Let 
be a Hilbert space and 
an orthonormal basis for 
. The set of all products of two Hilbert-Schmidt
operators is denoted 
, and its elements are called trace-class operators. This
set is a self-adjoint two-sided ideal of 
and coincides with the set of those operators 
for which 
where 
is the absolute value of 
in the 
-algebra 
. If 
, then 
with this norm is a Banach algebra in which 
is dense. Furthermore, 
.
Trace-Class Operator
See also
Hilbert-Schmidt OperatorThis entry contributed by Mohammad Sal Moslehian
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References
Gohberg, I. C. and Kreǐn, M. G. Introduction to the Theory of Linear Nonselfadjoint Operators. Providence, RI: Amer. Math. Soc., 1969.Murphy, G. J. C-*-Algebras and Operator Theory. New York: Academic Press, 1990.Referenced on Wolfram|Alpha
Trace-Class OperatorCite this as:
Moslehian, Mohammad Sal. "Trace-Class Operator." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Trace-ClassOperator.html