Graduate What Is the Spectrum of a Linear Operator in L2 Spaces?

[maxandri]

http://<img 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] 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

 

[Ssnow]

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 $t$?

 

[S.G. Janssens]

http://<img 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] 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

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.