Nonanalytic functional calculi and spectral maximal spaces

A classical result due to M. Eidelheit and B. Yood states that the standard algebra norm on the algebra of bounded linear operators on a Banach space is minimal, in the sense that the norm must be less than a multiple of any other submultiplicative norm on the same algebra. This definition does not assume that the arbitrary algebra norm is complete. In cases when the standard algebra norm is, in addition, maximal, it is therefore unique up to equivalence. More recently, M. Meyer showed that the Calkin algebras of a very restricted class of Banach spaces also have unique algebra norms. We generalise the Eidelheit-Yood method of proof, to show that the conventional quotient norm on a larger class of Calkin algebras is minimal. Since maximality of the norm is a presumed property for the class, the norm is also unique. We thus extend the result of Meyer. In particular, we establish that the Calkin algebras of canonical Banach spaces such as James’ space and Tsirelson’s space have unique algebra norms, without assuming completeness. We also prove uniqueness of norm for quotients of the algebras of operators on classical non-separable spaces, the closed ideals of which were previously studied by M. Daws. One aspect of the Eidelheit-Yood method is a dependence on the uniform boundedness principle. As a component of our generalisation, we prove an analogue of that principle which applies to Calkin algebra elements rather than bounded linear operators. In order to translate the uniform boundedness principle into this new setting, we take the perspective that non-compact operators map certain wellseparated sequences to other well-separated sequences. We analyse the limiting separation of such sequences, using these values to measure the non-compactness of operators and define the requisite notion of a bounded set of non-compact operators. In the cases when the underlying Banach space has a Schauder basis, we are able to restrict attention to seminormalised block basic sequences. As a consequence, our main uniqueness of norm result for Calkin algebras relies on the existence of bounded mappings between, and projections onto, the spans of block basic sequences in the relevant Banach spaces.

If E and F are real Banach spaces let Cp,q(E, F) O ≤ q ≤ p ≤ ∞, denote those maps from E to F which have p continuous Frechet derivatives of which the first q derivatives are bounded. A Banach space E is defined to be Cp,q smooth if Cp,q(E,R) contains a nonzero function with bounded support. This generalizes the standard Cp smoothness classification. If an Lp space, p ≥ 1, is Cq smooth then it is also Cq,q smooth so that in particular Lp for p an even integer is C∞,∞ smooth and Lp for p an odd integer is Cp-1,p-1 smooth. In general, however, a Cp smooth B-space need not be Cp,p smooth. Co is shown to be a non-C2,2 smooth B-space although it is known to be C∞ smooth. It is proved that if E is Cp,1 smooth then Co(E) is Cp,1 smooth and if E has an equivalent Cp norm then co(E) has an equivalent Cp norm. Various consequences of Cp,q smoothness are studied. If f ϵ Cp,q(E,F), if F is Cp,q smooth and if E is non-Cp,q smooth, then the image under f of the boundary of any bounded open subset U of E is dense in the image of U. If E is separable then E is Cp,q smooth if and only if E admits Cp,q partitions of unity; E is Cp,psmooth, p ˂∞, if and only if every closed subset of E is the zero set of some CP function. f ϵ Cq(E,F), 0 ≤ q ≤ p ≤ ∞, is said to be Cp,q approximable on a subset U of E if for any ϵ ˃ 0 there exists a g ϵ Cp(E,F) satisfying sup/xϵU, O≤k≤q ‖ Dk f(x) - Dk g(x) ‖ ≤ ϵ. It is shown that if E is separable and Cp,q smooth and if f ϵ Cq(E,F) is Cp,q approximable on some neighborhood of every point of E, then F is Cp,q approximable on all of E. In general it is unknown whether an arbitrary function in C1(l2, R) is C2,1 approximable and an example of a function in C1(l2, R) which may not be C2,1 approximable is given. A weak form of C∞,q, q≥1, to functions in Cq(l2, R) is proved: Let {Uα} be a locally finite cover of l2 and let {Tα} be a corresponding collection of Hilbert-Schmidt operators on l2. Then for any f ϵ Cq(l2,F) such that for all α sup ‖ Dk(f(x)-g(x))[Tαh]‖ ≤ 1. xϵUα,‖h‖≤1, 0≤k≤q

It is a classical result that every Bade σ-complete Boolean algebra of (selfadjoint) projections in a separable Hilbert space coincides with the projections forming the resolution of the identity of some bounded selfadjoint operator. This result is extended to the setting of separable Frechet spaces. Namely, every Bade σ-complete Boolean algebra of projections in such a space coincides with the resolution of the identity of some (continuous) scalar-type spectral operator having spectrum a compact subset ofℝ.

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.

In this paper, apply an established transference principle to obtain the boundedness of certain functional calculi for the sequence of semigroup generators. It is proved that if be the sequence generates 0- semigroups on a Hilbert space, then for each the sequence of operators has bounded calculus for the closed ideal of bounded holomorphic functions on right half–plane. The bounded of this calculus grows at most logarithmically as. As a consequence decay at ∞. Then showed that each sequence of semigroup generator has a so-called (strong) m-bounded calculus for all m∈ℕ, and that this property characterizes the sequence of semigroup generators. Similar results are obtained if the underlying Banach space is a UMD space. Upon restriction to so-called semigroups, the Hilbert space results actually hold in general Banach spaces.

