The polynomial cluster value problem for Banach spaces

We introduce and study Banach spaces which have property CWO, i.e., every finite convex combination of relatively weakly open subsets of their unit ball is open in the relative weak topology of the unit ball. Stability results of such spaces are established, and we introduce and discuss a geometric condition---property (co)---on a Banach space. Property (co) essentially says that the operation of taking convex combinations of elements of the unit ball is, in a sense, an open map. We show that if a finite dimensional Banach space $X$ has property (co), then for any scattered locally compact Hausdorff space $K$, the space $C_0(K,X)$ of continuous $X$-valued functions vanishing at infinity has property CWO. Several Banach spaces are proved to possess this geometric property; among others: 2-dimensional real spaces, finite dimensional strictly convex spaces, finite dimensional polyhedral spaces, and the complex space $\ell_1^n$. In contrast to this, we provide an example of a $3$-dimensional real Banach space $X$ for which $C_0(K,X)$ fails to have property CWO. We also show that $c_0$-sums of finite dimensional Banach spaces with property (co) have property CWO. In particular, this provides examples of such spaces outside the class of $C_0(K,X)$-spaces.

We study classes of mappings between finite and infinite dimensional Banach spaces that are monotone and mappings which are differences of monotone mappings (DM). We prove a Radó–Reichelderfer estimate for monotone mappings in finite dimensional spaces that remains valid for DM mappings. This provides an alternative proof of the Fréchet differentiability a.e. of DM mappings. We establish a Morrey-type estimate for the distributional derivative of monotone mappings. We prove that a locally DM mapping between finite dimensional spaces is also globally DM. We introduce and study a new class of the so-called UDM mappings between Banach spaces, which generalizes the concept of curves of finite variation.

Sequences and Series in Banach Spaces

Let X be a Banach space, (fi, X, X) a finite measure space, and 1 < p < oo . It is shown that LP (I, X) has the complete continuity property if and only if X has it. A similar result about LlA(G, X) is also given. I. Introduction Let X be a Banach space, let (Q, X, X) be a finite measure space, and let 1 < p < oo. We denote by LP(X, X) the Banach space of all (class of) X- valued p-Bochner A-integrable functions (class of) with its usual norm. If X is the scalar field then LP(X, X) will be denoted by LP(X). A Banach space X is said to have the complete continuity property if for ev- ery finite measure space (K, 3r, p), every bounded operator T: LX(K, 3r, p) —► X is a Dunford-Pettis operator. Any Banach space with the Radon-Nikodym property (RNP) has the complete continuity property. In particular, any LP(X), 1 < p < oo , has the complete continuity property. It is well known (see (DU)) that if X has the (RNP) then LP(X, X) has the same property. Recently, Saab and Saab (SS) observed that if X is a dual Banach space that has the complete continuity property then LP(X, X) enjoys the same property. They also asked (SS, Question 13) whether IP(X, X) has the complete continuity property whenever X does. In this paper we will show that the answer is always affirmative. The question of when a property passes from the Banach space X to LP(X, X) was exten- sively studied by several authors in the past. Let us recall that Kwapien (Kw) showed that LP(X, X) (1 < p < oo) contains a copy of en if and only if X contains a copy of Co . Talagrand (T) showed that if X is weakly sequentially complete then LP(X, X) (1 <p < oo) is weakly sequentially complete. Kalton, Saab, and Saab (KSS) were able to prove that the property (u) also passes from X to LP(X, X) (1 < p < oo). Mendoza (M) succeeded in showing that X contains a complemented copy of lx if and only if LP(X, X) (1 < p < oo)

Continuity properties of the ball hull mapping

Liapounoff, in 1940, proved that the range of a countably additive bounded measure with values in a finite dimensional vector space is compact and, in the nonatomic case, is convex. Later, in 1945, Liapounoff showed, by counterexample, that neither the convexity nor compactness need hold in the infinite dimensional case. The next step was taken by Halmos who in 1948 gave simplified proofs of Liapounoff's results for the finite dimensional case. In 1951, Blackwell [I] considered the case of a measure represented by a finite dimensional vector integral and obtained results similar to those of Liapounoff for these measures. Various versions of Liapounoff's theorem appeared in the 1950's and 1960's, and in 1966, Lindenstrauss [8] gave a very elegant short proof of Liapounoff's earlier result. Finally, in 1968, Olech [9] considered the case of an unbounded measure with range in a finite dimensional vector space. The purpose of this note is to demonstrate that the closure of the range of a measure of bounded variation with values in a Banach space, which is either a reflexive space or a separable dual space, is compact and, in the nonatomic case, is convex. To this end, let Q be a point set and z be a a-field of subsets of U. If X is a Banach space, then an p-valued measure is a countably additive function F defined on 2 with values in X. F is of bounded variation if

The connection between Stokes's Integral Theorem and the Frobenius-Cartan Integration Theorem concerning Pfaffian systems has been noted a long time. In this note, we generalize Stokes's theorem to implicit vector valued differential forms and derive from it a general Frobenius theorem concerning mappings in Banach spaces. The only difficulty in the proof arises in the need to show differentiability with respect to a parameter of solutions of a certain differential equation, but is is easily overcome. The generality of the theorem seems to be necessary for applications to the new subjects of infinite groups and of differential geometry in infinitely many dimensions. E.g., it allows us to associate a local group to any infinite-dimensional Lie algebra in a Banach space. For finite dimensional vector spaces we obtain the classical theorem with nearly minimal differentiability conditions [4]. Also for finite dimensional spaces, one might derive from it parts of the Cartan-Kahler theory of integral manifolds [3 ] for not completely integrable CI systems.

