{"history":[],"article":{"id":"f05b9156-6047-4b5b-97bf-13e999d0ddd1","title":"Trace class","slug":"trace-class","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), specifically [functional analysis](https://bliptext.com/articles/functional-analysis), a **[trace](https://bliptext.com/articles/trace-linear-algebra)-class** operator is a linear operator for which a [trace](https://bliptext.com/articles/trace-linear-algebra) may be defined, such that the [trace](https://bliptext.com/articles/trace-linear-algebra) is a finite number independent of the choice of basis used to compute the [trace](https://bliptext.com/articles/trace-linear-algebra).\nThis trace of trace-class operators generalizes the trace of matrices studied in linear algebra.\nAll trace-class operators are [compact operators](https://bliptext.com/articles/compact-operator).\nIn quantum mechanics, [quantum states](https://bliptext.com/articles/quantum-state) are described by [density matrices](https://bliptext.com/articles/density-matrix), which are certain trace class operators.\nTrace-class operators are essentially the same as [nuclear operators](https://bliptext.com/articles/nuclear-operator), though many authors reserve the term \"trace-class operator\" for the special case of [nuclear operators](https://bliptext.com/articles/nuclear-operator) on [Hilbert spaces](https://bliptext.com/articles/hilbert-space) and use the term \"nuclear operator\" in more general [topological vector spaces](https://bliptext.com/articles/topological-vector-space) (such as [Banach spaces](https://bliptext.com/articles/banach-space)).\nNote that the [trace operator](https://bliptext.com/articles/trace-operator) studied in partial differential equations is an unrelated concept.\n\n# Definition\nLet H be a [separable](https://bliptext.com/articles/separable-space) [Hilbert space](https://bliptext.com/articles/hilbert-space), an [orthonormal basis](https://bliptext.com/articles/orthonormal-basis) and A : H \\to H a [positive](https://bliptext.com/articles/positive-operator-hilbert-space) [bounded linear operator](https://bliptext.com/articles/bounded-linear-operator) on H. The **trace** of A is denoted by and defined as independent of the choice of [orthonormal basis](https://bliptext.com/articles/orthonormal-basis).\nA (not necessarily positive) bounded linear operator is called **trace class** if and only if\nwhere denotes the positive-semidefinite [Hermitian](https://bliptext.com/articles/conjugate-transpose) [square root](https://bliptext.com/articles/square-root-of-a-matrix).\nThe **trace-norm** of a trace class operator T is defined as\nOne can show that the trace-[norm](https://bliptext.com/articles/norm-mathematics) is a [norm](https://bliptext.com/articles/norm-mathematics) on the space of all trace class operators B_1(H) and that B_1(H), with the trace-[norm](https://bliptext.com/articles/norm-mathematics), becomes a [Banach space](https://bliptext.com/articles/banach-space).\nWhen H is finite-dimensional, every (positive) operator is trace class and this definition of trace of A coincides with the definition of the [trace of a matrix](https://bliptext.com/articles/trace-matrix).\nIf H is complex, then A is always [self-adjoint](https://bliptext.com/articles/self-adjoint-operator) (i.e. A=A^*=|A|) though the converse is not necessarily true.\n\n# Equivalent formulations\nGiven a bounded linear operator T : H \\to H, each of the following statements is equivalent to T being in the trace class:\n\n# Examples\n\n## Spectral theorem\nLet T be a bounded self-adjoint operator on a Hilbert space.\nThen T^2 is trace class if and only if T has a [pure point spectrum](https://bliptext.com/articles/spectrum-functional-analysis) with eigenvalues such that\n\n## Mercer's theorem\n[Mercer's theorem](https://bliptext.com/articles/mercer-s-theorem) provides another example of a trace class operator.\nThat is, suppose K is a continuous symmetric [positive-definite kernel](https://bliptext.com/articles/positive-definite-kernel) on L^2([a,b]), defined as then the associated [Hilbert–Schmidt integral operator](https://bliptext.com/articles/hilbert-schmidt-integral-operator) T_K is trace class, i.e.,\n\n## Finite-rank operators\nEvery [finite-rank operator](https://bliptext.com/articles/finite-rank-operator) is a trace-class operator.\nFurthermore, the space of all finite-rank operators is a dense subspace of B_1(H) (when endowed with the trace norm).\nGiven any x, y \\in H, define the operator by Then x \\otimes y is a continuous linear operator of rank 1 and is thus trace class;\nmoreover, for any bounded linear operator A on H (and into H),\n\n# Properties\n

If A : H \\to H is a non-negative [self-adjoint operator](https://bliptext.com/articles/self-adjoint-operator), then A is trace-class [if and only if](https://bliptext.com/articles/if-and-only-if) Therefore, a [self-adjoint operator](https://bliptext.com/articles/self-adjoint-operator) A is trace-class [if and only if](https://bliptext.com/articles/if-and-only-if) its positive part A^{+} and negative part A^{-} are both trace-class.\n(The positive and negative parts of a self-adjoint operator are obtained by the [continuous functional calculus](https://bliptext.com/articles/continuous-functional-calculus).)
