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Compact Operators

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Hilbert-Schmidt Operators

Integral Operator

On , we can define an operator by the formula we call it an integral operator with associated kernel , where is a measurable function such that for almost every .

Hilbert-Schmidt Operator

An integral operator is called a Hilbert-Schmidt operator if the associated kernel is an element of .

Proposition

Let be a Hilbert-Schmidt operator on with kernel . Then

  • for every , and almost every , the function is integrable.
  • is bounded, and .
  • the adjoint of is also a Hilbert-Schmidt operator with kernel .

Compact Operators

We shall use the notion of sequential compactness in a separable Hilbert space. An example would be the closed unit ball in , which is not compact unless is finite-dimensional. If is finite-dimensional, then is compact by the Hein-Borel theorem. Otherwise, is infinite-dimensional. Suppose is an orthonormal basis, then no subsequence of the sequence will converge to any point in , because for all .

Compact Operator

Suppose and are normed spaces. An operator is compact if all bounded sequences in have a subsequence such that converges in . Equivalently, is compact if the closure of the image of the unit ball in under is compact in .

e.g.

  • is compact if (i.e. dimension of the image) is finite.
  • Hilbert-Schmidt operators are compact. (See the corollary)

Proposition

Composition of a compact operator of bounded operator is compact.

Theorem

Suppose is a bounded linear operator on Hilbert space , then

  • If is a compact operator on , then and are also compact.
  • If is a family of compact linear operators with as tends to infinity, then is compact.
  • Conversely, if is compact, there is a sequence of operators of finite rank such that .
  • is compact if and only if is compact.

Corollary

Every Hilbert-Schmidt operator is compact.

Spectral Theorem

Lemma

Suppose is a bounded self-adjoint linear operator on a Hilbert space . Then any eigenvalue of is real, and if and are eigenvectors corresponding to two distinct eigenvalues, then and are orthogonal.

Spectral Theorem of Compact Operators

Suppose is a compact self-adjoint operator on a separable Hilbert space . Then there exits an orthonormal basis of , that consists of eigenvectors of . Moreover, if then and as . Conversely, every operator of the above form is compact and self-adjoint. The collection is called the spectrum of .

The Trace Class Operators

Trace Class Operator

A bounded linear operator on a separable Hilbert space is called a trace class operator if where the positive operator , and are the singular values of . In this case, the trace norm is defined as

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