Compact Operator On Hilbert Spaces
An equivalent definition of compact operators on a Hilbert space may be given as follows.
An operator on a Hilbert space
is said to be compact if it can be written in the form
where and are (not necessarily complete) orthonormal sets. Here, is a sequence of positive numbers, called the singular values of the operator. The singular values can accumulate only at zero. If the sequence becomes stationary at zero, that is for some, then the operator has finite rank resp. a finite-dimenisional range and can be written as
The bracket is the scalar product on the Hilbert space; the sum on the right hand side converges in the operator norm.
An important subclass of compact operators are the trace-class or nuclear operators.
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