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A Spectrum of linear operator

  1. Aug 10, 2016 #1
    http://<img [Broken] src="https://latex.codecogs.com/gif.latex?L_{2}[0,1]->L_{2}[0,1]\int_{0}^{1}\left&space;|t-s&space;\right&space;|f(s)ds" title="L_{2}[0,1]->L_{2}[0,1]\int_{0}^{1}\left |t-s \right |f(s)ds" />[/PLAIN] [Broken] I have many doubts on linear operator. How I can find a spectrum of a linear operator? For example:
    $$L_{2}[0,1]->L_{2}[0,1]\int_{0}^{1}\left |t-s \right |f(s)ds$$
    ?? Thank you
     
    Last edited by a moderator: May 8, 2017
  2. jcsd
  3. Aug 10, 2016 #2

    Ssnow

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    Gold Member

    Hi, I don't understand the form of this operator, he takes ##f\in L^{2}([0,1])## and assign ##\int_{0}^{1}|t-s|f(s)ds## ? What is ##t##? Is this operator depending by ##t##?
     
  4. Aug 10, 2016 #3

    Krylov

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    Science Advisor
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    I suppose the operator is ##T : L^2(0,1) \to L^2(0,1)## defined by
    $$
    (Tf)(t) := \int_0^1{|t-s|f(s)\,ds} \qquad \forall\,t \in [0,1]
    $$
    This is a Fredholm integral operator and the archetypical example of a compact self-adjoint operator. Its spectrum ##\sigma(T)## consists of isolated eigenvalues of finite algebraic multiplicity, possibly accumulating at ##0 \in \sigma(T)##. It can be nicely approximated by operators of finite rank.

    You can find a treatment in most introductory functional analysis books. In case you fancy a recommendation, there is for example the book "Basic Classes of Linear Operators" by Gohberg, Goldberg and Kaashoek. It strikes a good balance between theoretical development and computation and it is easy to read. For this particular operator, have a look at Chapter V.
     
    Last edited by a moderator: May 8, 2017
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