trace class

Let $H$ be a separable Hilbert space, $\backslash left\backslash \{e\_k\backslash right\backslash \}\_\{k=1\}^\{\backslash infty\}$ an orthonormal basis and $A\; :\; H\; \backslash to\; H$ a positive bounded linear operator on $H$. The trace of $A$ is denoted by $\backslash operatorname\{Tr\}\; (A)$ and defined as{{sfn | Conway | 2000 | p=86}}{{sfn | Reed | Simon | 1980 | p=206}}

:$\backslash operatorname\{Tr\}\; (A)\; =\; \backslash sum\_\{k=1\}^\{\backslash infty\}\; \backslash left\backslash langle\; A\; e\_k,\; e\_k\; \backslash right\backslash rangle,$

independent of the choice of orthonormal basis. A (not necessarily positive) bounded linear operator $T:H\backslash rightarrow\; H$ is called trace class if and only if

:

where $|T|\; :=\; \backslash sqrt\{T^*\; T\}$ denotes the positive-semidefinite Hermitian square root.{{sfn | Reed | Simon | 1980 | p=196}}

The trace-norm of a trace class operator {{mvar|T}} is defined as

$$\backslash |T\backslash |\_1\; :=\; \backslash operatorname\{Tr\}\; (|T|).$$

One can show that the trace-norm is a norm on the space of all trace class operators $B\_1(H)$ and that $B\_1(H)$, with the trace-norm, becomes a Banach space.

When $H$ is finite-dimensional, every (positive) operator is trace class. For $A$ this definition coincides with that of the trace of a matrix. If $H$ is complex, then $A$ is always self-adjoint (i.e. $A=A^*=|A|$) though the converse is not necessarily true.{{sfn | Reed | Simon | 1980 | p=195}}

Given a bounded linear operator $T\; :\; H\; \backslash to\; H$, each of the following statements is equivalent to $T$ being in the trace class:

• $\backslash operatorname\{Tr\}\; (|T|)\; =\backslash sum\_k\; \backslash left\backslash langle\; |T|\; \backslash ,\; e\_k,\; e\_k\; \backslash right\backslash rangle$ is finite for every orthonormal basis $\backslash left(e\_k\backslash right)\_\{k\}$ of {{mvar|H}}.{{sfn|Conway|2000|p=86}}
• {{mvar|T}} is a nuclear operator.{{sfn|Trèves|2006|p=494}}{{sfn|Conway|2000|p=89}}
• : There exist two orthogonal sequences $\backslash left(x\_i\backslash right)\_\{i=1\}^\{\backslash infty\}$ and $\backslash left(y\_i\backslash right)\_\{i=1\}^\{\backslash infty\}$ in $H$ and positive real numbers $\backslash left(\backslash lambda\_i\backslash right)\_\{i=1\}^\{\backslash infty\}$ in $\backslash ell^1$ such that and
• ::$x\; \backslash mapsto\; T(x)\; =\; \backslash sum\_\{i=1\}^\{\backslash infty\}\; \backslash lambda\_i\; \backslash left\backslash langle\; x,\; x\_i\; \backslash right\backslash rangle\; y\_i,\; \backslash quad\; \backslash forall\; x\; \backslash in\; H,$
• :where $\backslash left(\backslash lambda\_i\backslash right)\_\{i=1\}^\{\backslash infty\}$ are the singular values of {{mvar|T}} (or, equivalently, the eigenvalues of $|T|$), with each value repeated as often as its multiplicity.{{sfn | Reed | Simon | 1980 | pp=203-204,209}}
• {{mvar|T}} is a compact operator with
• :If {{mvar|T}} is trace class then{{sfn|Conway|1990|p=268}}
• ::$\backslash |T\backslash |\_1\; =\; \backslash sup\; \backslash left\backslash \{\; |\backslash operatorname\{Tr\}\; (C\; T)|\; :\; \backslash |C\backslash |\; \backslash leq\; 1\; \backslash text\{\; and\; \}\; C\; :\; H\; \backslash to\; H\; \backslash text\{\; is\; a\; compact\; operator\; \}\; \backslash right\backslash \}.$
• {{mvar|T}} is an integral operator.{{sfn|Trèves|2006|pp=502-508}}
• {{mvar|T}} is equal to the composition of two Hilbert-Schmidt operators.{{sfn|Conway|1990|p=267}}
• $\backslash sqrt$

