \(C\). The second chapter continues with a main focus on the notion of the affine dual of a closed, convex, bounded set. Using a theorem of M. Krein and D. Milman from Studia Mathematica 1940, one can show that certain spaces like \(c_0\) and \(L^1[0, 1]\) are not dual spaces. However, it turns out that we can see them as affine dual spaces. In the third part of this thesis we provide a new proof that compactness in for closed, bounded, convex sets is equivalent with the fixed point property for cascading nonexpansive mappings. We also prove an analogue of this result in \(L^1[0, 1]\). The last part is dedicated to the study of the stability constant of the weak-fixed point property for the dual of separable Lindenstrauss spaces. Initiated in 1980 and 1982 by P. Soardi and T.C. Lim for the space \(c_0\), we will now find a precise formula in the general case of an arbitrary predual of \(ℓ_1\) that depends only on a geometrical property of the unit ball of \(ℓ_1\) with respect to the predual considered. Therefore, this formula establishes a quantitative result in terms of geometric properties.