1,736,831 research outputs found
Optimal control of stochastic partial differential equations in Banach spaces
Get PDFIn 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
An amalgamation of the Banach spaces associated with James and Schreier, Part I : Banach-space structure.
Get PDFWe 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
Reverses of the triangle inequality in Banach spaces
No full textRecent reverses for the discrete generalised triangle inequality and its continuous version for vector-valued integrals in Banach spaces are surveyed. New results are also obtained. Particular instances of interest in Hilbert spaces and for complex numbers and functions are pointed out as well
Maximal ideals in the algebra of operators on certain Banach spaces.
Get PDFFor 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
Several Inequalities of Frechet Spaces
No full textSeveral inequalities of Frechet spaces are given in this paper, which
can be regarded as the Frechet spaces versions of the well-known polarization
identity occurring in Hilbert spaces. Our results generalize many inequalities
in Banach spaces. The inequalities developed here have various applications
in a number of fields. By using these inequalities, many recent results can be
easily generalized from Banach spaces to Frechet spaces
Genericity and amalgamation of classes of Banach spaces
No full textAbstractWe study universality problems in Banach space theory. We show that if A is an analytic class, in the Effros–Borel structure of subspaces of C([0,1]), of non-universal separable Banach spaces, then there exists a non-universal separable Banach space Y, with a Schauder basis, that contains isomorphs of each member of A with the bounded approximation property. The proof is based on the amalgamation technique of a class C of separable Banach spaces, introduced in the paper. We show, among others, that there exists a separable Banach space R not containing L1(0,1) such that the indices β and rND are unbounded on the set of Baire-1 elements of the ball of the double dual R∗∗ of R. This answers two questions of H.P. Rosenthal.We also introduce the concept of a strongly bounded class of separable Banach spaces. A class C of separable Banach spaces is strongly bounded if for every analytic subset A of C there exists Y∈C that contains all members of A up to isomorphism. We show that several natural classes of separable Banach spaces are strongly bounded, among them the class of non-universal spaces with a Schauder basis, the class of reflexive spaces with a Schauder basis, the class of spaces with a shrinking Schauder basis and the class of spaces with Schauder basis not containing a minimal Banach space X
On the existence of representer theorems in Banach spaces
Get PDFWe consider general regularisation and regularised interpolation problems for learning parameter vectors from data. In particular in Hilbert spaces regularisation methods have been applied very successfully, largely due to the well known representer theorem. Classical formulations of the theorem state that under certain conditions on the regulariser there exists a solution of the optimisation problem which is contained in a linear subspace spanned by the data points. This is at the core of kernel methods in machine learning as it significantly reduces the dimensionality and thus makes the problem computationally tractable. Most of the literature only deals with sufficient conditions on the regulariser for a representer theorem to hold, mostly in Hilbert spaces with some generalisations to certain classes of Banach spaces. In this work we give an essentially complete answer to the question of existence of representer theorems in general Banach spaces. This question had previously been answered for Hilbert spaces with an intuitive characterisation for differentiable regularisers. We show how the necessary and sufficient conditions extend to arbitrary Banach spaces and give the more intuitive geometric characterisation for a variety of classes of Banach spaces, which contain all spaces we know of which are commonly used in applications. We conjecture that the same characterisation can also be given for any other Banach space not currently covered by those classes. We further show that, if the learning relies on the linear representer theorem, in most cases the solution is actually independent of the regulariser and determined by the function space alone. This is interesting for two reasons. Firstly it means one is free to choose whichever regulariser is most suitable for the application at hand, whether this is computational efficiency or ease of calculations. Moreover it shows the importance of extending classical elements of learning theory such as kernel methods from Hilbert spaces to Banach spaces
Admissibility and Non-Uniform Dichotomy for Differential Systems
No full textThe problem of nonuniform exponential dichotomy of linear differential
systems in Banach spaces is discussed. It is established a connection
between the admissibility of a pair of certain function spaces which are translations
invariant, on one hand, and the nonuniform exponential dichotomy
of differential systems, on the other. Also, Some results due to Hartman,
Massera, Schäffer and Coppel are generalized as well
Banach spaces with few operators and multiplier results
Get PDFThe construction of a non-separable reflexive Banach space on which every operator is the sum of a scalar multiple of the identity operator and an operator of separable range is presented. Using a result of Rao, a sufficient condition is given for Banach spaces with smooth norms to be decomposable. It is shown that operators on Banach spaces of co-dimension one in their biduals are the sum of a scalar multiple of the identity operator and a weakly compact operator. The Banach spaces of bounded operators L(11, 1p) (1p, 1r), 1 < p ≤ r ≤ p1 < ꝏ, where 1/p + 1/p1 = 1, are shown to be primary. The spaces of bounded diagonal operators and compact diagonal operators on a seminormalized Schauder basis β, the multiplier algebras Ld(X, β) and Kd(X, β), are introduced and studied. New examples of these multiplier algebras are presented and a theorem of Sersouri is extended. A necessary and sufficient condition is given for co to embed in Kd(X, β). A sufficient condition is given on a semi-normalized Schauder basis β of a reflexive hereditarily indecomposable Banach space Y to ensure that Kd(Y, β) has the RNP. It is shown that the algebra Ld(X, β) is semisimple and that on the algebra Kd(X, β) derivations are automatically continuous. By representing diagonal operators as stochastic processes a general method of constructing multiplier algebras is given. A non trivial multiplier invariance for the normalized Haar basis of L1[0,1] is proved
A Variational Characterization of Reflexivity and Strict Convexity
No full textIn this paper we give a variational characterization of reflexivity and strict convexity which is related to James and Krein theorems in Geometry of Banach spaces
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