The trace is a linear functional over the space of trace-class operators, that is,\nThe bilinear map is an [inner product](https://bliptext.com/articles/inner-product) on the trace class; the corresponding norm is called the [Hilbert–Schmidt](https://bliptext.com/articles/hilbert-schmidt-operator) norm.\nThe completion of the trace-class operators in the Hilbert–Schmidt norm are called the Hilbert–Schmidt operators.
is a positive linear functional such that if T is a trace class operator satisfying then T = 0.
If T : H \\to H is trace-class then so is T^* and
If A : H \\to H is bounded, and T : H \\to H is trace-class, then AT and TA are also trace-class (i.e. the space of trace-class operators on H is an [ideal](https://bliptext.com/articles/ideal-ring-theory) in the algebra of bounded linear operators on H), and\nFurthermore, under the same hypothesis, and\nThe last assertion also holds under the weaker hypothesis that A and T are Hilbert–Schmidt.
If and are two orthonormal bases of H and if T is trace class then
If A is trace-class, then one can define the [Fredholm determinant](https://bliptext.com/articles/fredholm-determinant) of I + A: where is the spectrum of A. The trace class condition on A guarantees that the infinite product is finite: indeed,\nIt also implies that if and only if (I + A) is invertible.
If A : H \\to H is trace class then for any [orthonormal basis](https://bliptext.com/articles/orthonormal-basis) of H, the sum of positive terms is finite.
If A = B^* C for some [Hilbert-Schmidt operators](https://bliptext.com/articles/hilbert-schmidt-operator) B and C then for any normal vector e \\in H, holds.

\n\n## Lidskii's theorem\nLet A be a trace-class operator in a separable Hilbert space H, and let be the eigenvalues of A. Let us assume that are enumerated with algebraic multiplicities taken into account (that is, if the algebraic multiplicity of \\lambda is k, then \\lambda is repeated k times in the list ).\nLidskii's theorem (named after [Victor Borisovich Lidskii](https://bliptext.com/articles/victor-borisovich-lidskii)) states that\nNote that the series on the right converges absolutely due to [Weyl's inequality](https://bliptext.com/articles/weyl-s-inequality)\nbetween the eigenvalues and the [singular values](https://bliptext.com/articles/singular-value) of the compact operator A.\n\n## Relationship between common classes of operators\nOne can view certain classes of bounded operators as noncommutative analogue of classical [sequence spaces](https://bliptext.com/articles/sequence-space), with trace-class operators as the noncommutative analogue of the [sequence space](https://bliptext.com/articles/sequence-space) \\ell^1(\\N).\nIndeed, it is possible to apply the [spectral theorem](https://bliptext.com/articles/spectral-theorem) to show that every normal trace-class operator on a separable Hilbert space can be realized in a certain way as an \\ell^1 sequence with respect to some choice of a pair of Hilbert bases.\nIn the same vein, the bounded operators are noncommutative versions of the [compact operators](https://bliptext.com/articles/compact-operator-on-hilbert-space) that of c_0 (the sequences convergent to 0), Hilbert–Schmidt operators correspond to \\ell^2(\\N), and [finite-rank operators](https://bliptext.com/articles/finite-rank-operator) to c_{00} (the sequences that have only finitely many non-zero terms).\nTo some extent, the relationships between these classes of operators are similar to the relationships between their commutative counterparts.\nRecall 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 of non-negative numbers with such that\nMaking the above heuristic comments more precise, we have that T is trace-class iff the series is convergent, T is Hilbert–Schmidt iff is convergent, and T is finite-rank iff the sequence has only finitely many nonzero terms.\nThis allows to relate these classes of operators.\nThe following inclusions hold and are all proper when H is infinite-dimensional:\nThe trace-class operators are given the trace norm The norm corresponding to the Hilbert–Schmidt inner product is\nAlso, the usual [operator norm](https://bliptext.com/articles/operator-norm) is By classical inequalities regarding sequences,\nfor appropriate T.\nIt is also clear that finite-rank operators are dense in both trace-class and Hilbert–Schmidt in their respective norms.\n\n## Trace class as the dual of compact operators\nThe [dual space](https://bliptext.com/articles/dual-space) of c_0 is \\ell^1(\\N).