In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of trace-class operators generalizes the trace of matrices studied in linear algebra. All trace-class operators are compact operators. In quantum mechanics, quantum states are described by density matrices, which are certain trace class operators. 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 use the term "nuclear operator" in more general topological vector spaces (such as Banach spaces).
Definition Let H {\displaystyle H} be a separable Hilbert space, { e k } k = 1 ∞ {\displaystyle \left\{e_{k}\right\}_{k=1}^{\infty }} an orthonormal basis and A : H → H {\displaystyle A:H\to H} a positive bounded linear operator on H {\displaystyle H} . The trace of A {\displaystyle A} is denoted by Tr ( A ) {\displaystyle \operatorname {Tr} (A)} and defined as
Tr ( A ) = ∑ k = 1 ∞ ⟨ A e k , e k ⟩ , {\displaystyle \operatorname {Tr} (A)=\sum _{k=1}^{\infty }\left\langle Ae_{k},e_{k}\right\rangle ,}
independent of the choice of orthonormal basis. A (not necessarily positive) bounded linear operator T : H → H {\displaystyle T:H\rightarrow H} is called trace class if and only if
Tr ( | T | ) < ∞ , {\displaystyle \operatorname {Tr} (|T|)<\infty ,}
where | T | := T ∗ T {\displaystyle |T|:={\sqrt {T^{*}T}}} denotes the positive-semidefinite Hermitian square root. The trace-norm of a trace class operator T is defined as
‖ T ‖ 1 := Tr ( | T | ) . {\displaystyle \|T\|_{1}:=\operatorname {Tr} (|T|).}
One can show that the trace-norm is a norm on the space of all trace class operators B 1 ( H ) {\displaystyle B_{1}(H)} and that B 1 ( H ) {\displaystyle B_{1}(H)} , with the trace-norm, becomes a Banach space. When H {\displaystyle H} is finite-dimensional, every (positive) operator is trace class. For A {\displaystyle A} this definition coincides with that of the trace of a matrix. If H {\displaystyle H} is complex, then A {\displaystyle A} is always self-adjoint (i.e. A = A ∗ = | A | {\displaystyle A=A^{*}=|A|} ) though the converse is not necessarily true.
Equivalent formulations Given a bounded linear operator T : H → H {\displaystyle T:H\to H} , each of the following statements is equivalent to T {\displaystyle T} being in the trace class:
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