Estimates of the weak distance between finite-dimensional Banach spaces

We prove a variant of a theorem of N. Alon and V. D. Milman. Using it we construct for everyn-dimensional Banach spacesX andY a measure space Ω and two operator-valued functionsT: Ω→L(X, Y),S: Ω→L(Y, X) so that ∫_Ω S(ω)oT(ω)dω is the identity operator inX and ∫_Ω||S(ω)||·||T(ω)||dω=O(n ^α ) for some absolute constantα<1. We prove also that any subset of the unitn-cube which is convex, symmetric with respect to the origin and has a sufficiently large volume possesses a section of big dimension isomorphic to ak-cube. Research supported in part by a grant of the Israel Academy of Sciences.     

Free distance ratio mappings in Banach spaces

Monatshefte für Mathematik - In this paper, we introduce a class of mappings related to the distance ratio metric and study its connection to the freely quasiconformal mapping in Banach...     

Subdifferentials of perturbed distance functions in Banach spaces

In general Banach space setting, we study the perturbed distance function d_S^J(·){d_S^J(cdot)} determined by a closed subset S and a lower semicontinuous function J (·). In particular, we show that the Fréchet subdifferential and the proximal subdifferential of a perturbed distance function are representable by virtue of corresponding normal cones of S and subdifferentials of J (·).     

The Banach-Mazur distance between symmetric spaces

We show that the Banach-Mazur distance betweenN-dimensional symmetric spacesE andF satisfies , wherec is a numerical constant. IfE is a symmetric space, then max (M ⁽²⁾(E),M ₍₂₎(E)), whereM ⁽²⁾(E) (resp.M ₍₂₎(E)) denotes the 2-convexity (resp. the 2-concavity) constant ofE. We also give an example of a spaceF with an 1-unconditional basis and enough symmetries that satisfiesd(F, l ₂ ^dimF )=M ⁽²⁾(F)M ₍₂₎(F). Partially supported by NSF Grant MCS-8201044.     

Integral mappings between Banach spaces

We consider the classes of “Grothendieck-integral” (G-integral) and “Pietsch-integral” (P-integral) linear and multilinear operators (see definitions below), and we prove that a multilinear operator between Banach spaces is G-integral (resp. P-integral) if and only if its linearization is G-integral (resp. P-integral) on the injective tensor product of the spaces, together with some related results concerning certain canonically associated linear operators. As an application we give a new proof of a result on the Radon-Nikodym property of the dual of the injective tensor product of Banach spaces. Moreover, we give a simple proof of a characterization of the G-integral operators on C(K,X) spaces and we also give a partial characterization of P-integral operators on C(K,X) spaces.     

Approximation of the limit distance function in Banach spaces

In this paper we study the behavior of the limit distance function d(x)=limdist(x,Cn) defined by a nested sequence (Cn) of subsets of a real Banach space X. We first present some new criteria for the non-emptiness of the intersection of a nested sequence of sets and of their ε-neighborhoods from which we derive, among other results, Dilworth's characterization [S.J. Dilworth, Intersections of centred sets in normed spaces, Far East J. Math. Sci. (Part II) (1988) 129-136 (special volume)] of Banach spaces not containing c₀ and Marino's result [G. Marino, A remark on intersection of convex sets, J. Math. Anal. Appl. 284 (2003) 775-778]. Passing then to the approximation of the limit distance function, we show three types of results: (i) that the limit distance function defined by a nested sequence of non-empty bounded closed convex sets coincides with the distance function to the intersection of the weak^∗-closures in the bidual; this extends and improves the results in [J.M.F. Castillo, P.L. Papini, Distance types in Banach spaces, Set-Valued Anal. 7 (1999) 101-115]; (ii) that the convexity condition is necessary; and (iii) that in spaces with separable dual, the distance function to a weak^∗-compact convex set is approximable by a (non-necessarily nested) sequence of bounded closed convex sets of the space.     

The distance between fixed points of some pairs of maps in Banach spaces and applications to differential systems

Let T be a λ-contraction on a Banach space Y and let S be an almost λ-contraction, i.e. sum of an (ε, λ)-contraction with a continuous, bounded function which is less than ε in norm. According to the contraction principle, there is a unique element u in Y for which u = Tu: If moreover there exists v in Y with v = Sv, then we will give estimates for ‖u−v‖. Finally, we establish some inequalities related to the Cauchy problem.     

Isometries between Banach spaces of Lipschitz functions

The Banach spaces Lip ^a (S, Δ), lip ^a (S, Δ), Lip ^a (S, Δ;s ₀) and lip ^a (S, Δ;s ₀) of Lipschitz functions are defined. We shall identify the extreme points of the unit balls in their corresponding dual spaces and make use of them to present a complete characterization of the isometries between these function spaces. This paper is a part of the author’s M.Sc. thesis which was prepared under the guidance of Dr. Y. Benyamini.     

The stability of Banach frames in Banach spaces

In this paper, we give some equivalent conditions on a Banach frame for a Banach space by using the pseudoinverse operator. We also consider the stability of a Banach frame for a Banach space X with respect to Xd or an Xd-frame for a Banach space X under perturbation. These results generalize and improve the related works of Balan, Casazza, Christensen, Stoeva and Jian et al.     

Closed linear operators between nonarchimedean Banach spaces

In this work we establish some basic properties of closed linear operators between nonarchimedean Banach spaces. As a consequence, we characterize the operators with closed range, and we establish the state diagram for closed linear operators when the underlying spaces satisfy a Hahn-Banach property.     

Rank theorems of operators between Banach spaces

Let E and F be Banach spaces, and B(E,F) all of bounded linear operators on E into F. Let T0∈B(E,F) with an outer inverse T#0∈B(F,E). Then a characteristic condition of S=(I+T#0(T-T0))-1T#0 with T∈B(E,F) and ‖T#0(T-T0)‖<1, being a generalized inverse of T, is presented, and hence, a rank theorem of operators on E into F is established (which generalizes the rank theorem of matrices to Banach spaces). Consequently, an improved finite rank theorem and a new rank theorem are deduced. These results will be very useful to nonlinear functional analysis.     

The trace formula in Banach spaces

A classical result of Grothendieck and Lidskii says that the trace formula (that the trace of a nuclear operator is the sum of its eigenvalues provided the sequence of eigenvalues is absolutely summable) holds in Hilbert spaces. In 1988, Pisier proved that weak Hilbert spaces satisfy the trace formula. We exhibit a much larger class of Banach spaces, called Γ-spaces, that satisfy the trace formula. A natural class of asymptotically Hilbertian spaces, including some spaces that are ?₂ sums of finite-dimensional spaces, are Γ-spaces. One consequence is that the direct sum of two Γ-spaces need not be a Γ-space.     

The RAGE theorem in Banach spaces

A generalization of the so called RAGE theorems from scattering theory in Hilbėrt spaces to a more general class of uniformly boundedC ₀-semigroups on Banach spaces is given. The extension is based on the notion of weak almost periodicity (in the sense of Eberlein). This research is based on parts of Chapter 3 of the author’s dissertation at Essen University. The work of the author was supported by the Deutsche Forschungsgemeinschaft (DFG).