  T

  is a Hilbert-Schmidt operator.{{sfn|Conway|1990|p=267}}

Let $T$ be a bounded self-adjoint operator on a Hilbert space. Then $T^2$ is trace class if and only if $T$ has a pure point spectrum with eigenvalues $\backslash left\backslash \{\backslash lambda\_i(T)\backslash right\backslash \}\_\{i=1\}^\{\backslash infty\}$ such that{{sfn|Simon|2010|page=21}}

:

Mercer's theorem provides another example of a trace class operator. That is, suppose $K$ is a continuous symmetric positive-definite kernel on $L^2([a,b])$, defined as

:$K(s,t)\; =\; \backslash sum\_\{j=1\}^\backslash infty\; \backslash lambda\_j\; \backslash ,\; e\_j(s)\; \backslash ,\; e\_j(t)$

then the associated Hilbert–Schmidt integral operator $T\_K$ is trace class, i.e.,

:$\backslash operatorname\{Tr\}(T\_K)\; =\; \backslash int\_a^b\; K(t,t)\backslash ,dt\; =\; \backslash sum\_i\; \backslash lambda\_i.$

Every finite-rank operator is a trace-class operator. Furthermore, the space of all finite-rank operators is a dense subspace of $B\_1(H)$ (when endowed with the trace norm).{{sfn|Conway|1990|p=268}}

Given any $x,\; y\; \backslash in\; H,$ define the operator $$

x \otimes y : H \to H by $(x\; \backslash otimes\; y)(z)\; :=\; \backslash langle\; z,\; y\; \backslash rangle\; x.$

Then $x\; \backslash otimes\; y$ 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), $\backslash operatorname\{Tr\}(A(x\; \backslash otimes\; y))\; =\; \backslash langle\; A\; x,\; y\; \backslash rangle.${{sfn|Conway|1990|p=268}}

1. If $A\; :\; H\; \backslash to\; H$ is a non-negative self-adjoint operator, then $A$ is trace-class if and only if 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.)
2. The trace is a linear functional over the space of trace-class operators, that is, $$\backslash operatorname\{Tr\}(aA\; +\; bB)\; =\; a\; \backslash operatorname\{Tr\}(A)\; +\; b\; \backslash operatorname\{Tr\}(B).$$

   The bilinear map $$\backslash langle\; A,\; B\; \backslash rangle\; =\; \backslash operatorname\{Tr\}(A^*\; B)$$ 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.

3. $\backslash operatorname\{Tr\}\; :\; B\_1(H)\; \backslash to\; \backslash Complex$ is a positive linear functional such that if $T$ is a trace class operator satisfying $T\; \backslash geq\; 0\; \backslash text\{\; and\; \}\backslash operatorname\{Tr\}\; T\; =\; 0,$ then $T\; =\; 0.${{sfn|Conway|1990|p=267}}
4. If $T\; :\; H\; \backslash to\; H$ is trace-class then so is $T^*$ and $\backslash |T\backslash |\_1\; =\; \backslash left\backslash |T^*\backslash right\backslash |\_1.${{sfn|Conway|1990|p=267}}
5. If $A\; :\; H\; \backslash to\; H$ is bounded, and $T\; :\; H\; \backslash to\; H$ is trace-class, then $AT$ and $TA$ are also trace-class (i.e. the space of trace-class operators on H is a two-sided ideal in the algebra of bounded linear operators on H), and{{sfn|Conway|1990|p=267}}{{sfn | Reed | Simon | 1980 | p=218}}

   $$\backslash |A\; T\backslash |\_1\; =\; \backslash operatorname\{Tr\}(|A\; T|)\; \backslash leq\; \backslash |A\backslash |\; \backslash |T\backslash |\_1,\; \backslash quad\; \backslash |T\; A\backslash |\_1\; =\; \backslash operatorname\{Tr\}(|T\; A|)\; \backslash leq\; \backslash |A\backslash |\; \backslash |T\backslash |\_1.$$

   Furthermore, under the same hypothesis,{{sfn|Conway|1990|p=267}} $$\backslash operatorname\{Tr\}(A\; T)\; =\; \backslash operatorname\{Tr\}(T\; A)$$ and $|\backslash operatorname\{Tr\}(A\; T)|\; \backslash leq\; \backslash |A\backslash |\; \backslash |T\backslash |.$

   The last assertion also holds under the weaker hypothesis that A and T are Hilbert–Schmidt.