\nSimilarly, we have that the dual of compact operators, denoted by K(H)^*, is the trace-class operators, denoted by B_1.\nThe argument, which we now sketch, is reminiscent of that for the corresponding sequence spaces.\nLet we identify f with the operator T_f defined by\nwhere S_{x,y} is the rank-one operator given by\nThis identification works because the finite-rank operators are norm-dense in K(H).\nIn the event that T_f is a positive operator, for any orthonormal basis u_i, one has\nwhere I is the identity operator:\nBut this means that T_f is trace-class.\nAn appeal to [polar decomposition](https://bliptext.com/articles/polar-decomposition) extend this to the general case, where T_f need not be positive.\nA limiting argument using finite-rank operators shows that Thus K(H)^* is isometrically isomorphic to B_1.\n\n## As the predual of bounded operators\nRecall that the dual of \\ell^1(\\N) is In the present context, the dual of trace-class operators B_1 is the bounded operators B(H).\nMore precisely, the set B_1 is a two-sided [ideal](https://bliptext.com/articles/ideal-ring-theory) in B(H).\nSo given any operator T \\in B(H), we may define a [continuous](https://bliptext.com/articles/continuous-function-topology) [linear functional](https://bliptext.com/articles/linear-functional) \\varphi_T on B_1 by This correspondence between bounded linear operators and elements \\varphi_T of the [dual space](https://bliptext.com/articles/dual-space) of B_1 is an isometric [isomorphism](https://bliptext.com/articles/isomorphism).\nIt follows that B(H) the dual space of B_1.\nThis can be used to define the [weak-* topology](https://bliptext.com/articles/weak-star-operator-topology) on B(H).","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T18:36:56.623Z","updated_at":"2025-01-25T18:36:56.623Z","search_vector":"'class':2 'trace':1","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), specifically [functional analysis](https://bliptext.com/articles/functional-analysis), a **[trace](https://bliptext.com/articles/trace-linear-algebra)-class** operator is a linear operator for which a [trace](https://bliptext.com/articles/trace-linear-algebra) may be defined, such that the [trace](https://bliptext.com/articles/trace-linear-algebra) is a finite number independent of the choice of basis used to compute the [trace](https://bliptext.com/articles/trace-linear-algebra).\nThis trace of trace-class operators generalizes the trace of matrices studied in linear algebra.\nAll trace-class operators are [compact operators](https://bliptext.com/articles/compact-operator).\nIn quantum mechanics, [quantum states](https://bliptext.com/articles/quantum-state) are described by [density matrices](https://bliptext.com/articles/density-matrix), which are certain trace class operators.\nTrace-class operators are essentially the same as [nuclear operators](https://bliptext.com/articles/nuclear-operator), though many authors reserve the term \"trace-class operator\" for the special case of [nuclear operators](https://bliptext.com/articles/nuclear-operator) on [Hilbert spaces](https://bliptext.com/articles/hilbert-space) and use the term \"nuclear operator\" in more general [topological vector spaces](https://bliptext.com/articles/topological-vector-space) (such as [Banach spaces](https://bliptext.com/articles/banach-space)).\nNote that the [trace operator](https://bliptext.com/articles/trace-operator) studied in partial differential equations is an unrelated concept.\n\n# Definition\nLet H be a [separable](https://bliptext.com/articles/separable-space) [Hilbert space](https://bliptext.com/articles/hilbert-space), an [orthonormal basis](https://bliptext.com/articles/orthonormal-basis) and A : H \\to H a [positive](https://bliptext.com/articles/positive-operator-hilbert-space) [bounded linear operator](https://bliptext.com/articles/bounded-linear-operator) on H. The **trace** of A is denoted by and defined as independent of the choice of [orthonormal basis](https://bliptext.com/articles/orthonormal-basis).\nA (not necessarily positive) bounded linear operator is called **trace class** if and only if\nwhere denotes the positive-semidefinite [Hermitian](https://bliptext.com/articles/conjugate-transpose) [square root](https://bliptext.com/articles/square-root-of-a-matrix).