Functional Calculus, Integral Representations, and Banach Space Geometry

CONTENTS Introduction Chapter I. Preliminary information from the compactification theory of topological spaces § 1. The concept of a topological compactification § 2. Some methods of obtaining compactifications § 3. Some special compactifications Chapter II. The compactification theory of topological spaces and the integral representation of linear functionals § 1. The Riesz representation theorem for compact and locally compact spaces § 2. A functional representation by measures on compactifications § 3. An integral representation of functionals on the space of all continuous functions § 4. An integral representation of functionals on spaces of continuous functions of bounded support § 5. An integral representation of functionals on spaces without a topology § 6. The problem of extending measures with values in vector lattices Chapter III. Invariant Banach limits § 1. The classical statement of the problem § 2. The problem in a more general topological setting § 3. The support of the representative measure of an invariant permanent functional Chapter IV. Summability theory § 1. The classical statement of the problem § 2. The Stone-Čech compactification and multiplicative summability methods § 3. The Stone-Čech compactification and compatibility criteria § 4. The Stone-Čech compactification of non-discrete countable spaces and summability § 5. Summability problems in the more general setting of compactification theory and measure theory References

In this paper we are concerned with the problem of constructing the functional calculus for an operator with continuous spectrum from a given collection of weak eigenvectors.A functional calculus is a homomorphism/- Tf of an algebra of functions defined over the 'spectrum' V into a subalgebra of bounded linear operators acting in a banach space E; we may regard an 'eigenfunction expansion' for such a functional calculus as an expression for all Tf over a banach space D dense in E of the form Tf= vf(X)Txdm(X), where m is a measure over Fand Tx is a bounded linear operator from D into the space of 'distributions' D'.In general, it is hoped that the range of Tx, lmTx, will consist of rather precisely specified weak eigenvectors (specified, for example, by boundary values).Since a family of weak eigenvectors is often available, a problem that often appears in applications is the following: Given a family of weak eigenvectors at X, when is it true that this family coincides with lmTxl There are many results in the literature for a closed operator with dense domain in a banach space with continuous spectrum which guarantee a rich functional calculus and a representation of the functional calculus in terms of an eigenfunction expansion where, in fact, ImTA consists of weak eigenvectors.F. Browder [3] established such results for a large class of nonsymmetric partial differential operators, and Gel'fand and Silov [5] were the first to consider the problem of the existence of eigenfunction expansions in general operator theoretic terms.However there are few results that tell us which weak eigenvectors to choose for the eigenfunction expansion.Most results of this nature are confined to differential operators with strong regularity theorems, the most classical being Weyl's construction of the Plancherel formula for a singular second order ordinary differential operator [10].The crucial step is the 'Weyl lemma' which asserts that certain weak eigenvectors are actually infinitely differentiable, and so, classical uniqueness theorems may be invoked to determine precisely which weak eigenvectors contribute to the eigenfunction expansion.The results given here, in contrast, are well within the context of functional