6. If $\backslash left(e\_k\backslash right)\_\{k\}$ and $\backslash left(f\_k\backslash right)\_\{k\}$ are two orthonormal bases of H and if T is trace class then $\backslash sum\_\{k\}\; \backslash left|\; \backslash left\backslash langle\; T\; e\_k,\; f\_k\; \backslash right\backslash rangle\; \backslash right|\; \backslash leq\; \backslash |T\backslash |\_\{1\}.${{sfn|Conway|1990|p=268}}
7. If A is trace-class, then one can define the Fredholm determinant of $I\; +\; A$: $$\backslash det(I\; +\; A)\; :=\; \backslash prod\_\{n\; \backslash geq\; 1\}[1\; +\; \backslash lambda\_n(A)],$$ where $\backslash \{\backslash lambda\_n(A)\backslash \}\_n$ is the spectrum of $A.$ The trace class condition on $A$ guarantees that the infinite product is finite: indeed, $$\backslash det(I\; +\; A)\; \backslash leq\; e^\{\backslash |A\backslash |\_1\}.$$

   It also implies that $\backslash det(I\; +\; A)\; \backslash neq\; 0$ if and only if $(I\; +\; A)$ is invertible.

8. If $A\; :\; H\; \backslash to\; H$ is trace class then for any orthonormal basis $\backslash left(e\_k\backslash right)\_\{k\}$ of $H,$ the sum of positive terms $\backslash sum\_k\; \backslash left|\; \backslash left\backslash langle\; A\; \backslash ,\; e\_k,\; e\_k\; \backslash right\backslash rangle\; \backslash right|$ is finite.{{sfn|Conway|1990|p=267}}
9. If $A\; =\; B^*\; C$ for some Hilbert-Schmidt operators $B$ and $C$ then for any normal vector $e\; \backslash in\; H,$ $|\backslash langle\; A\; e,\; e\; \backslash rangle|\; =\; \backslash frac\{1\}\{2\}\; \backslash left(\backslash |B\; e\backslash |^2\; +\; \backslash |C\; e\backslash |^2\backslash right)$ holds.{{sfn|Conway|1990|p=267}}

Let $A$ be a trace-class operator in a separable Hilbert space $H,$ and let $\backslash \{\backslash lambda\_n(A)\backslash \}\_\{n=1\}^\{N\backslash leq\; \backslash infty\}$ be the eigenvalues of $A.$ Let us assume that $\backslash lambda\_n(A)$ are enumerated with algebraic multiplicities taken into account (that is, if the algebraic multiplicity of $\backslash lambda$ is $k,$ then $\backslash lambda$ is repeated $k$ times in the list $\backslash lambda\_1(A),\; \backslash lambda\_2(A),\; \backslash dots$). Lidskii's theorem (named after Victor Borisovich Lidskii) states that

$$\backslash operatorname\{Tr\}(A)=\backslash sum\_\{n=1\}^N\; \backslash lambda\_n(A)$$

Note that the series on the right converges absolutely due to Weyl's inequality

$$\backslash sum\_\{n=1\}^N\; \backslash left|\backslash lambda\_n(A)\backslash right|\; \backslash leq\; \backslash sum\_\{m=1\}^M\; s\_m(A)$$

between the eigenvalues $\backslash \{\backslash lambda\_n(A)\backslash \}\_\{n=1\}^N$ and the singular values $\backslash \{s\_m(A)\backslash \}\_\{m=1\}^M$ of the compact operator $A.$Simon, B. (2005) Trace ideals and their applications, Second Edition, American Mathematical Society.

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 $\backslash ell^1(\backslash N).$

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 $\backslash ell^1$ sequence with respect to some choice of a pair of Hilbert bases. In the same vein, the bounded operators are noncommutative versions of $\backslash ell^\{\backslash infty\}(\backslash N),$ the compact operators that of $c\_0$ (the sequences convergent to 0), Hilbert–Schmidt operators correspond to $\backslash ell^2(\backslash N),$ and finite-rank operators to $c\_\{00\}$ (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: there exist orthonormal bases $(u\_i)\_i$ and $(v\_i)\_i$ and a sequence $\backslash left(\backslash alpha\_i\backslash right)\_\{i\}$ of non-negative numbers with $\backslash alpha\_i\; \backslash to\; 0$ such that

$$T\; x\; =\; \backslash sum\_i\; \backslash alpha\_i\; \backslash langle\; x,\; v\_i\backslash rangle\; u\_i\; \backslash quad\; \backslash text\{\; for\; all\; \}\; x\backslash in\; H.$$

