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.
Read more about this topic: Compact Operator On Hilbert Space
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