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
Famous quotes containing the words compact and/or normal:
“What compact mean you to have with us?
Will you be pricked in number of our friends,
Or shall we on, and not depend on you?”
—William Shakespeare (1564–1616)
“We have been weakened in our resistance to the professional anti-Communists because we know in our hearts that our so-called democracy has excluded millions of citizens from a normal life and the normal American privileges of health, housing and education.”
—Agnes E. Meyer (1887–1970)