183,270 research outputs found
An amalgamation of the Banach spaces associated with James and Schreier, Part I : Banach-space structure.
Full text linkWe create a new family of Banach spaces, the James-Schreier spaces, by amalgamating two important classical Banach spaces: James' quasi-reflexive Banach space on the one hand and Schreier's Banach space giving a counterexample to the Banach-Saks property on the other. We then investigate the properties of these James-Schreier spaces, paying particular attention to how key properties of their `ancestors' (that is, the James space and the Schreier space) are expressed in them. Our main results include that each James-Schreier space is c_0-saturated and that no James-Schreier space embeds in a Banach space with an unconditional basis
Maximal ideals in the algebra of operators on certain Banach spaces.
Full text linkFor a Banach space , let denote the Banach algebra of all continuous linear operators on . First, we study the lattice of closed ideals in , where and is the th James space. Our main result is that the ideal of weakly compact operators is the unique maximal ideal in . Applications of this result include the following. (i) The Brown–McCoy radical of , which by definition is the intersection of all maximal ideals in , cannot be turned into an operator ideal. This implies that there is no ‘Brown–McCoy’ analogue of Pietsch’s construction of the operator ideal of inessential operators from the Jacobson radical of . (ii) For each natural number and each -tuple in , there is a Banach space such that has exactly maximal ideals, and these maximal ideals have codimensions in , respectively; the Banach space is a finite direct sum of James spaces and -spaces. Second, building on the work of Gowers and Maurey, we obtain further examples of Banach spaces such that all the maximal ideals in can be classified. We show that the ideal of strictly singular operators is the unique maximal ideal in for each hereditarily indecomposable Banach space , and we prove that there are distinct maximal ideals in , where is the Banach space constructed by Gowers to solve Banach’s hyperplane problem
Composition mechanisms for retrenchment
No full textRetrenchment is a flexible model evolution formalism that arose as a reaction to the limitations imposed by refinement, and for which the proof obligations feature additional predicates for accommodating design data. Composition mechanisms for retrenchment are studied. Vertical, horizontal, dataflow, parallel and fusion compositions are described. Of particular note are the means by which the additional predicates compose. It is argued that all of the compositions introduced are associative, and that they are mutually coherent. Composition of retrenchment with refinement, so important for the smooth interworking of the two techniques, is discussed. Decomposition, allowing finer grained retrenchments to be extracted from a single large grained retrenchment, is also investigated
Optimal control of stochastic partial differential equations in Banach spaces
Full text linkIn this thesis we study optimal control problems in Banach spaces for stochastic partial differential equations. We investigate two different approaches. In the first part we study Hamilton-Jacobi-Bellman equations (HJB) in Banach spaces associated with optimal feedback control of a class of non-autonomous semilinear stochastic evolution equations driven by additive noise. We prove the existence and uniqueness of mild solutions to HJB equations using the smoothing property of the transition evolution operator associated with the linearized stochastic equation. In the second part we study an optimal relaxed control problem for a class of autonomous semilinear stochastic stochastic PDEs on Banach spaces driven by multiplicative noise. The state equation is controlled through the nonlinear part of the drift coefficient and satisfies a dissipative-type condition with respect to the state variable. The main tools of our study are the factorization method for stochastic convolutions in UMD type-2 Banach spaces and certain compactness properties of the factorization operator and of the class of Young measures on Suslin metrisable control sets
Generators of maximal left ideals in Banach algebras
Full text linkIn 1971, Grauert and Remmert proved that a commutative, complex, Noetherian Banach algebra is necessarily finite-dimensional. More precisely, they proved that a commutative, complex Banach algebra has finite dimension over C whenever all the closed ideals in the algebra are (algebraically) finitely generated. In 1974, Sinclair and Tullo obtained a non-commutative version of this result. In 1978, Ferreira and Tomassini improved the result of Grauert and Remmert by showing that the statement is also true if one replaces `closed ideals' by `maximal ideals in the Shilov boundary of A'. We give a shorter proof of this latter result, together with some extensions and related examples. We study the following conjecture. Suppose that all maximal left ideals in a unital Banach algebra A are finitely generated. Then A is finite-dimensional
On -reflexive Banach spaces
Full text linksummary:A Banach space is called {\it -reflexive\/} if for any cover \Cal U of by weakly open sets there is a finite subfamily \Cal V\subset\Cal U covering some ball of radius 1 centered at a point with . We prove that an infinite-dimensional separable Banach space is -reflexive (-reflexive for some ) if and only if each -net for has an accumulation point (resp., contains a non-trivial convergent sequence) in the weak topology of . We show that the quasireflexive James space is -reflexive for no . We do not know if each -reflexive Banach space is reflexive, but we prove that each separable -reflexive Banach space has Asplund dual. As a by-product of the proof we obtain a covering characterization of the Asplund property of Banach spaces
Ornstein-Uhlenbeck processes in Banach spaces and their spectral representations.
Full text linkFor Q the variance of some centred Gaussian random vector in a separable Banach space it is shown that, necessarily, Q factors through as a product of 2-summing operators. This factorization condition is sufficient when the Banach space is of Gaussian type 2. The stochastic integral of a deterministic family of operators with respect to a Q-Wiener process is shown to exist under a continuity condition involving the 2-summing norm. A Langevin equation \rd\bm{Z}_t+\sLa\bm{Z}_t\,\rd t=\rd\bm{B}_t, with values in a separable Banach space, is studied. The operator \sLa is closed and densely defined. A weak solution , where is centred, Gaussian and stationary, while is a Q-Wiener process, is given when \ri\sLa and \ri\sLa^* generate groups and the resolvent of \sLa is uniformly bounded on the imaginary axis. Both and are stochastic integrals with respect to a spectral Q-Wiener process. AMS 2000 Mathematics subject classification: Primary 60G15. Secondary 46E40; 47B10; 47D03; 60H1
Elementary evolutions in Banach algebra
Full text linkAn elementary class of evolutions in unital Banach algebras is obtained by integrating product functions over Guichardet's symmetric measure space on the half-line. These evolutions, along with a useful subclass, are characterised and a Lie-Trotter product formula is proved. The class is rich enough to form the basis for a recent approach to quantum stochastic evolutions
Equilibria in reflexive Banach lattices with a continuum of agents.
Full text linkWe consider exchange economies with a measure space of agents and for which the commodity space is a separable and reflexive Banach lattice. Under assumptions imposing uniform bounds on marginal rates of substitution, positive results on core-Walras equivalence were established in Rustichini-Yannelis [27] and Podczeck [25]. In this paper we prove that under similar assumptions on marginal rates of substitution, the set of competitive equilibria (and thus the core) is non-empty.Competitive equilibria; Continuum of agents; Reflexive Banach lattice commodity spaces; Uniform properness;
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