Examples of spectral decompositions

Question

I would like examples of spectral decompositions and how they are obtained for normal compact operators and normal non-compact operators on an infinite dimensional hilbert space. I have googled it, but all I can find are proofs of the theorems; no concrete examples. I can't imagine it is that hard, for example for normal compact operator we only need the eigenvalues and eigenvectors of the operator.

Thanks

Answer 1

Consider $H=L^2(\Omega,\mu)$ where $\Omega\subseteq\mathbb C$ is compact, and multiplication operator $T:H\to H,\ (Tf)(z)=zf(z).$ Then $T$ is bounded and normal. By the spectral theorem $T=\int_{\mathbb C}\lambda dE(\lambda).$ In this case you can give an explicit formula for the spectral measure $E:$

$$(E(X)f)(z)=\chi_X(z)f(z),$$ where $\chi_X$ is the indicator function of $X\subseteq\mathbb C.$

One version of a spectral theorem says that every bounded normal operator is unitarily equivalent to a direct sum of such operators $T$.

Answer 2

The spectral theorem is a general argument. For a specific normal operators, there may be a concrete "diagonalizing" unitary.

Take the shift $S$ on $l^2(\mathbb{Z})$. In this concrete example, the diagonalizing unitary is the Fourier transform

$$ \mathcal{F} : l^2(\mathbb{Z}) \rightarrow L^2(\mathbb{T}). $$

The operator

$$ \mathcal{F} S\mathcal{F}^{-1} : L^2(\mathbb{T}) \rightarrow L^2(\mathbb{T}) $$

is multiplication by $z$.

The Fourier transform on $\mathbb{R}$ is itself a unitary, with discrete spectrum, the fourth roots of unity, with eigenvectors being the Hermite polynomials.