Making the above heuristic comments more precise, we have that $T$ is trace-class iff the series $\backslash sum\_i\; \backslash alpha\_i$ is convergent, $T$ is Hilbert–Schmidt iff $\backslash sum\_i\; \backslash alpha\_i^2$ is convergent, and $T$ is finite-rank iff the sequence $\backslash left(\backslash alpha\_i\backslash right)\_\{i\}$ has only finitely many nonzero terms. This allows to relate these classes of operators. The following inclusions hold and are all proper when $H$ is infinite-dimensional:$$\backslash \{\; \backslash text\{\; finite\; rank\; \}\; \backslash \}\; \backslash subseteq\; \backslash \{\; \backslash text\{\; trace\; class\; \}\; \backslash \}\; \backslash subseteq\; \backslash \{\; \backslash text\{\; Hilbert--Schmidt\; \}\; \backslash \}\; \backslash subseteq\; \backslash \{\; \backslash text\{\; compact\; \}\; \backslash \}.$$

The trace-class operators are given the trace norm $\backslash |T\backslash |\_1\; =\; \backslash operatorname\{Tr\}\; \backslash left[\backslash left(T^*\; T\backslash right)^\{1/2\}\backslash right]\; =\; \backslash sum\_i\; \backslash alpha\_i.$ The norm corresponding to the Hilbert–Schmidt inner product is

$$\backslash |T\backslash |\_2\; =\; \backslash left[\backslash operatorname\{Tr\}\; \backslash left(T^*\; T\backslash right)\backslash right]^\{1/2\}\; =\; \backslash left(\backslash sum\_i\; \backslash alpha\_i^2\backslash right)^\{1/2\}.$$

Also, the usual operator norm is $\backslash |\; T\; \backslash |\; =\; \backslash sup\_\{i\}\; \backslash left(\backslash alpha\_i\backslash right).$ By classical inequalities regarding sequences,

$$\backslash |T\backslash |\; \backslash leq\; \backslash |T\backslash |\_2\; \backslash leq\; \backslash |T\backslash |\_1$$

for appropriate $T.$

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

The dual space of $c\_0$ is $\backslash ell^1(\backslash N).$ Similarly, we have that the dual of compact operators, denoted by $K(H)^*,$ is the trace-class operators, denoted by $B\_1.$ The argument, which we now sketch, is reminiscent of that for the corresponding sequence spaces. Let $f\; \backslash in\; K(H)^*,$ we identify $f$ with the operator $T\_f$ defined by

$$\backslash langle\; T\_f\; x,\; y\; \backslash rangle\; =\; f\backslash left(S\_\{x,y\}\backslash right),$$

where $S\_\{x,y\}$ is the rank-one operator given by

$$S\_\{x,y\}(h)\; =\; \backslash langle\; h,\; y\; \backslash rangle\; x.$$

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

$$\backslash sum\_i\; \backslash langle\; T\_f\; u\_i,\; u\_i\; \backslash rangle\; =\; f(I)\; \backslash leq\; \backslash |f\backslash |,$$

where $I$ is the identity operator:

$$I\; =\; \backslash sum\_i\; \backslash langle\; \backslash cdot,\; u\_i\; \backslash rangle\; u\_i.$$

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 $\backslash |T\_f\backslash |\_1\; =\; \backslash |f\backslash |.$ Thus $K(H)^*$ is isometrically isomorphic to $B\_1.$

Recall that the dual of $\backslash ell^1(\backslash N)$ is $\backslash ell^\{\backslash infty\}(\backslash N).$ In the present context, the dual of trace-class operators $B\_1$ is the bounded operators $B(H).$ More precisely, the set $B\_1$ is a two-sided ideal in $B(H).$ So given any operator $T\; \backslash in\; B(H),$ we may define a continuous linear functional $\backslash varphi\_T$ on $B\_1$ by $\backslash varphi\_T(A)\; =\; \backslash operatorname\{Tr\}\; (AT).$ This correspondence between bounded linear operators and elements $\backslash varphi\_T$ of the dual space of $B\_1$ is an isometric isomorphism. It follows that $B(H)$ {{em|is}} the dual space of $B\_1.$ This can be used to define the weak-* topology on $B(H).$

• {{annotated link|Nuclear operators between Banach spaces}}

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• {{Trèves François Topological vector spaces, distributions and kernels}}

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