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<h2 id="definition">Definition</h2> <p>Let H {\displaystyle H} be a <a href="/facts/Separable_space/2bmU73bk">separable</a> <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a>, { e k } k = 1 ∞ {\displaystyle \left\{e_{k}\right\}_{k=1}^{\infty }} an <a href="/facts/Orthonormal_basis/pzfKHlJG">orthonormal basis</a> and A : H → H {\displaystyle A:H\to H} a <a href="/facts/Positive_operator_(Hilbert_space)/kvTXwFHj">positive</a> <a href="/facts/Bounded_linear_operator/LYmDTIqR">bounded linear operator</a> on H {\displaystyle H} . The trace of A {\displaystyle A} is denoted by Tr ( A ) {\displaystyle \operatorname {Tr} (A)} and defined as<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> 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 ,} <p>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 <i>if and only if</i> </p> Tr ( | T | ) < ∞ , {\displaystyle \operatorname {Tr} (|T|)<\infty ,} <p>where | T | := T ∗ T {\displaystyle |T|:={\sqrt {T^{*}T}}} denotes the positive-semidefinite <a href="/facts/Conjugate_transpose/7eBWtqVG">Hermitian</a> <a href="/facts/Square_root_of_a_matrix/uWGkQQqN">square root</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p><p>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 <a href="/facts/Norm_(mathematics)/xIbR4uE1">norm</a> 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 <a href="/facts/Banach_space/sDEIk7EC">Banach space</a>. </p><p>When H {\displaystyle H} is finite-dimensional, every (positive) operator is trace class. For A {\displaystyle A} this definition coincides with that of the <a href="/facts/Trace_(matrix)/dchFFvHt">trace of a matrix</a>. If H {\displaystyle H} is complex, then A {\displaystyle A} is always <a href="/facts/Self-adjoint_operator/VrH1nEAK">self-adjoint</a> (i.e. A = A ∗ = | A | {\displaystyle A=A^{*}=|A|} ) though the converse is not necessarily true.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p> <h2 id="equivalent-formulations">Equivalent formulations</h2> <p>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: </p> <ul><li> Tr ( | T | ) = ∑ k ⟨ | T | e k , e k ⟩ {\textstyle \operatorname {Tr} (|T|)=\sum _{k}\left\langle |T|\,e_{k},e_{k}\right\rangle } is finite for every <a href="/facts/Orthonormal_basis/pzfKHlJG">orthonormal basis</a> ( e k ) k {\displaystyle \left(e_{k}\right)_{k}} of H.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a></li> <li>T is a <a href="/facts/Nuclear_operator/kxnx5bIJ">nuclear operator</a>.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a><a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> There exist two <a href="/facts/Orthogonal_(mathematics)/eKmrl6oQ">orthogonal</a> sequences ( x i ) i = 1 ∞ {\displaystyle \left(x_{i}\right)_{i=1}^{\infty }} and ( y i ) i = 1 ∞ {\displaystyle \left(y_{i}\right)_{i=1}^{\infty }} in H {\displaystyle H} and positive <a href="/facts/Real_number/R02gw5Pb">real numbers</a> ( λ i ) i = 1 ∞ {\displaystyle \left(\lambda _{i}\right)_{i=1}^{\infty }} in <a href="/facts/Sequence_space/2HkcM12K"> ℓ 1 {\displaystyle \ell ^{1}} </a> such that ∑ i = 1 ∞ λ i < ∞ {\textstyle \sum _{i=1}^{\infty }\lambda _{i}<\infty } and x ↦ T ( x ) = ∑ i = 1 ∞ λ i ⟨ x , x i ⟩ y i , ∀ x ∈ H , {\displaystyle x\mapsto T(x)=\sum _{i=1}^{\infty }\lambda _{i}\left\langle x,x_{i}\right\rangle y_{i},\quad \forall x\in H,} where ( λ i ) i = 1 ∞ {\displaystyle \left(\lambda _{i}\right)_{i=1}^{\infty }} are the <a href="/facts/Singular_value/0QWHnqGG">singular values</a> of T (or, equivalently, the eigenvalues of | T | {\displaystyle |T|} ), with each value repeated as often as its multiplicity.