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).

DefinitionEdit

Definition: The trace, denoted by  , of linear operator A to be the sum of the series[1]
 ,

where this sum is independent of the choice of the orthonormal basis { ek }k of H and where this sum equals if it does not converge.

If H is finite-dimensional then Tr A is equal to the usual definition of the trace.

Definition: For any bounded linear operator T : HH over a Hilbert space H, we define its absolute value, denoted by |T|, to be the positive square root of  , i.e.   is the unique bounded positive operator on H such that  .

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]

Definition: A bounded linear operator T : HH over a Hilbert space H is said to be in the trace class if any of the following equivalent conditions is satisfied:
  1. T is a nuclear operator.
  2. T is equal to the composition of two Hilbert-Schmidt operators.[1]
  3.   is a Hilbert-Schmidt operator.[1]
  4. T is an integral operator.[2]
  5. there exist weakly closed and equicontinuous (and thus weakly compact) subsets   and   of   and  , respectively, and some positive Radon measure   on   of total mass ≤ 1 such that for all xH and  :
     .
  6. there exist two orthogonal sequences   and   in H and a sequence   in l1 such that for all xH,  .[3]
    • Here, the infinite sum means that the sequence of partial sums   converges to T(x) in H.
  7. T is a compact operator and  , 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 IdH, where IdH : HH is the identity map.
  8. for some orthonormal basis (ek)k of H, the sum of positive terms   is finite.
  9. the above condition but with the word "some" replaced by "every".
  10. the transpose map   is trace class (according to any defining condition other than this one), in which case  .[4]
    • Recall that the transpose of T is defined by  , for all   belonging to the continuous dual space   of H. The subscript b indicates that   has its usual norm topology.
  11.  .[1]

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

  1. the operator |T| is trace class (according to any defining condition other than this one).

Trace-normEdit

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

(where it may be shown that the last equality necessarily holds). We denote the space of all trace class linear operators on H by B1(H).

If T is trace class then

 .[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

 

where this sum is independent of the choice of the orthonormal basis {ek}k of H.

ExamplesEdit

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   norm).[5] The composition of two Hilbert-Schmidt operators is a trace class operator.[1]

Given any x and y in H, define   by (xy)(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),  .[5]

PropertiesEdit

  1. If A : HH 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.)
  2. The trace is a linear functional over the space of trace-class operators, i.e.
     

    The bilinear map

     
    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. If T : HH is trace-class then so is T* and  .[1]
  4. If A : HH is bounded, and T : HH is trace-class, AT and TA are also trace-class, and[6][1]
     [1]
    Furthermore, under the same hypothesis,
      and  .[1]
    The last assertion also holds under the weaker hypothesis that A and T are Hilbert–Schmidt.
  5. The space of trace-class operators on H is an ideal in the space of bounded linear operators on H.[1]
  6. If {ek}k and {fk}k are two orthonormal bases of H and if T is trace class then  .[5]
  7. If A is trace-class, then one can define the Fredholm determinant of 1 + A:
     
    where   is the spectrum of  . The trace class condition on   guarantees that the infinite product is finite: indeed,
     
    It also implies that   if and only if (I + A) is invertible.
  8. If A : HH is trace class then for any orthonormal basis {ek}k of H, the sum of positive terms   is finite.[1]

Lidskii's theoremEdit

Let   be a trace-class operator in a separable Hilbert space  , and let     be the eigenvalues of  . Let us assume that   are enumerated with algebraic multiplicities taken into account (i.e. if the algebraic multiplicity of   is  , then   is repeated   times in the list  ). Lidskii's theorem (named after Victor Borisovich Lidskii) states that

 

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

 

between the eigenvalues   and the singular values   of a compact operator  .[7]

Relationship between some classes of operatorsEdit

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 1(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 1 sequence with respect to some choice of a pair of Hilbert bases. In the same vein, the bounded operators are noncommutative versions of (N), the compact operators that of c0 (the sequences convergent to 0), Hilbert–Schmidt operators correspond to 2(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:

 

for some orthonormal bases {ui} and {vi}. Making the above heuristic comments more precise, we have that T is trace-class if the series ∑i αi is convergent, T is Hilbert–Schmidt if ∑i αi2 is convergent, and T is finite-rank if the sequence {αi} 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)1/2] = ∑i αi. The norm corresponding to the Hilbert–Schmidt inner product is ||T||2 = [Tr(T*T)]1/2 = (∑iαi2)1/2. Also, the usual operator norm is ||T|| = supi(αi). By classical inequalities regarding sequences,

 

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 operatorsEdit

The dual space of c0 is 1(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 fK(H)*, we identify f with the operator Tf defined by

 

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

 

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

 

where I is the identity operator:

 

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

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

As the predual of bounded operatorsEdit

Recall that the dual of 1(N) is (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 φT on   by φT(A) = Tr(AT). This correspondence between bounded linear operators and elements φT of the dual space of   is an isometric isomorphism. It follows that B(H) is the dual space of  . This can be used to define the weak-* topology on B(H).

See alsoEdit

NotesEdit

  1. ^ a b c d e f g h i j k l m n o p Conway 1990, p. 267.
  2. ^ Trèves 2006, pp. 502-508.
  3. ^ Trèves 2006, p. 494.
  4. ^ Trèves 2006, p. 484.
  5. ^ a b c d 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.

ReferencesEdit

  • Conway, John (1990). A course in functional analysis. New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908.
  • Dixmier, J. (1969). Les Algebres d'Operateurs dans l'Espace Hilbertien. Gauthier-Villars.
  • 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.
  • Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.