Strongly irreducible operators on Banach spaces

Over the past years, a significant interest has developed in the study of holomorphic functions defined on a domain in an infinite-dimensional Banach space and of their constituents (via Taylor expansions), the homogeneous polynomials. Many of the questions that have been studied have arisen from considerations of infinitedimensional topology and from standard function algebra questions [CCG]. Recently, there has been an interest in connecting the well-developed theory of the geometry of Banach spaces with the function theory questions that have been studied classically, and some progress has been made in this direction [ACG; D; F; CCG; CGJ]. In addition, connections between properties of polynomials and geometry of the unit ball has been of interest (see [GJL] for a survey of this topic). The present work is an attempt to study some of the properties of bounded analytic functions on the unit ball of an infinite-dimensional Banach space. In particular, we are interested in understanding something of boundary behavior; we combine techniques from the several fields to investigate it, especially with regard to the interplay with convexity and smoothness. Many of the results here apply to the classical “nice” reflexive spaces, such as lp and Lp (1 < p <∞). It is almost certain that there is much more to be learned even about the Hilbert space case. We consider the boundary behavior of H∞ functions on B, the open unit ball of an infinite-dimensional complex Banach space that has the geometric properties of uniform convexity, uniform smoothness, or both. By uniform smoothness, we mean uniform (real) Frechet differentiability of the norm, with the space considered as a real Banach space. Uniform convexity will mean that the dual is uniformly smooth; since spaces with either property are reflexive, this definition is complete. To be specific however, we state the following (after [LT]).

The multiplicities of the powers of a bounded linear operator T T , acting on a complex separable infinite-dimensional Banach space X \mathfrak {X} , satisfy the inequalities \[ ( ∗ ∗ ) μ ( T n ) ⩽ μ ( T h n ) ⩽ h μ ( T n ) for all h , n ⩾ 1. ( * * )\qquad \mu ({T^n}) \leqslant \mu ({T^{hn}}) \leqslant h\mu ({T^n})\quad {\text {for}}\;{\text {all}}\;h,n \geqslant 1. \] Nothing else can be said, in general, because simple examples show that for each sequence { μ n } n = 1 ∞ \{ {\mu _n}\} _{n = 1}^\infty , satisfying the inequalities ( ∗ ∗ ) ( * * ) , there exists T T acting on X \mathfrak {X} such that μ ( T n ) = μ n \mu ({T^n}) = {\mu _n} for all n ⩾ 1 n \geqslant 1 .

Two homogeneous measures of noncompactness β and γ on an infinite dimensional Banach space X are called “equivalent” if there exist positive constants b and c such that bβ(S) ≤ γ(S) ≤ cβ(S) for all bounded sets \({S\subset X}\). If such constants do not exist, the measures of noncompactness are “inequivalent.” We ask a foundational question which apparently has not previously been considered: For what infinite dimensional Banach spaces do there exist inequivalent measures of noncompactness on X? We provide here the first examples of inequivalent measures of noncompactness. We prove that such inequivalent measures exist if X is a Hilbert space; or if (Ω, Σ, μ) is a general measure space, 1 ≤ p ≤ ∞, and X = Lp(Ω, Σ, μ); or if K is a compact Hausdorff space and X = C(K); or if K is a compact metric space, 0 < λ ≤ 1, and X = C0,λ(K), the Banach space of Holder continuous functions with Holder exponent λ. We also prove the existence of such inequivalent measures of noncompactness if Ω is an open subset of \({\mathbb{R}^n}\) and X is the Sobolev space Wm,p(Ω). Our motivation comes from questions about existence of eigenvectors of homogeneous, continuous, order-preserving cone maps f : C→C and from the closely related issue of giving the proper definition of the “cone essential spectral radius” of such maps. These questions are considered in the companion paper [28]; see, also, [27].

