Compact Operator On Hilbert Space - Compact Normal Operator

Compact Normal Operator

The family of Hermitian matrices is a proper subset of matrices that are unitarily diagonalizable. A matrix M is unitarily diagonalizable if and only if it is normal, i.e. M*M = MM*. Similar statements hold for compact normal operators.

Let T be compact and T*T = TT*. Apply the Cartesian decomposition to T: define

The self adjoint compact operators R and J are called the real and imaginary parts of T respectively. T is compact means T*, consequently R and J, are compact. Furthermore, the normality of T implies R and J commute. Therefore they can be simultaneously diagonalized, from which follows the claim.

A hyponormal compact operator (in particular, a subnormal operator) is normal.