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26078 results in Abstract analysis

Page 1158 of 1304

  • PREFACE

  • Ivan N. Erdelyi, Wang Shengwang
  • Book:
    A Local Spectral Theory for Closed Operators
    Published online:
    05 April 2013
    Print publication:
    01 August 1985, pp vii-viii

  • Summary

    This monograph has grown out of a recent sequel of jointly written papers by the authors. It proposes to present a unified treatment of the local spectral theory for closed operators acting on a complex Banach space.

    While working with closed operators, in a few instances it will be unavoidable to transgress the former concept. More to the point, the theory makes frequent use of operators coinduced by closed operators on a quotient space. Such operators, in general, are not even closable. Special efforts will be requested to exhibit conditions which make them, at least, closable and hence useful instruments in the subsequent theory.

    The plan of this work may be sketched as follows. After a brief presentation of the spectral decomposition problem, Chapter I introduces the notion of the single valued extension property with some of its implications. Subsequently, a general study of invariant subspaces is followed by the developing ideas for some special types of subspaces, such as v-spaces, μ-spaces, analytically invariant subspaces, T-absorbent spaces, spectral maximal and T-bounded spectral maximal spaces, T being the given closed operator.

    With Chapter II we come to the essence of the spectral theory: the general type of spectral decomposition property with its relationship to the unbounded decomposable operators.

    Chapter III is devoted to the spectral duality theory. After having overcome some difficulties due to the general type of the spectral decomposition problem, it was gratifying to have obtained a spectral duality theorem, under the natural constraints of the problem.

  • Appendix B - Some open problems

  • Ivan N. Erdelyi, Wang Shengwang
  • Book:
    A Local Spectral Theory for Closed Operators
    Published online:
    05 April 2013
    Print publication:
    01 August 1985, pp 161-161

  • Summary

    In the Notes and Comments on Chapter IV (p.106), a diagram of implications between various types of spectral resolvents is given. Are there more implications? Specifically:

    PROBLEM 1. Is every spectral resolvent of a given operator monotonic? Is every spectral resolvent analytically invariant?

    PROBLEM 2. Is every monotonic spectral resolvent strongly monotonic? Is every monotonic spectral resolvent analytically invariant? Is every almost localized spectral resolvent a strong spectral resolvent? Is every almost localized spectral resolvent analytically invariant?

    In [W.a] it was shown that there exists a complex Banach space X and an operator T ∈ B(X), which is decomposable, the adjoint T* is strongly decomposable but T is not strongly decomposable. This gives rise to

    PROBLEM 3. Does it exist a complex Banach space X and a strongly decomposable T ∈ B(X), with T* decomposable but not strongly decomposable?

    PROBLEM 4. Is it true that T is strongly decomposable iff T** is strongly decomposable?

    In the study of operators with the SDI, Corollary 17.10 asserts that if a closed operator is endowed with the SDI then it has the SSDP. What about the converse?

  • On Hayman's alternative

  • Part of
  • J. K. Langley
  • Journal:
    Mathematika / Volume 32 / Issue 1 / June 1985
    Published online by Cambridge University Press:
    26 February 2010, pp. 139-146
    Print publication:
    June 1985

  • Suppose that f(z) is non-constant and meromorphic in the plane and that, for some k≥= 1, a0(z),…, ak(z) are meromorphic in the plane with

    for j' = 0,…, k. Here, using standard notation from [3], S(r,f) denotes any quantity satisfying S(r,f) = o(T(r,f)) as r→ ∞, possibly outside a set of finite linear measure. Then, setting

    we have ([3, p. 57])

    Theorem A. Suppose that f(z) is non-constant and meromorphic in the plane, and thatψ (z) given by (1.2) and (1.1) and is non-constant. Then

    where N0(r, l/ψ') counts only zeros of ψ' which are not zeros of ψ − 1, and thecounting functions count points without regard to multiplicity.

  • MTK volume 32 issue 1 Cover and Front matter

  • Journal:
    Mathematika / Volume 32 / Issue 1 / June 1985
    Published online by Cambridge University Press:
    26 February 2010, pp. f1-f4
    Print publication:
    June 1985
    • Article
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  • Singularities of bessel-zeta functions and Hawkins' polynomials

  • Kenneth B. Stolarsky
  • Journal:
    Mathematika / Volume 32 / Issue 1 / June 1985
    Published online by Cambridge University Press:
    26 February 2010, pp. 96-103
    Print publication:
    June 1985

  • The asymptotic behaviour of a sequence of polynomials cm = cm(v) satisfying

    is established. These polynomials occur in Hawkins' formula for the residues of a Bessel-zeta function at its possible poles in the left half plane. The results imply that cm(v)/cm(0) converges uniformly to cos πV on compact sets. This in turn implies that, for v not a half odd integer, all but finitely many of the possible poles are actual poles.

  • Zeros of generalized Airy functions

  • P. Baldwin
  • Journal:
    Mathematika / Volume 32 / Issue 1 / June 1985
    Published online by Cambridge University Press:
    26 February 2010, pp. 104-117
    Print publication:
    June 1985

  • Some interlacing properties of the zeros of the generalized Airy functions A1(z, p) are given for non-positive integral values of p. The result that A1 (z,p) has no real zero for is extended to show that all the zeros of A1(z,p) are real and simple if . It is also shown that all the zeros of the functions Bk(z,p, 1) for k = 1, 2, 3 are simple for non-positive integral p.

Page 1158 of 1304