Functional Calculus for the Series of Semigroup Generators Via Transference

Let –iA be the generator of a C 0-group \({(U(s))_{s\in \mathbb {R}}}\) on a Banach space X and ω > θ(U), the group type of U. We prove a transference principle that allows to estimate \({||{f(A)}||}\) in terms of the \({{\rm{L}}^{p}(\mathbb {R};X)}\)-Fourier multiplier norm of \({f(\cdot \pm i \omega)}\). If X is a Hilbert space this yields new proofs of important results of McIntosh and Boyadzhiev–de Laubenfels. If X is a UMD space, one obtains a bounded \({{\rm{H}}^\infty_1}\)-calculus of A on horizontal strips. Related results for sectorial and parabola-type operators follow. Finally it is proved that each generator of a cosine function on a UMD space has bounded \({{\rm{H}}^\infty}\)-calculus on sectors.

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 nuclear operator T on a Banach space is an operator admitting a representation T = (SIGMA) T(,n) where each operator T(,n) has rank one and (SIGMA) (VERT.BAR)(VERT.BAR)(TAU)(,n)(VBAR)(VBAR) < (INFIN). A. Grothendieck proved that the sequence of eigenvalues, repeated according to their multiplicities, of such a nuclear operator must be square summable and that, in fact, two is the best possible exponent of summability. This work uses operators of diagonal type, i.e., operators which admit a diagonal representation with respect to a biorthogonal system, to construct examples of eigenvalue behavior. In particular we show that all of the known results concerning eigenvalue summability of nuclear operators can be obtained using operators of diagonal type. In Chapter 1, we present the terminology and notation to be used in this work. Chapter 2 gives a summary of results concerning nuclear (trace class) operators on Hilbert space. In Chapter 3, some of the important known results concerning eigenvalue behavior of nuclear operators on general Banach spaces are presented. The convolution operators on L(,p)(0,1), 1 < p < (INFIN), p (NOT=) 2, are diagonal operators since the trigonometric system (e('2(pi)inx)) is a conditional basis of these spaces. In Chapter 4 we use these important operators to exhibit extremal behavior of eigenvalue summability of nuclear operators on the L(,p)-spaces. In Chapter 5, we study nuclear cyclic diagonal operators. If X(,1),... ,X(,n) are sequence spaces, an operator D on (CRPLUS)X(,i) is a cyclic diagonal operator if its restriction to each X(,i) is a diagonal operator into X(,i+1) (if i = n into X(,1)). Certain eigenspaces of these operators have important applications to Banach space theory. Finally, in Chapter 6, we use a recently discovered space of G. Pisier to answer affirmatively a long outstanding question of. Pe l cynski and Saphar which is converse to Grothendieck's result: Given a (nonzero) sequence ((lamda)(,n)) in l(,2), is there a Banach space X and a nuclear operator on X whose eigenvalue sequence is ((lamda)(,n))?

Abstract. The necessary and sufficient conditions for stability of abstract difference schemes in Hilbert and Banach spaces are formulated. Contrary to known stability results we give stability conditions for schemes with non-self-adjoint operator coefficients in a Hilbert space and with strongly positive operator coefficients in a Banach space. It is shown that the parameters of the sectorial spectral domain play the crucial role. As an application we consider the Richardson iteration scheme for an operator equation in a Banach space, in particulary the Richardson iteration with precondition for a finite element scheme for a non-selfadjoint operator. The theoretical results are also the basis when using the regularization principle to construct stable difference schemes. For this aim we start from some simple scheme (even unstable) and derive stable schemes by perturbing the initial operator coefficients and by taking into account the stability conditions. Our approach is also valid for schemes with unbounded operator coefficients.

Characterizations of the metric and generalized metric projections on subspaces of Banach 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.

We prove a version of Carleson's Theorem in the Walsh model for vector-valued functions: For $1<p< \infty$, and a UMD space $Y$, the Walsh-Fourier series of $f \in L ^{p}(0,1;Y)$ converges pointwise, provided that $Y$ is a complex interpolation space $Y=[X,H]_\theta$ between another UMD space $X$ and a Hilbert space $H$, for some $\theta\in(0,1)$. Apparently, all known examples of UMD spaces satisfy this condition.

