Trace class

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 topological vector spaces (such as Banach spaces).

Definition

Define the trace, denoted by $\operatorname {Tr} A$ , of linear operator A to be the sum of the series[1]

\operatorname {Tr} A=\sum _{k}\left\langle Ae_{k},e_{k}\right\rangle

,

where this sum is independent of the choice of the orthonormal basis {e_k}_k of H and where this sum equals $\infty$ if it does not converge. If H is finite-dimensional then $\operatorname {Tr} A$ is equal to the usual definition of the trace.

For any bounded linear operator T : H → H over a Hilbert space H, we define its absolute value, denoted by |T|, to be the positive square root of $T^{*}T$ , i.e. $\left|T\right|:={\sqrt {T^{*}T}}$ is the unique bounded positive operator on H such that $|T|\circ |T|=T^{*}\circ T$ . It may be shown that a bounded linear operator on a Hilbert space is trace class if and only if its absolute value is trace class.[1]

A bounded linear operator T : H → H over a Hilbert space H is said to be in the trace class if any of the following equivalent conditions is satisfied:

T is a nuclear operator.
T is equal to the composition of two Hilbert-Schmidt operators.[1]
${\sqrt {\left|T\right|}}$ is a Hilbert-Schmidt operator.[1]
T is an integral operator.[2]
there exist weakly closed and equicontinuous (and thus weakly compact) subsets $A^{\prime }$ and $B^{\prime \prime }$ of $H^{\prime }$ and $H^{\prime \prime }$ , respectively, and some positive Radon measure $\mu$ on $A^{\prime }\times B^{\prime \prime }$ of total mass ≤ 1 such that for all x ∈ H and $y^{\prime }\in H^{\prime }$ :
$y^{\prime }\left(T(x)\right)=\int _{A^{\prime }\times B^{\prime \prime }}x^{\prime }\left(x\right)\cdot y^{\prime \prime }\left(y^{\prime }\right)\operatorname {d} \mu \left(x^{\prime },y^{\prime \prime }\right)$ .
there exist two orthogonal sequences $\left(x_{i}\right)_{i=1}^{\infty }$ $\text{[math]}$ and $\left(y_{i}\right)_{i=1}^{\infty }$ $\text{[math]}$ in H and a sequence $\left(\lambda _{i}\right)_{i=1}^{\infty }$ $\text{[math]}$ in l¹ such that for all x in H, $T(x)=\sum _{i=1}^{\infty }\lambda _{i}\left\langle x,x_{i}\right\rangle y_{i}$ $\text{[math]}$ .[3]
- Here, the infinite sum means that the sequence of partial sums $\left(\sum _{i=1}^{N}\lambda _{i}\left\langle x,x_{i}\right\rangle y_{i}\right)_{N=1}^{\infty }$ converges to T(x) in H.
T is a compact operator and $\sum _{i=1}^{\infty }l_{i}<\infty$ $\text{[math]}$ , where l₁, l₂, ... are the eigenvalues of T with each eigenvalue repeated as often as its multiplicity.[1]
- Recall that the multiplicity of an eigenvalue r is the dimension of the kernel of T - r Id_H, where Id_H : H → H is the identity map.
for some orthonormal basis (e_k)_k of H, the sum of positive terms $\sum _{k}\left\langle \left|T\right|\,e_{k},e_{k}\right\rangle$ is finite.
the above condition but with the word "some" replaced by "every".
the transpose map ${}^{t}T:H_{b}^{\prime }\to H_{b}^{\prime }$ $\text{[math]}$ is trace class (according to any defining condition other than this one), in which case $\|{}^{t}T\|_{1}=\|T\|_{1}$ $\text{[math]}$ .[4]
- Recall that the transpose of T is defined by $\left({}^{t}T\right)\left(y^{\prime }\right):=y^{\prime }\circ T$ , for all $y^{\prime }$ belonging to the continuous dual space $H^{\prime }$ of H. The subscript b indicates that $H^{\prime }$ has its usual norm topology.
$\operatorname {Tr} \left(\left|T\right|\right)<\infty$ .[1]

and if T is not already a positive operator then we may add to this list:

the operator $\left|T\right|$ is trace class (according to any defining condition other than this one).

Trace-norm

If T is trace class then we define the trace-norm of a trace class operator T to be the common value

\left\|T\right\|_{1}:=\sum _{k}\left\langle \left|T\right|\,e_{k},e_{k}\right\rangle =\operatorname {Tr} \left(\left|T\right|\right)

(where it may be shown that the last equality necessarily holds) and we denote the space of all trace class linear operators on H by B₁(H). If T is trace class then

\left\|T\right\|_{1}=\sup \left\{\left|\operatorname {Tr} \left(CT\right)\right|:\|C\|\leq 1{\text{ and }}C:H\to H{\text{ is a compact operator }}\right\}

.[5]

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

\sum _{k}\left\langle Ae_{k},e_{k}\right\rangle .

where this sum is independent of the choice of the orthonormal basis {e_k}_k of H.