\nThe **trace-norm** of a trace class operator T is defined as\nOne can show that the trace-[norm](https://bliptext.com/articles/norm-mathematics) is a [norm](https://bliptext.com/articles/norm-mathematics) on the space of all trace class operators B_1(H) and that B_1(H), with the trace-[norm](https://bliptext.com/articles/norm-mathematics), becomes a [Banach space](https://bliptext.com/articles/banach-space).\nWhen H is finite-dimensional, every (positive) operator is trace class and this definition of trace of A coincides with the definition of the [trace of a matrix](https://bliptext.com/articles/trace-matrix).\nIf H is complex, then A is always [self-adjoint](https://bliptext.com/articles/self-adjoint-operator) (i.e. A=A^*=|A|) though the converse is not necessarily true.\n\n# Equivalent formulations\nGiven a bounded linear operator T : H \\to H, each of the following statements is equivalent to T being in the trace class:\n\n# Examples\n\n## Spectral theorem\nLet T be a bounded self-adjoint operator on a Hilbert space.\nThen T^2 is trace class if and only if T has a [pure point spectrum](https://bliptext.com/articles/spectrum-functional-analysis) with eigenvalues such that\n\n## Mercer's theorem\n[Mercer's theorem](https://bliptext.com/articles/mercer-s-theorem) provides another example of a trace class operator.\nThat is, suppose K is a continuous symmetric [positive-definite kernel](https://bliptext.com/articles/positive-definite-kernel) on L^2([a,b]), defined as then the associated [Hilbert–Schmidt integral operator](https://bliptext.com/articles/hilbert-schmidt-integral-operator) T_K is trace class, i.e.,\n\n## Finite-rank operators\nEvery [finite-rank operator](https://bliptext.com/articles/finite-rank-operator) is a trace-class operator.\nFurthermore, the space of all finite-rank operators is a dense subspace of B_1(H) (when endowed with the trace norm).\nGiven any x, y \\in H, define the operator by Then x \\otimes y is a continuous linear operator of rank 1 and is thus trace class;\nmoreover, for any bounded linear operator A on H (and into H),\n\n# Properties\n \n If A : H \\to H is a non-negative [self-adjoint operator](https://bliptext.com/articles/self-adjoint-operator), then A is trace-class [if and only if](https://bliptext.com/articles/if-and-only-if) Therefore, a [self-adjoint operator](https://bliptext.com/articles/self-adjoint-operator) A is trace-class [if and only if](https://bliptext.com/articles/if-and-only-if) its positive part A^{+} and negative part A^{-} are both trace-class.\n(The positive and negative parts of a self-adjoint operator are obtained by the [continuous functional calculus](https://bliptext.com/articles/continuous-functional-calculus).) \n The trace is a linear functional over the space of trace-class operators, that is,\nThe bilinear map is an [inner product](https://bliptext.com/articles/inner-product) on the trace class; the corresponding norm is called the [Hilbert–Schmidt](https://bliptext.com/articles/hilbert-schmidt-operator) norm.\nThe completion of the trace-class operators in the Hilbert–Schmidt norm are called the Hilbert–Schmidt operators. \n is a positive linear functional such that if T is a trace class operator satisfying then T = 0. \n If T : H \\to H is trace-class then so is T^* and \n If A : H \\to H is bounded, and T : H \\to H is trace-class, then AT and TA are also trace-class (i.e. the space of trace-class operators on H is an [ideal](https://bliptext.com/articles/ideal-ring-theory) in the algebra of bounded linear operators on H), and\nFurthermore, under the same hypothesis, and\nThe last assertion also holds under the weaker hypothesis that A and T are Hilbert–Schmidt. \n If and are two orthonormal bases of H and if T is trace class then \n If A is trace-class, then one can define the [Fredholm determinant](https://bliptext.com/articles/fredholm-determinant) of I + A: where is the spectrum of A. The trace class condition on A guarantees that the infinite product is finite: indeed,\nIt also implies that if and only if (I + A) is invertible. \n If A : H \\to H is trace class then for any [orthonormal basis](https://bliptext.com/articles/orthonormal-basis) of H, the sum of positive terms is finite. \n If A = B^* C for some [Hilbert-Schmidt operators](https://bliptext.com/articles/hilbert-schmidt-operator) B and C then for any normal vector e \\in H, holds. \n \n\n## Lidskii's theorem\nLet A be a trace-class operator in a separable Hilbert space H, and let be the eigenvalues of A. Let us assume that are enumerated with algebraic multiplicities taken into account (that is, if the algebraic multiplicity of \\lambda is k, then \\lambda is repeated k times in the list ).