Quantitative affine approximation for UMD targets, Discrete Analysis 2016:6, 48 pp. Let $Y$ be a Banach space. A _martingale difference sequence_ in $Y$ is a sequence of $Y$-valued random variables such that $\mathbb{E}[d_i|d_1,\dots,d_{i-1}]=0$ for every $i$ (and $\mathbb{E}(d_1)=0$). Given $1<p<\infty$, we say that $Y$ satisfies the UMD$_p$ property if there is a constant $C_p$ such that $$\mathbb{E}\|\sum_{i=1}^n\epsilon_id_i\|^p\leq C_p^p\mathbb{E}\|\sum_{i=1}^nd_i\|^p$$ for every martingale difference sequence $d_1,\dots,d_n$ and every choice of signs $\epsilon_1,\dots,\epsilon_n\in\{-1,1_{}\}$. An _unconditional martingale difference_ space, or UMD space for short, is a Banach space $Y$ that satisfies the UMD$_p$ property for some $1<p<\infty$, which can be shown to imply that it satisfies the UMD$_p$ property for all such $p$. It is easy to show that when $Y$ is a Hilbert space, it has the UMD$_p$ property with a constant of 1. In the context of stochastic analysis, UMD spaces can be thought of as spaces that have many of the good properties of Hilbert spaces. The question considered in this paper concerns a kind of differentiability property of Lipschitz functions. Let $X$ be an $n$-dimensional normed space with unit ball $B_X$ and let $Y$ be a UMD Banach space. Given a 1-Lipschitz function $f:B_X\to Y$ and $\epsilon>0$, the aim is to find a sub-ball $B'\subset B$ of radius $\rho$, with $\rho$ as large as possible, and an affine function $\Lambda:B'\to Y$, such that $\|f(x)-\Lambda(x)\|\leq\epsilon\rho$ for every $x\in B'$. Importantly, the lower bound on $\rho$ must be independent of the function $f$. The paper obtains a lower bound of $\exp(-(1/\epsilon)^{cn})$ for a constant $c$ that depends on $Y$ only. This improves on the previously best known bound even when $Y$ is a Hilbert space. In the other direction, the best known upper bound (on how large $\rho$ is in the worst case) is singly exponential in $n$, so there is still a large and very interesting gap. There are several other attractive open problems in the paper. The proof of the main result of the paper is not easy, but it contains many new ideas in the geometry of Banach spaces and in harmonic analysis, some of which are interesting results in their own right.

We study the unconditional convergence of series in Banach spaces. We consider series of special type (Hadamard series), obtain the condition of their unconditional convergence, and discuss some of their applications. Further, we examine the almost sure unconditional convergence of random series in Banach spaces and, in the case of Gaussian series, we establish the relationship between the almost sure unconditional convergence and the geometry of Banach spaces. We also consider some probabilistic problems related to the convergence of series.

The purpose of this work is to study various geometric properties of Banach spaces, with particular emphasis on their stability with respect to the formation of (infinite) absolute sums and Köthe-Bochner spaces of vector- valued functions. The first three chapters are devoted to the class of acs (``alternatively convex or smooth'') Banach spaces and their relatives, the luacs and uacs spaces. These notions were introduced by Kadets et al. in 2000 (in connection with the so called anti-Daugavet property) and form common generalisations of the usual convexity and smoothness properties of Banach spaces. In the first chapter, these notions (as well as some other related properties introduced by the author) are studied from a general point of view. For example, some equivalent characterisations are established and results on duality and quotient spaces are discussed. In the second chapter, various results on (mainly infinite) absolute sums of acs spaces and their relatives are proved, while in the third chapter, acs properties of Köthe-Bochner spaces of vector-valued functions are discussed. The fourth chapter is devoted to the so called Opial property and its variants, as well as some other geometric notions (WORTH property and Garcia-Falset coefficient) of Banach spaces, which are all related to the fixed point property for nonexpansive mappings. Some stability results for these properties in certain infinite absolute sums are established. Furthemore, some properties analogous to the Opial property in Lebesgue-Bochner spaces and Ces`aro spaces of vector-valued functions are discussed. The fifth chapter concerns Banach spaces with the so called ``ball generated property'' (BGP). The stability of this property with respect to certain sums of two spaces is proved. The last chapter deals with generalised lush (GL) spaces, which were introduced by Huang et al. in 2013 in connection with the Mazur-Ulam extension problem. Some results on GL-spaces are established. In particular, every M-ideal in a GL-space is again a GL-space.

We study in this paper the finite Jung constant, its interplay with Kottman’s constant and its meaning regarding the geometry of Banach spaces.

We consider two Pythagorean modulus introduced by Gao (2005, 2006) recently. The exact values concerning these modulus for some classical Banach spaces are determined. Some applications in geometry of Banach spaces are also obtained.

