Simple example of operator spectral analysis.

Question

I'm studying on Rudin - Functional Analysis the part related to Banach algebra Bounded/Unbounded operator. Specifically i've studied the part of Banach algebra (where the concept of spectrum is actually mentioned...) probably in the next parts will be some more technical detail... However i had a glance to the chapter i haven't studied yet and i haven't seen a specific example of spectral analysis (the book is quite abstract, like mostly of them focused on the subject). Can you provide me an example of how a spectral analysis is performed?

At present more than the isomorphism between banach algebra and $\mathbb{C}$ (under some condition of course) i haven't seen that much... since i'm looking forward for the stuff i mentioned can you give me an example?

Answer 1

Not sure what you mean by "performing a spectral analysis", but here are some examples that might help you understand the ideas involved:

1. If $T$ is an operator on a finite dimensional Hilbert space, then its spectrum is simply the collection of all eigen-values

2. If $T: C[0,1] \to C[0,1]$ is the operator $$ T(f)(x) = \int_0^x f(t)dt $$ Then $T$ is injective, but not surjective (use the fundamental theorem of calculus to check this). Hence, $0$ is a spectral value, but not an eigen-value.

3. If $f \in C[0,1]$ (treated as a Banach algebra with the sup norm), then $$ \sigma(f) = f([0,1]) $$ because a function $g\in C[0,1]$ is invertible in $C[0,1]$ if and only if $g(t) \neq 0$ for all $t\in [0,1]$

4. If $x:= (\lambda_n) \in \ell^{\infty}$ (once again, a Banach algebra with the sup-norm and point-wise multiplication), then $$ \sigma(x) = \overline{\{\lambda_n\}} $$ (ie. the closure of the set $\{\lambda_n\} \subset \mathbb{C}$) Try proving this, and you should get the hang of how much of this works.

Answer 2

The motivation for Spectral Theory came from Fourier's separation of variables technique that he invented to solve his heat equation. There was one notable example in particular that Historians say triggered the birth of Spectral Theory. In this example, Fourier was looking at the accuracy of a thermometer and, in particular, the "cooling off" problem for a sphere, where the heat in the spherical end would satisfy $$ \frac{\partial v}{\partial t} = k\left(\frac{\partial^{2}v}{\partial x^{2}}+\frac{2}{x}\frac{\partial v}{\partial x}\right) $$ with "boundary conditions" that $v(x,t)$ must remain finite when the distance from the center $x$ tends to $0$, and must satisfy the following conduction condition at the boundary $x=r$ of the sphere: $$ \frac{\partial v}{\partial x}+hv = 0,\;\;\; x=r,\; t \ge 0. $$ Both $h$ and $k$ are physical constants.

Using his separation of variables, Fourier obtained solutions of the form $$ u(x,t) = \frac{1}{x}e^{-k\lambda^{2}t}\sin(\lambda x), $$ where the parameter $\lambda$ may be any solution of the transcendental equation $$ \frac{\lambda r}{\tan(\lambda r)}=1-hr. $$ Fourier easily proves that the equation has an infinity of real solutions $\lambda_n$ tending to $+\infty$. To obtain a solution of the cooling problem with boundary condition, and such that the initial temperature distribution $u(x,0)$ is a given function $f(x)$ at $t=0$, he must write $$ xf(x) = \sum_{n=1}^{\infty}c_n \sin(\lambda_n x). $$ Fourier shows that one has the "orthogonality" relations $$ \int_{0}^{r}\sin\lambda_n x\sin\lambda_m x dx = 0,\;\;\; m \ne n, $$ from which he deduces that $$ c_n = \frac{\int_{0}^{r}xf(x)\sin\lambda_n x dx} {\int_{0}^{r}\sin^{2}\lambda_n x dx}. $$ People were already bothered by Fourier's conjecture that the ordinary Fourier series would hold for all reasonable functions. But in this same treatise on Heat Conduction, he further proposed that an even stranger, non-harmonic series of $\sin$ functions could be used to expand all reasonable functions. All of this was so controversial that Fourier's work was not allowed to be published for the next 10-15 years, until Fourier become prominent enough to force its publication.

Fourier's problem can be boiled down to the study of eigenvalue problem for $Lf = -f''$ on the domain $\mathcal{D}(L)$ consisting of all twice continuously differentiable functions on $[0,r]$ for which $$ f(0)=0,\;\;\; f'(r)+\left(h-\frac{1}{r}\right)f(r)=0. $$ The general solution of $-f''=\lambda^{2} f$ is $A\sin\lambda x+B\cos\lambda x$, and the condition at $x=0$ then gives $B=0$. In order to satisfy the second endpoint condition with a non-zero solution, one must have $$ \lambda\cos\lambda r +\left(h-\frac{1}{r}\right)\sin\lambda r = 0. $$ Fourier recast this equation for the separation parameter $\lambda$ as $$ \frac{\lambda r}{\tan(\lambda r)}=1-hr. $$ In today's language, the eigenvalue is $\sqrt{\lambda}$, and the set of all such eigenvalues is the "spectrum" of the operator. The normalized eigenfunctions $A_n\sin\sqrt{\lambda_n}x$ form a complete orthonormal basis of $L^{2}[0,r]$, and that allows the type of expansion that Fourier conjectured.