Introduction. This paper was motivated by a discovery that most finite dimensional subalgebras of (B(H), for H an infinite dimensional Hilbert space, are reflexive, with the only obstructions to reflexivity being finite-rank considerations. Proofs (section 2) do not depend on multiplicative structure, nor on topology, so extend to linear subspaces of transformations in an abstract setting. Abstract reflexivity has been studied in [3, 4, 7], primarily for singly generated algebras. Reflexivity properties can be interpreted as linear interpolation properties, and we shall adopt this point of view. The results of section 2 extend, in section 3, to algebraic reflexivity counterparts for countably generated (algebraically) linear subspaces of bounded linear transformations acting on a Banach space. As consequences, in section 4 we obtain generalizations of two single operator results. One is a multivariate version of the result, due to Kaplansky [10, Theorem 15], that a bounded locally algebraic operator acting on a Banach space is algebraic. The other is a multivariate version of the result, due to Douglas and Foias [5] for Hilbert space and to Hadwin [7] for a general Banach space, extending work of Fillmore [6], that a bounded non-algebraic operator acting on a Banach space is (topologically) algebraically reflexive. In section 5 we give an application, kindly suggested by D. Hadwin, concerning joint strong similarity orbits of n-tupules of operators. We wish to thank E. Azoff, K. Davidson, J. Erdos, D. Hadwin, J. Kraus, C. Lance and A. Sourour for conversations, and some correspon-

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.

Milutin's Theorem states that if X and Y are uncountable metrizable compact Hausdorff spaces, then C(X) and C(Y) are isomorphic as Banach spaces [15, p. 379]. Thus there is only one isomorphism class of such Banach spaces. There is also an extensive theory of the Banach–Mazur distance between various classes of classical Banach spaces with the deepest results depending on probabilistic and asymptotic estimates [18]. Lindenstrauss, Haagerup and possibly others know that as Banach spaceswhere H is the infinite dimensional separable Hilbert space, R is the injective II 1-factor on H, and ≈ denotes Banach space isomorphism. Haagerup informed us of this result, and suggested considering completely bounded isomorphisms; it is a pleasure to acknowledge his suggestion. We replace Banach space isomorphisms by completely bounded isomorphisms that preserve the linear structure and involution, but not the product. One of the two theorems of this paper is a strengthened version of the above result: if N is an injective von Neumann algebra with separable predual and not finite type I of bounded degree, then N is completely boundedly isomorphic to B(H). The methods used are similar to those in Banach space theory with complete boundedness needing a little care at various points in the argument. Extensive use is made of the conditional expectation available for injective algebras, and the methods do not apply to the interesting problems of completely bounded isomorphisms of non-injective von Neumann algebras (see [4] for a study of the completely bounded approximation property).

This work studies the structural properties and convergence approach of chance-constrained optimization of boundary-value elliptic partial differential equation systems (CCPDEs). The boundary conditions are random input functions deliberated from the boundary of the partial differential equation (PDE) system and in the infinite-dimensional reflexive and separable Banach space. The structural properties of the chance constraints studied in this paper are continuity, closedness, compactness, convexity, and smoothness of probabilistic uniform or pointwise state constrained functions and their parametric approximations. These are open issues even in the finite-dimensional Banach space. Thus, it needs finite-dimensional and smooth parametric approximation representations. We propose a convex approximation approach to nonconvex CCPDE problems. When the approximation parameter goes to zero from the right, the solutions of the relaxation and compression approximations converge asymptotically to the optimal solution of the original CCPDE. Due to the convexity of the problem, a global solution exists for the proposed approximations. Numerical results are provided to demonstrate the plausibility and applicability of the proposed approach.

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.

Unconditional Basis and Gordon–Lewis Constants for Spaces of Polynomials

In the paper we establish some conditions under which a given sequence of polynomials on a Banach space X supports entire functions of unbounded type, and construct some counter examples. We show that if X is an infinite dimensional Banach space, then the set of entire functions of unbounded type can be represented as a union of infinite dimensional linear subspaces (without the origin). Moreover, we show that for some cases, the set of entire functions of unbounded type generated by a given sequence of polynomials contains an infinite dimensional algebra (without the origin). Some applications for symmetric analytic functions on Banach spaces are obtained.

SOME RESULTS RELATED TO THE SEPARABLE QUOTIENT PROBLEM

Asymptotically isometric copies of c0 and l1 in certain Banach spaces

Retracting a ball onto a sphere in some Banach spaces

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

The celebrated result of Lomonosov [6] on the existence of invariant subspaces for operators commuting with a compact operator have been generalized in different directions (for example see [2], [7], [8], [9]). The main result of [9] (see also [7]) is: If is a norm closed algebra of (bounded) operators on an infinite dimensional (complex) Banach space , if K is a nonzero compact operator on , and if then has a non-trivial (closed) invariant subspace. In [7], it is mentioned that the above result holds if instead of compactness for K we assume that K is a non-invertible injective operator with a non-zero eigenvalue belonging to the class of decomposable, hyponormal, or subspectral operators.