Lemma
If H a finite-dimensional Hilbert space, and a commutative set of operators, each of which is diagonalisable; then the operators can be simultaneously diagonalized.
Suppose all the operators in are compact. Then every closed non-zero -invariant sub-space has a common eigenvector for .
Theorem 1
functional analysis concept
Compact operator on Hilbert space
Theorem 3
Συμπαγής τελεστής σε χώρο Hilbert
If all the operators in are compact then the operators can be simultaneously diagonalized.
Theorem 2
If there is an injective compact operator in ; then the operators can be simultaneously diagonalized.
ヒルベルト空間上のコンパクト作用素