Examples

Every bounded linear operator that has a finite-dimensional range (i.e. operators of finite-rank) is trace class;[1] furthermore, the space of all finite-rank operators is a dense subspace of B₁(H) (when endowed with the $\|\cdot \|_{1}$ norm).[5] The composition of two Hilbert-Schmidt operators is a trace class operator.[1]

Given any x and y in H, define $x\otimes y:H\to H$ by (x ⊗ y)(z) = <z, y> x, which is a continuous linear operator of rank 1 and is thus trace class; moreover, for any bounded linear operator A on H (and into H), $\operatorname {Tr} \left(A\left(x\otimes y\right)\right)=\left\langle Ax,y\right\rangle$ .[5]

Properties

If A : H → H is a non-negative self-adjoint, then A is trace-class if and only if Tr(A) < ∞. Therefore, a self-adjoint operator A is trace-class if and only if its positive part A⁺ and negative part A⁻ are both trace-class. (The positive and negative parts of a self-adjoint operator are obtained by the continuous functional calculus.)
The trace is a linear functional over the space of trace-class operators, i.e.
$\operatorname {Tr} (aA+bB)=a\operatorname {Tr} (A)+b\operatorname {Tr} (B).$

The bilinear map

$\langle A,B\rangle =\operatorname {Tr} (A^{*}B)$
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 are called the Hilbert–Schmidt operators.
If T : H → H is trace-class then so is T^* and $\left\|T\right\|_{1}=\left\|T^{*}\right\|_{1}$ .[1]
If A : H → H is bounded, and T : H → H is trace-class, AT and TA are also trace-class, and[6][1]
$\|AT\|_{1}=\operatorname {Tr} (|AT|)\leq \|A\|\|T\|_{1},\quad \|TA\|_{1}=\operatorname {Tr} (|TA|)\leq \|A\|\|T\|_{1}.$ [1]

Furthermore, under the same hypothesis,

$\operatorname {Tr} (AT)=\operatorname {Tr} (TA)$ and $\left|\operatorname {Tr} \left(AT\right)\right|\leq \left\|A\right\|\left\|T\right|$ .[1]
The last assertion also holds under the weaker hypothesis that A and T are Hilbert–Schmidt.
The space of trace-class operators on H is an ideal in the space of bounded linear operators on H.[1]
If {e_k}_k and {f_k}_k are two orthonormal bases of H and if T is trace class then $\sum _{k}\left|\left\langle Te_{k},f_{k}\right\rangle \right|\leq \left\|T\right\|_{1}$ .[5]
If A is trace-class, then one can define the Fredholm determinant of 1 + A:
$\det(I+A):=\prod _{n\geq 1}[1+\lambda _{n}(A)],$

where $\{\lambda _{n}(A)\}_{n}$ is the spectrum of $A$ . The trace class condition on $A$ guarantees that the infinite product is finite: indeed,

$\det(I+A)\leq e^{\|A\|_{1}}.$
It also implies that $\det(I+A)\neq 0$ if and only if (I + A) is invertible.
If A : H → H is trace class then for any orthonormal basis {e_k}_k of H, the sum of positive terms $\sum _{k}\left|\left\langle A\,e_{k},e_{k}\right\rangle \right|$ is finite.[1]

Lidskii's theorem

Let $A$ be a trace-class operator in a separable Hilbert space $H$ , and let $\{\lambda _{n}(A)\}_{n=1}^{N},$ $N\leq \infty$ be the eigenvalues of $A$ . Let us assume that $\lambda _{n}(A)$ are enumerated with algebraic multiplicities taken into account (i.e. if the algebraic multiplicity of $\lambda$ is $k$ , then $\lambda$ is repeated $k$ times in the list $\lambda _{1}(A),\lambda _{2}(A),\dots$ ). Lidskii's theorem (named after Victor Borisovich Lidskii) states that

\sum _{n=1}^{N}\lambda _{n}(A)=\operatorname {Tr} (A).

Note that the series in the left converges absolutely due to Weyl's inequality

\sum _{n=1}^{N}{\big |}\lambda _{n}(A){\big |}\leq \sum _{m=1}^{M}s_{m}(A)

between the eigenvalues $\{\lambda _{n}(A)\}_{n=1}^{N}$ and the singular values $\{s_{m}(A)\}_{m=1}^{M}$ of a compact operator $A$ .[7]

Relationship between some classes of operators

One can view certain classes of bounded operators as noncommutative analogue of classical sequence spaces, with trace-class operators as the noncommutative analogue of the sequence space ℓ¹(N).