Characterization of simultaneous similarity for commuting m m - tuples of operators is an open problem even in finite-dimensional spaces; known as “A wild problem in linear algebra”. In this paper we offer a criterion for simultaneous similarity of m m -tuples of k k -cyclic commuting operators on an arbitrary Banach space. Moreover, we obtain an additional equivalence condition in the case of finite dimensional Banach spaces, which extends the result found by Shekhtman [Math. Stat. 1 (2013), pp. 157–161] for pairs of cyclic commuting matrices. We also present two applications of our results, one in the case of general multiplication operators on Banach spaces of analytic function, and one for m m -tuples of commuting square matrices.

We derive new representations for the generalised Jacobian of a locally Lipschitz map between finite dimensional real Euclidean spaces as the lower limit (i.e., limit inferior) of the classical derivative of the map where it exists. The new representations lead to significantly shorter proofs for the basic properties of the subgradient and the generalised Jacobian including the chain rule. We establish that a sequence of locally Lipschitz maps between finite dimensional Euclidean spaces converges to a given locally Lipschitz map in the L-topology—that is, the weakest refinement of the sup norm topology on the space of locally Lipschitz maps that makes the generalised Jacobian a continuous functional—if and only if the limit superior of the sequence of directional derivatives of the maps in a given vector direction coincides with the generalised directional derivative of the given map in that direction, with the convergence to the limit superior being uniform for all unit vectors. We then prove our main result that the subspace of Lipschitz C ∞ maps between finite dimensional Euclidean spaces is dense in the space of Lipschitz maps equipped with the L-topology, and, for a given Lipschitz map, we explicitly construct a sequence of Lipschitz C ∞ maps converging to it in the L-topology, allowing global smooth approximation of a Lipschitz map and its differential properties. As an application, we obtain a short proof of the extension of Green’s theorem to interval-valued vector fields. For infinite dimensions, we show that the subgradient of a Lipschitz map on a Banach space is upper continuous, and, for a given real-valued Lipschitz map on a separable Banach space, we construct a sequence of Gateaux differentiable functions that converges to the map in the sup norm topology such that the limit superior of the directional derivatives in any direction coincides with the generalised directional derivative of the Lipschitz map in that direction.

Birkhoff integral for multi-valued functions

A necessary and sufficient condition is given to ensure the boundedness of Fourier–Haar multiplier operators from $$L^1 ([0, 1], X)$$ to $$L^1 ([0, 1], Y)$$ , where X is an arbitrary finite dimensional Banach space and Y is an arbitrary Banach space. The Fourier–Haar multiplier sequences come not from $${\mathbb {R}}$$ , as in the classical case, but from the space of bounded operators from the Banach space X to the Banach space Y. Moreover, it is shown that this condition characterises the finite dimensionality of the Banach space X.

Numerical peak holomorphic functions on Banach spaces

monograph by T. Kato is an excellent textbook in the theory of linear operators in Banach and Hilbert spaces. It is a thoroughly worthwhile reference work both for graduate students in functional analysis as well as for researchers in perturbation, spectral, and scattering theory. In chapters 1, 3, 5 operators in finite-dimensional vector spaces, Banach spaces and Hilbert spaces are introduced. Stability and perturbation theory are studied in finite-dimensional spaces (chapter 2) and in Banach spaces (chapter 4). Sesquilinear forms in Hilbert spaces are considered in detail (chapter 6), analytic and asymptotic perturbation theory is described (chapter 7 and 8). The fundamentals of semigroup theory are given in chapter 9. The supplementary notes appearing in the second edition of the book gave mainly additional information concerning scattering theory described in chapter 10. The first edition is now 30 years old. The revised edition is 20 years old. Nevertheless it is a standard textbook for the theory of linear operators. It is user-friendly in the sense that any sought after definitions, theorems or proofs may be easily located. In the last two decades much progress has been made in understanding some of the topics dealt with in the book, for instance in semigroup and scattering theory. However the book has such a high didactical and scientific standard that I can recomment it for any mathematician or physicist interested in this field. Zentralblatt MATH, 836

Frame potential for finite-dimensional Banach spaces

According to the spatial dimension, equation type, and time sequence, the latest progress in controllability of stochastic linear systems and some unsolved problems are introduced. Firstly, the exact controllability of stochastic linear systems in finite dimensional spaces is discussed. Secondly, the exact, exact null, approximate, approximate null, and partial approximate controllability of stochastic linear systems in infinite dimensional spaces are considered. Thirdly, the exact, exact null and impulse controllability of stochastic singular linear systems in finite dimensional spaces are investigated. Fourthly, the exact and approximate controllability of stochastic singular linear systems in infinite dimensional spaces are studied. At last, the controllability and observability for a type of time-varying stochastic singular linear systems are studied by using stochastic GE-evolution operator in the sense of mild solution in Banach spaces, some necessary and sufficient conditions are obtained, the dual principle is proved to be true, an example is given to illustrate the validity of the theoretical results obtained in this part, and a problem to be solved is introduced. The main purpose of this paper is to facilitate readers to fully understand the latest research results concerning the controllability of stochastic linear systems and the problems that need to be further studied, and attract more scholars to engage in this research.