Compact Operators - Key Properties

The spectrum of a compact operator has a particularly simple structure, resembling the spectrum of finite-dimensional operators.

TheoremSpectrum of Compact Operators

Let $X$ be an infinite-dimensional Banach space and $T \in \mathcal{K}(X)$ a compact operator. Then:

The spectrum $\sigma(T)$ is at most countable
$0 \in \sigma(T)$
Every nonzero $\lambda \in \sigma(T)$ is an eigenvalue with finite-dimensional eigenspace
If $\sigma(T)$ is infinite, then its only accumulation point is $0$

This theorem shows that compact operators have "discrete" spectra outside of $0$ , just like matrices.

ExampleSpectral Properties

Hilbert-Schmidt: For $(Kf)(x) = \int_0^1 k(x,y) f(y) \, dy$ with $k \in L^2([0,1]^2)$ , the eigenvalues $\lambda_n$ satisfy $\sum |\lambda_n|^2 < \infty$
Diagonal: For $T((x_n)) = (\lambda_n x_n)$ on $\ell^2$ with $\lambda_n \to 0$ , we have $\sigma(T) = \{\lambda_n\} \cup \{0\}$
Volterra Operator: $(Vf)(x) = \int_0^x f(y) \, dy$ on $L^2[0,1]$ has $\sigma(V) = \{0\}$ (only eigenvalue is $0$ )

TheoremFredholm Alternative

Let $X$ be a Banach space and $K \in \mathcal{K}(X)$ compact. For $\lambda \neq 0$ , exactly one of the following holds:

The equation $(I - \lambda K)x = y$ has a unique solution for every $y \in X$
The equation $(I - \lambda K)x = 0$ has a nontrivial solution

Moreover, $\dim \ker(I - \lambda K) = \dim \ker(I - \lambda K^*) < \infty$ .

This is analogous to the classical result for systems of linear equations: either a unique solution exists or the homogeneous equation has nontrivial solutions.

Proof

Since $\lambda \neq 0$ , we have $\lambda \in \sigma(K)$ if and only if $1/\lambda \in \sigma(I - K/\lambda)$ . For compact $K$ , nonzero spectral points are eigenvalues with finite-dimensional eigenspaces.

If $(I - \lambda K)$ is not invertible, then $1 \in \sigma(\lambda K) = \lambda \sigma(K)$ , so $1/\lambda \in \sigma(K)$ is an eigenvalue. Thus $(I - \lambda K)x = 0$ has nontrivial solutions.

If $(I - \lambda K)$ is injective, the range is closed (by compactness arguments) and has finite codimension. For compact operators, injectivity implies surjectivity for $\lambda \neq 0$ .

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Remark

The Fredholm alternative is fundamental to solving integral equations of the second kind: $\phi(x) - \lambda \int_a^b k(x,y) \phi(y) \, dy = f(x)$ Either this has a unique solution for every $f$ , or the homogeneous equation ( $f = 0$ ) has nontrivial solutions.

This theory provides a complete analog of finite-dimensional linear algebra for equations involving compact operators.