Operator-valued Fourier multiplier theorems on Lp( X) and geometry of Banach spaces

&lt;p&gt;&lt;b&gt;We investigate the geometry of effective Banach spaces, namely a sequenceof approximation properties that lies in between a Banach space having a basis and the approximation property.&lt;/b&gt;&lt;/p&gt; &lt;p&gt;We establish some upper bounds on suchproperties, as well as proving some arithmetical lower bounds. Unfortunately,the upper bounds obtained in some cases are far away from the lower bound.&lt;/p&gt; &lt;p&gt;However, we will show that much tighter bounds will require genuinely newconstructions, and resolve long-standing open problems in Banach space theory.&lt;/p&gt; &lt;p&gt;We also investigate the effectivisations of certain classical theorems in Banachspaces.&lt;/p&gt;

This paper is devoted to the multivariable \(H^\infty \) functional calculus associated with a finite commuting family of sectorial operators on Banach space. First we prove that if \((A_1,\ldots , A_d)\) is such a family, if \(A_k\) is R-sectorial of R-type \(\omega _k\in (0,\pi )\), \(k=1,\ldots ,d\), and if \((A_1,\ldots , A_d)\) admits a bounded \(H^\infty (\Sigma _{\theta _1}\times \cdots \times \Sigma _{\theta _d})\) joint functional calculus for some \(\theta _k\in (\omega _k,\pi )\), then it admits a bounded \(H^\infty (\Sigma _{\theta _1}\times \cdots \times \Sigma _{\theta _d})\) joint functional calculus for all \(\theta _k\in (\omega _k,\pi )\), \(k=1,\ldots ,d\). Second we introduce square functions adapted to the multivariable case and extend to this setting some of the well-known one-variable results relating the boundedness of \(H^\infty \) functional calculus to square function estimates. Third, on K-convex reflexive spaces, we establish sharp dilation properties for d-tuples \((A_1,\ldots , A_d)\) which admit a bounded \(H^\infty (\Sigma _{\theta _1}\times \cdots \times \Sigma _{\theta _d})\) joint functional calculus for some \(\theta _k<\frac{\pi }{2}\).

In this paper we shall construct a certain spectral theoryfor closed linear operators on a Banach space.These operators have a suitable spectral behaviour on subsets of theirspectra but we must eliminate some residual part which do not offer informationabout the intimate structure of the considered objects, at least from our pointof view.It will be easy to see that this theory contains many examples of operators,bounded or not, having a functional calculus on their spectrum [1], [2], [3], [6],[8], [9].A permanent model for our construction will be the theory of decomposableoperators on a Banach space [7], [2].Throughout this paper the sets of points will be taken in CΌo^Cuf

This note deals with estimating the volume of the set of separable mixed quantum states when the dimension of the state space grows to infinity. This has been studied recently for qubits; here we consider larger particles and conclude that, in all cases, the proportion of the states that are separable is super-exponentially small in the dimension of the set. We also show that the partial transpose criterion becomes imprecise when the dimension increases, and that the lower bound $6^{-N/2}$ on the (Hilbert-Schmidt) inradius of the set of separable states on N qubits obtained recently by Gurvits and Barnum is essentially optimal. We employ standard tools of classical convexity, high-dimensional probability and geometry of Banach spaces. One relatively non-standard point is a formal introduction of the concept of projective tensor products of convex bodies, and an initial study of this concept. PACS numbers: 03.65.Ud, 03.67.Mn, 03.65.Db, 02.40.Ft, 02.50.Cw MSC-class: 46B28, 47B10, 47L05, 52A38, 81P68

In this paper the properties of convex set and its inscribed polygons in a two dimensional Euclidean space axe studied. Consequently we applied our results in solving a problem in geometry of Banach space.

Holomorphic Functions on Nuclear Sequence Spaces

We investigate the set (a) of positive, trace preserving maps acting on density matrices of size N and a sequence of its nested subsets: the sets of maps which are (b) decomposable, (c) completely positive, and (d) extended by identity impose positive partial transpose and (e) are superpositive. Working with the Hilbert–Schmidt (Euclidean) measure, we derive tight explicit two-sided bounds for the volumes of all five sets. A sample consequence is the fact that, as N increases, a generic positive map becomes not decomposable and, a fortiori, not completely positive. Due to the Jamiołkowski isomorphism, the results obtained for quantum maps are closely connected to similar relations between the volume of the set of quantum states and the volumes of its subsets (such as states with positive partial transpose or separable states) or supersets. Our approach depends on the systematic use of duality to derive quantitative estimates and on various tools of classical convexity, high-dimensional probability, and geometry of Banach spaces, some of which are not standard.