Compact Operator - Compact Operator On Hilbert Spaces

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.