Trace-class operator
In mathematics, a trace class operator is a compact operator for which a trace may be defined, such that the trace is finite and independent of the choice of basis. Trace class operators are essentially the same as nuclear operators, though many authors reserve the term "trace class operator" for the special case of nuclear operators on Hilbert spaces, and reserve "nuclear operator" for usage in more general Banach spaces.
Mimicking the definition for matrices, a bounded linear operator A over a separable Hilbert space H is said to be in the trace class if for some (and hence all) orthonormal bases {ek}k of H the sum of positive terms
is finite. In this case, the sum
is absolutely convergent and is independent of the choice of the orthonormal basis. This value is called the trace of A. When H is finite-dimensional, every operator is trace class and this definition of trace of A coincides with the definition of the trace of a matrix.
By extension, if A is a non-negative self-adjoint operator, we can also define the trace of A as an extended real number by the possibly divergent sum
The bilinear map
is an inner product on the trace class; the corresponding norm is called the Hilbert–Schmidt norm. The completion of the trace class operators in the Hilbert–Schmidt norm are called the Hilbert–Schmidt operators.
and furthermore, under the same hypothesis,
The last assertion also holds under the weaker hypothesis that and are Hilbert Schmidt.
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