Indeed, it is possible to apply the spectral theorem to show that every normal trace-class operator on a separable Hilbert space can be realized in a certain way as an ℓ¹ sequence with respect to some choice of a pair of Hilbert bases. In the same vein, the bounded operators are noncommutative versions of ℓ^∞(N), the compact operators that of c₀ (the sequences convergent to 0), Hilbert–Schmidt operators correspond to ℓ²(N), and finite-rank operators (the sequences that have only finitely many non-zero terms). To some extent, the relationships between these classes of operators are similar to the relationships between their commutative counterparts.

Recall that every compact operator T on a Hilbert space takes the following canonical form:

\forall h\in H,\;Th=\sum _{i=1}\alpha _{i}\langle h,v_{i}\rangle u_{i},\quad {\text{where}}\quad \alpha _{i}\geq 0,\ \alpha _{i}\to 0,

for some orthonormal bases {u_i} and {v_i}. Making the above heuristic comments more precise, we have that T is trace-class if the series ∑_i α_i is convergent, T is Hilbert–Schmidt if ∑_i α_i² is convergent, and T is finite-rank if the sequence {α_i} has only finitely many nonzero terms.

The above description allows one to obtain easily some facts that relate these classes of operators. For example, the following inclusions hold and are all proper when H is infinite-dimensional: {finite rank} ⊂ {trace class} ⊂ {Hilbert–Schmidt} ⊂ {compact}.

The trace-class operators are given the trace norm ||T||₁ = Tr[(T*T)^½] = ∑_i α_i. The norm corresponding to the Hilbert–Schmidt inner product is ||T||₂ = [Tr(T*T)]^½ = (∑_iα_i²)^½. Also, the usual operator norm is ||T|| = sup_i(α_i). By classical inequalities regarding sequences,

\|T\|\leq \|T\|_{2}\leq \|T\|_{1}

for appropriate T.

It is also clear that finite-rank operators are dense in both trace-class and Hilbert–Schmidt in their respective norms.

Trace class as the dual of compact operators

The dual space of c₀ is ℓ¹(N). Similarly, we have that the dual of compact operators, denoted by K(H)*, is the trace-class operators, denoted by C₁. The argument, which we now sketch, is reminiscent of that for the corresponding sequence spaces. Let f ∈ K(H)*, we identify f with the operator T_f defined by

\langle T_{f}x,y\rangle =f(S_{x,y}),

where S_x,y is the rank-one operator given by

S_{x,y}(h)=\langle h,y\rangle x.

This identification works because the finite-rank operators are norm-dense in K(H). In the event that T_f is a positive operator, for any orthonormal basis u_i, one has

\sum _{i}\langle T_{f}u_{i},u_{i}\rangle =f(I)\leq \|f\|,

where I is the identity operator:

I=\sum _{i}\langle \cdot ,u_{i}\rangle u_{i}.

But this means that T_f is trace-class. An appeal to polar decomposition extend this to the general case, where T_f need not be positive.

A limiting argument using finite-rank operators shows that ||T_f||₁ = ||f||. Thus K(H)* is isometrically isomorphic to C₁.

As the predual of bounded operators

Recall that the dual of ℓ¹(N) is ℓ^∞(N). In the present context, the dual of trace-class operators C₁ is the bounded operators B(H). More precisely, the set C₁ is a two-sided ideal in B(H). So given any operator T in B(H), we may define a continuous linear functional φ_T on $C_{1}$ by φ_T(A) = Tr(AT). This correspondence between bounded linear operators and elements φ_T of the dual space of $C_{1}$ is an isometric isomorphism. It follows that B(H) is the dual space of $C_{1}$ . This can be used to define the weak-* topology on B(H).

Notes

Conway 1990, p. 267.
Treves 2006, pp. 502-508.
Treves 2006, p. 494.
Treves 2006, p. 484.
Conway 1990, p. 268.
M. Reed and B. Simon, Functional Analysis, Exercises 27, 28, page 218.
Simon, B. (2005) Trace ideals and their applications, Second Edition, American Mathematical Society.

References

Conway, John (1990). A course in functional analysis. New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908.
Schaefer, Helmut H. (1999). Topological Vector Spaces. GTM. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.CS1 maint: ref=harv (link)

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

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[FOOTNOTEConway1990267-1] Conway 1990, p. 267.

[FOOTNOTETreves2006502-508-2] Treves 2006, pp. 502-508.

[FOOTNOTETreves2006494-3] Treves 2006, p. 494.

[FOOTNOTETreves2006484-4] Treves 2006, p. 484.

[FOOTNOTEConway1990268-5] Conway 1990, p. 268.

[6] M. Reed and B. Simon, Functional Analysis, Exercises 27, 28, page 218.

[7] Simon, B. (2005) Trace ideals and their applications, Second Edition, American Mathematical Society.

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