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Trace class operators are essentially the same as Nuclear Operator s, though many authors reserve the term "trace class operator" for the special case of nuclear operators on Hilbert Space s, and reserve nuclear (=trace class) operators for more general Banach Space s.


DEFINITION

A Bounded Linear Operator ''A'' over a Hilbert Space ''H'' is said to be in the trace class if for some (and hence all) Orthonormal Bases {''e''''k''}''k'' of ''H'' the sum of positive terms
  • A)^{1/2} \, e_k, e_k angle

    • is finite.

In this case, the sum
:\sum_{k} \langle A e_k, e_k angle
is absolutely convergent and is independent of the choice of the orthonormal basis. This value is called the trace of ''A'', denoted by Tr(''A'').

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
:\sum_{k} \langle A e_k, e_k angle.


PROPERTIES


If ''A'' is a non-negative self-adjoint, ''A'' is trace class iff Tr(''A'') < ∞. Therefore a self adjoint operator ''A'' is trace class Iff its positive part ''A''+ and negative part ''A''− are both trace class.

When ''H'' is finite-dimensional, then the trace of ''A'' is just the Trace Of A Matrix and the last property stated above is roughly saying that trace is invariant under Similarity .

The trace is a linear functional over the space of trace class operators, meaning
:\operatorname{Tr}(aA+bB)=a\,\operatorname{Tr}(A)+b\,\operatorname{Tr}(B).
The bilinear map
  • B)

    • 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 can also be considered as a class of operators, the Hilbert-Schmidt operators.


For infinite dimensional spaces, the class of Hilbert-Schmidt operators is strictly larger than that of trace class operators. The heuristic is that Hilbert-Schmidt is to trace class as ''l''2(N) is to ''l''1(N).

  • Topology on B(''H'').



SEE ALSO



REFERENCES


#Dixmier, J. (1969). ''Les Algebres d'Operateurs dans l'Espace Hilbertien''. Gauthier-Villars.