Bases play a prominent role in the analysis of vector spaces, as well in the finite-dimensional as in the infinite-dimensional case. The idea is the same in both cases, namely to consider a family of elements such that all vectors in the considered space can be expressed in a unique way as a linear combination of these elements. In the infinite-dimensional case the situation is complicated: we are forced to work with infinite series, and different concepts of a basis are possible, depending on how we want the series to converge. For example, are we asking for the series to converge with respect to a fixed order of the elements (conditional convergence) or do we want it to converge regardless of how the elements are ordered (unconditional convergence)? We define the relevant types of bases in general Banach spaces in Section 3.1, but besides this we mainly consider Hilbert spaces. In Section 3.4 we discuss the most important properties of orthonormal bases in Hilbert spaces; we expect the reader to have some basic knowledge about this subject. A slight (but useful) modification leads to the definition of Riesz bases, which are treated in detail in Section 3.6. Orthonormal bases and Riesz bases satisfy the so-called Bessel inequality, which is the key to the observation that they deliver unconditionally convergent expansions and can be ordered in an arbitrary way. Sequences satisfying the Bessel inequality are therefore discussed already in Section 3.2.

This chapter develops the beginnings of abstract functional analysis, a subject designed to study properties of functions by treating the functions as the members of a space and formulating the properties as properties of the space. Section 1 defines Banach spaces as complete normed linear spaces and gives a number of examples of these. The space of bounded linear operators from one normed linear space to another is a normed linear space, and it is a Banach space if the range is a Banach space. Sections 2–3 concern Hilbert spaces. These are Banach spaces whose norms are induced by inner products. Section 2 shows that closed vector subspaces of such a space have orthogonal complements, and it shows the role of orthonormal bases for such a space. Section 3 concentrates on bounded linear operators from a Hilbert space to itself and constructs the adjoint of each such operator. Sections 4–6 prove the three main abstract theorems about the norm topology of general normed linear spaces—the Hahn–Banach Theorem, the Uniform Boundedness or Banach–Steinhaus Theorem, and the Interior Mapping Principle. A number of consequences of these theorems are given. The second and third of the theorems require some hypothesis of completeness. The topic of Hilbert and Banach spaces continues in Chapter IV of the companion volume, Advanced Real Analysis.

0. Introduction. Many features of the theory of commutative W*-algebras in Hilbert space have been shown to extend to appropriate operator algebras in Banach spaces (cf. [2; 3; 7]). Recently an important gap in the theory was closed by Dieudonne [4] who obtained a multiplicity theory in the case that the adjoint X* of the underlying Banach space X (and hence also X) was separable. In this paper a new approach is developed which does not require assumptions of separability and allows us to construct a multiplicity theory also for the algebra of adjoints in X* where the relevant topology is the weak star or %-topology rather than the norm topology. Of the various treatments of multiplicity theory in Hilbert space our development is perhaps closest to that of Kelley [10]. The analogue of a commutative W*-algebra in a Banach space X is the operator algebra a generated in the uniform topology by a complete Boolean algebra 23 of projections in X. For given xeX, the cyclic subspace determined by x is the manifold 9Y(x) =cdm {Ax ACI }. A multiplicity function for f is a function m on e to cardinal numbers, the value m(E) being roughly the smallest number of cyclic subspaces required to span the range of E. The aim of multiplicity theory in this setting is to decompose the identity I in S into the sum of disjoint projections E of uniform multiplicity, i.e. projections such that m(F) =m(E), whenever 0 5 F<E. In ?2 such a decomposition theorem is proved to hold for any function on an abstract B.A. to cardinals which satisfies an appropriate condition of order continuity. Multiplicity functions are constructed respectively for Q3 in X anid Q3* in X* in ??3 and 6 and the cointiniuity condition is verified. Following a detailed analysis of the range of a projectioni of finite uniiform multiplicity, one of the main results of the paper (Corollary 8.5) is obtainied: A projection E in 23 has finite multiplicity n if anid oinly if its adjoint E* in Q* has multiplicity n. We have been unable to decide whether this result remains true for projections of infinite multiplicity. Finally an application is made to the theory of spectral operators of scalar tvpe dleveloped by Dunford [7; 8]. If the resolution of the identity E(.) of

Let \{a_k\}_{k\geq0} be a sequence of complex numbers. We obtain the necessary and sufficient conditions for the convergence of n^{-1}\sum_{k=0}^na_kT^kx for every contraction T on a Hilbert space H and every x \in H. It is shown that a natural strengthening of the conditions does not yield convergence for all weakly almost periodic operators in Banach spaces, and the relations between the conditions are exhibited. For a strictly increasing sequence of positive integers \{k_j\}, we study the problem of when n^{-1}\sum_{j=1}^nT^{k_j}x converges to a T-fixed point for every weakly almost periodic T or for every contraction in a Hilbert space and not for every weakly almost periodic operator.

