A trace-class operator on a Hilbert space 
is a product of two Hilbert-Schmidt
operators. The set of these operators is denoted 
. Write 
for the C*-algebra
of bounded operators on 
, and let 
be a Hilbert basis.
The set 
is a two-sided ideal
of 
closed under taking the adjoint operator. It
consists of the bounded operators 
satisfying 
. Here the operator modulus
is 
,
the positive operator square root of 
, where 
is the adjoint operator
(Bell 2016, p. 10).
With the norm 
, the set 
is a Banach
algebra in which 
, the set of bounded
operators of finite rank, is dense.
Furthermore, 
,
where 
is the set of Hilbert-Schmidt
operators and 
is the set of compact operators.