\nLidskii's theorem (named after [Victor Borisovich Lidskii](https://bliptext.com/articles/victor-borisovich-lidskii)) states that\nNote that the series on the right converges absolutely due to [Weyl's inequality](https://bliptext.com/articles/weyl-s-inequality)\nbetween the eigenvalues and the [singular values](https://bliptext.com/articles/singular-value) of the compact operator A.\n\n## Relationship between common classes of operators\nOne can view certain classes of bounded operators as noncommutative analogue of classical [[sequence space](https://bliptext.com/articles/sequence-space)s](https://bliptext.com/articles/sequence-space), with trace-class operators as the noncommutative analogue of the [sequence space](https://bliptext.com/articles/sequence-space) \\ell^1(\\N).\nIndeed, it is possible to apply the [spectral theorem](https://bliptext.com/articles/spectral-theorem) to show that every normal trace-class operator on a separable Hilbert space can be realized in a certain way as an \\ell^1 sequence with respect to some choice of a pair of Hilbert bases.\nIn the same vein, the bounded operators are noncommutative versions of the [compact operators](https://bliptext.com/articles/compact-operator-on-hilbert-space) that of c_0 (the sequences convergent to 0), Hilbert–Schmidt operators correspond to \\ell^2(\\N), and [finite-rank operators](https://bliptext.com/articles/finite-rank-operator) to c_{00} (the sequences that have only finitely many non-zero terms).\nTo some extent, the relationships between these classes of operators are similar to the relationships between their commutative counterparts.\nRecall 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 of non-negative numbers with such that\nMaking the above heuristic comments more precise, we have that T is trace-class iff the series is convergent, T is Hilbert–Schmidt iff is convergent, and T is finite-rank iff the sequence has only finitely many nonzero terms.\nThis allows to relate these classes of operators.\nThe following inclusions hold and are all proper when H is infinite-dimensional:\nThe trace-class operators are given the trace norm The norm corresponding to the Hilbert–Schmidt inner product is\nAlso, the usual [operator norm](https://bliptext.com/articles/operator-norm) is By classical inequalities regarding sequences,\nfor appropriate T.\nIt is also clear that finite-rank operators are dense in both trace-class and Hilbert–Schmidt in their respective norms.\n\n## Trace class as the dual of compact operators\nThe [dual space](https://bliptext.com/articles/dual-space) of c_0 is \\ell^1(\\N).\nSimilarly, we have that the dual of compact operators, denoted by K(H)^*, is the trace-class operators, denoted by B_1.\nThe argument, which we now sketch, is reminiscent of that for the corresponding sequence spaces.\nLet we identify f with the operator T_f defined by\nwhere S_{x,y} is the rank-one operator given by\nThis identification works because the finite-rank operators are norm-dense in K(H).\nIn the event that T_f is a positive operator, for any orthonormal basis u_i, one has\nwhere I is the identity operator:\nBut this means that T_f is trace-class.\nAn appeal to [polar decomposition](https://bliptext.com/articles/polar-decomposition) extend this to the general case, where T_f need not be positive.\nA limiting argument using finite-rank operators shows that Thus K(H)^* is isometrically isomorphic to B_1.\n\n## As the predual of bounded operators\nRecall that the dual of \\ell^1(\\N) is In the present context, the dual of trace-class operators B_1 is the bounded operators B(H).\nMore precisely, the set B_1 is a two-sided [ideal](https://bliptext.com/articles/ideal-ring-theory) in B(H).\nSo given any operator T \\in B(H), we may define a [continuous](https://bliptext.com/articles/continuous-function-topology) [linear functional](https://bliptext.com/articles/linear-functional) \\varphi_T on B_1 by This correspondence between bounded linear operators and elements \\varphi_T of the [dual space](https://bliptext.com/articles/dual-space) of B_1 is an isometric [isomorphism](https://bliptext.com/articles/isomorphism).\nIt follows that B(H) the dual space of B_1.\nThis can be used to define the [weak-* topology](https://bliptext.com/articles/weak-star-operator-topology) on B(H).","revisionCount":0},"hasMore":false,"nextCursor":null}

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