Many of the fundamental research problems in the geometry of normed linear spaces can be loosely phrased as: Given a Banach space X and a class of Banach spaces Y does X contain a subspace Y ∈ Y? As a Banach space X is determined by its unit ball B x ≡ { x ∈ X :‖ x ‖ ≤ 1 } the problem can be rephrased in terms of the geometry of convex sets: Can a given unit ball B x be sliced with a subspace to obtain a set in some given class of unit balls? A result of this type is the famous theorem of Dvoretzky (see also [L], [M6], [M4], [MS], [FLM]).

We analyze smooth nonlinear mappings for Hilbert and Banach spaces that carry small balls to convex sets, provided that the radii of the balls are small enough. We focus on the study of new and mildly sufficient conditions for the nonlinear mapping of Hilbert and Banach spaces to be locally convex, and address a suitably reformulated local convexity problem analyzed within the Leray–Schauder homotopy method approach for Hilbert spaces, and within the Lipschitz smoothness condition for both Hilbert and Banach spaces. Some of the results presented in this work prove to be interesting and novel, even for finite-dimensional problems. Open problems related to the local convexity property for nonlinear mappings of Banach spaces are also formulated.

We are concerned with the algebraic representation of the spectral maximal spaces for certain classes of decomposable operators on Banach spaces. The main emphasis will be on those operators which admit a functional calculus on a suitable algebra of functions such as, for instance, differentiable functions, Lipschitz functions, or functions with absolutely convergent Fourier series. Using appropriate partitions of unity, we shall give a unified approach to various previous results in this area as well as to certain extensions and variants thereof. In the case of spectral operators, we shall obtain the optimal results without any assumption on the underlying Banach space or the Boolean algebra of projections corresponding to the spectral measure.

Introduction. Let G be a locally compact topological group which is unimodular, that is, the left and right Haar measures coincide. We propose, in this and subsequent papers, to investigate the structure of its L2-system. By the we mean the (complex) Hilbert space formed from the complex-valued measurable functions f on G for which If| 2 is integrable with respect to the Haar measure, and with a product defined in terms of convolution. This product is not always defined (for that reason we call it an L2-system rather than an L2-algebra) but there are dense subsets (for example, L,nL2) on which multiplication always is defined so it is almost an algebra. Our theorems depend only on certain key properties of these L2-systems so we shall investigate general systems with these properties, calling them H-systems. We do this not only for generality but because when decomposing an L2-system into minimal parts we expect those parts to be H-systems though they need not be the L2-system of any group. This paper contains the definitions of an H-system and an L2-system; the proof that an L2-system is an H-system; theorems on idempotents and generation of ideals by idempotents; a functional calculus of self-adjoint elements in an H-system; a simple structure theorem for abelian H-systems; and an example of the L2-system of a special group, which has an interesting structure. Using the theorems of this paper we believe we shall be able to obtain the complete structure of H-systems--the ultimate theorems presumably asserting that every such system is a direct integral of simple systems, and that every simple system is a certain type of full matrix systembut with continuous rather than discrete matrices. At present these theorems are not proved but before even considering them carefully it seems necessary to develop the material of this paper as a foundation. One might ask why we choose to investigate this L2-system, in which multiplication is only partially defined, when there are other systems (for example, L1-the integrable functions) which are actually algebras and whose structure is still unknown. Our belief is that in spite of having only a partially defined multiplication our system may be easier to study, in somewhat the same way that certain questions are easier to answer about unbounded hypermaximal operators on Hilbert space than about bounded operators on a general Banach space. That there is a relation between that situation and ours is shown by the fact that left (or right) multiplication by a fixed element in an L2-system is, for many choices of the element, an unbounded hyper-