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.

A complete convergence theorem for arrays of rowwise independent random variables was proved by Sung, Volodin, and Hu [14]. In this paper, we extend this theorem to the Banach space without any geometric assumptions on the underlying Banach space. Our theorem also improves some known results from the literature.

For a compact Hausdorff space with a dense set of isolated points, we give a complete description of points of weak*-norm continuity in the dual unit ball of the space of Banach space valued functions that are continuous when the range has the weak topology. As an application we give a complete description of points of weak-norm continuity of the unit ball of the space of vector measures when the underlying Banach space has the Radon-Nikodym property.

It is an open problem whether an infinite-dimensional amenable Banach algebra exists whose underlying Banach space is reflexive. We give sufficient conditions for a reflexive, amenable Banach algebra to be finite-dimensional (and thus a finite direct sum of full matrix algebras). If $$ {\frak A} $$ is a reflexive, amenable Banach algebra such that for each maximal left ideal L of $$ {\frak A} $$ (i) the quotient $$ {\frak A}/L $$ has the approximation property and (ii) the canonical map from $$ {\frak A} \check{\otimes} L^\perp $ to $({\frak A} / L) \check{\otimes} L^\perp $$ is open, then $$ {\frak A} $$ is finite-dimensional. As an application, we show that, if $${\frak A}$$ is an amenable Banach algebra whose underlying Banach space is an Lp-space with $$ p\in (1,\infty) $$ such that for each maximal left ideal L the quotient $$ {\frak A}/L $$ has the approximation property, then $$ {\frak A} $$ is finite-dimensional.

The Lipschitz Condition Nonlinear spectral radius Uniformly lipschitzian mappings Basic Facts on Banach Spaces Convexity The operator norm Dual spaces, reexivity, the weak, and weak* topologies Mean Lipschitz Condition Nonexpansive and mean nonexpansive mappings in Banach spaces General case On the Lipschitz Constants for Iterates of Mean Lipschitzian Mappings A bound for Lipschitz constants of iterates A bound for the constant k (T) Moving averages in Banach spaces A bound for the constant k0(T) More about k(Tn), k0(T), and k (T) Subclasses Determined by p-Averages Basic definitions and observations A bound for k(Tn), k (T), and k0(T) On the moving p-averages Mean Contractions Classical Banach's contractions On characterizations of contractions On the rate of convergence of iterates Nonexpansive Mappings in Banach Space The asymptotic center technique Minimal invariant sets and normal structure Uniformly nonsquare, uniformly noncreasy, and reflexive Banach spaces Remarks on the stability of f.p.p. The case of l1 Mean Nonexpansive Mappings Some new results of stability type Sequential approximation of fixed points The case of n = 3 On the structure of the fixed points set Mean Lipschitzian Mappings with k > 1 Losing compactness in Brouwer's Fixed Point Theorem Retracting onto balls in Banach spaces Minimal displacement Optimal retractions Generalized characteristics of minimal displacement Bibliography Index

Analysis and Algebra on Differentiable Manifolds

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.

A Banach space is, by its nature, also a metric space. When we identify a Banach space with its underlying metric space, we choose to forget its linear structure. The fundamental question of nonlinear geometry is to determine to what extent the metric structure of a Banach space already determines its linear structure. What can be said of two Banach spaces that are Lipschitz-isomorphic? uniformly homeomorphic? or coarsely Lipschitz-isomorphic in the spirit of M. Gromov’s geometric theory of groups? These questions are still partly open, and investigating them requires some quite technical tools. In this chapter, we will consider only basic techniques whose purpose is to produce specific linear maps from nonlinear ones (typically, Lipschitz maps).

We study some properties of the logconvex quasi-Banach space QA defined by Arias-de-Reyna and show several applications to convergence of Fourier series. In particular, we describe the Banach envelope of QA and prove that there exists a Lorentz space strictly bigger than the Antonov space in which the almost everywhere convergence of the Fourier series holds. We also give a necessary condition for a Banach rearrangement invariant space X to be contained in QA. As an application, we show that for some classes of Banach spaces there is no the largest Banach space in a given class which is contained in QA.

We introduce a new monotone mapping in Banach spaces, which is an extension of the -monotone mapping studied by Nazemi (2012), and we generalize the variational inclusion involving the -monotone mapping. Based on the new monotone mapping, we propose a new proximal mapping which combines the proximal mapping studied by Nazemi (2012) with the mapping studied by Lan et al. (2011) and show its Lipschitz continuity. Based on the new proximal mapping, we give an iterative algorithm. Furthermore, we prove the convergence of iterative sequences generated by the algorithm under some appropriate conditions. Our results improve and extend corresponding ones announced by many others.