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25994 results in Abstract analysis

Page 825 of 1300

  • On a coagulation and fragmentation equation with mass loss

  • Jacek Banasiak, Wilson Lamb
  • Journal:
    Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 136 / Issue 6 / December 2006
    Published online by Cambridge University Press:
    12 July 2007, pp. 1157-1173
    Print publication:
    December 2006

  • A nonlinear integro-differential equation that models a coagulation and multiple fragmentation process in which continuous and discrete fragmentation mass loss can occur is examined using the theory of strongly continuous semigroups of operators. Under the assumptions that the coagulation kernel is constant, the fragmentation-rate function is linearly bounded, and the continuous mass-loss-rate function is locally Lipschitz, global existence and uniqueness of solutions that lose mass in accordance with the model are established. In the case when no coagulation is present and the fragmentation process is binary with constant fragmentation kernel and constant continuous mass loss, an explicit formula is given for the associated substochastic semigroup.

  • The category of exact sequences of Banach spaces

  • Edited by Jesus M. F. Castillo, Universidad de Extremadura, Spain, William B. Johnson, Texas A & M University
  • Book:
    Methods in Banach Space Theory
    Published online:
    04 May 2010
    Print publication:
    30 November 2006, pp 139-158

  • Summary

    To Atenea.

    Consider nature's magnificent foresight

    spreading seeds of madness everywhere.

    If mortals would refrain from no matter which contact

    with wisdom, even the old age would not exist.

    Life is not different from a dreaming game

    whose greatest gifts come to us through craziness.

    Consider nature's magnificent foresight

    in making the heart be always right.

    The purpose of this paper is to lay the foundations for the construction of the category of exact sequences of Banach spaces; the construction for quasi-Banach spaces is analogous and thus we omit it. The construction of a category associated to a theory, in addition to its intrinsic value, provides the right context to study, among others, isomorphic and universal objects. In our particular case, let us describe a couple of phenomena often encountered when working with exact sequences of Banach spaces for which the categorical approach provides rigorous explanations.

    If one “multiplies” an exact sequence 0 → Y → X → Z → 0 by the left (resp. right) by a given space E, the resulting exact sequence 0 → E ⊕ Y → E ⊕ X → Z → 0 (resp. 0 → Y → X ⊕ E → Z ⊕ E → 0) is “the same”. And this holds despite the fact that the original and the “multiplied” sequences are not equivalent under any known definition. The categorical approach provides the simplest explanation: the two sequences are isomorphic objects in the category.

  • Extension problems for C(K) spaces and twisted sums

  • Edited by Jesus M. F. Castillo, Universidad de Extremadura, Spain, William B. Johnson, Texas A & M University
  • Book:
    Methods in Banach Space Theory
    Published online:
    04 May 2010
    Print publication:
    30 November 2006, pp 159-168

  • Summary

    INTRODUCTION

    This article can be regarded as an update on the handbook article by Zippin [27]. In this article Zippin drew attention to problems surrounding extensions of linear operators with values in C(K)-spaces. The literature on this subject may be said to start with the work of Nachbin, Goodner and Kelley on the case when K is extremally disconnected around 1950. Thus the subject is over fifty years old, but it still seems that comparatively little is known in the general case. We are particularly interested in extending operators on separable Banach spaces when we can assume the range is C(K) for K a compact metric space. In this article we will sketch some recent progress on these problems.

    LINEAR EXTENSION PROBLEMS

    It is, by now, a very classical result that a Banach space X is 1-injective if and only if X is isometric to a space C(K) where K is extremally discon- nected; this is due to Nachbin, Goodner and Kelley [22], [11] and [17]. For a general compact Hausdorff space K the space C(K) is usually not injective (and, in particular, never if K is metrizable). However it is a rather interesting question to determine conditions when linear operators into arbitrary C(K)-spaces can be extended. This problem was first considered in depth by Lindenstrauss in 1964 [18].

    Let us introduce some notation. Suppose X is a Banach space and E is a closed subspace. Then, for λ ≥ 1; we will say that the pair (E;X) has the (λ, C)-extension property if whenever T0 : E → C(K) is a bounded operator then there is an extension T : X → C(K) with ||T|| ≤ λ ||T0||.

Page 825 of 1300