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a></li> <li>T is a <a href="/facts/Compact_operator/KNkcK4mE">compact operator</a> with Tr ( | T | ) < ∞ . {\displaystyle \operatorname {Tr} (|T|)<\infty .} If T is trace class then<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> ‖ T ‖ 1 = sup { | Tr ( C T ) | : ‖ C ‖ ≤ 1 and C : H → H is a compact operator } . {\displaystyle \|T\|_{1}=\sup \left\{|\operatorname {Tr} (CT)|:\|C\|\leq 1{\text{ and }}C:H\to H{\text{ is a compact operator }}\right\}.} </li> <li>T is an <a href="/facts/Integral_linear_operator/vmjadg6A">integral operator</a>.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a></li> <li>T is equal to the composition of two <a href="/facts/Hilbert-Schmidt_operator/Xhy17yn6">Hilbert-Schmidt operators</a>.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a></li> <li> | T | {\textstyle {\sqrt {|T|}}} is a Hilbert-Schmidt operator.<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a></li></ul> <h2 id="examples">Examples</h2> <h3>Spectral theorem</h3> <p>Let T {\displaystyle T} be a bounded self-adjoint operator on a Hilbert space. Then T 2 {\displaystyle T^{2}} is trace class <i>if and only if</i> T {\displaystyle T} has a <a href="/facts/Spectrum_(functional_analysis)/cA1evTo6">pure point spectrum</a> with eigenvalues { λ i ( T ) } i = 1 ∞ {\displaystyle \left\{\lambda _{i}(T)\right\}_{i=1}^{\infty }} such that<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> </p> Tr ( T 2 ) = ∑ i = 1 ∞ λ i ( T 2 ) < ∞ . {\displaystyle \operatorname {Tr} (T^{2})=\sum _{i=1}^{\infty }\lambda _{i}(T^{2})<\infty .} <h3>Mercer's theorem</h3> <p><a href="/facts/Mercer%2527s_theorem/6CmBHnHc">Mercer's theorem</a> provides another example of a trace class operator. That is, suppose K {\displaystyle K} is a continuous symmetric <a href="/facts/Positive-definite_kernel/Ws3zkxfl">positive-definite kernel</a> on L 2 ( [ a , b ] ) {\displaystyle L^{2}([a,b])} , defined as </p> K ( s , t ) = ∑ j = 1 ∞ λ j e j ( s ) e j ( t ) {\displaystyle K(s,t)=\sum _{j=1}^{\infty }\lambda _{j}\,e_{j}(s)\,e_{j}(t)} <p>then the associated <a href="/facts/Hilbert%25E2%2580%2593Schmidt_integral_operator/KsMYpV1m">Hilbert–Schmidt integral operator</a> T K {\displaystyle T_{K}} is trace class, i.e., </p> Tr ( T K ) = ∫ a b K ( t , t ) d t = ∑ i λ i . {\displaystyle \operatorname {Tr} (T_{K})=\int _{a}^{b}K(t,t)\,dt=\sum _{i}\lambda _{i}.} <h3>Finite-rank operators</h3> <p>Every <a href="/facts/Finite-rank_operator/jA2kCmCA">finite-rank operator</a> is a trace-class operator. Furthermore, the space of all finite-rank operators is a <a href="/facts/Dense_subspace/nPT9Yxao">dense subspace</a> of B 1 ( H ) {\displaystyle B_{1}(H)} (when endowed with the trace norm).<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> </p><p>Given any x , y ∈ H , {\displaystyle x,y\in H,} define the operator x ⊗ y : H → H {\displaystyle x\otimes y:H\to H} by ( x ⊗ y ) ( z ) := ⟨ z , y ⟩ x . {\displaystyle (x\otimes y)(z):=\langle z,y\rangle x.} Then x ⊗ y {\displaystyle x\otimes y} is a continuous linear operator of rank 1 and is thus trace class; moreover, for any bounded linear operator <i>A</i> on <i>H</i> (and into <i>H</i>), Tr ( A ( x ⊗ y ) ) = ⟨ A x , y ⟩ . {\displaystyle \operatorname {Tr} (A(x\otimes y))=\langle Ax,y\rangle .} <a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> </p> <h2 id="properties">Properties</h2> <ol> <li>If A : H → H {\displaystyle A:H\to H} is a non-negative <a href="/facts/Self-adjoint_operator/VrH1nEAK">self-adjoint operator</a>, then A {\displaystyle A} is trace-class if and only if Tr A < ∞ . {\displaystyle \operatorname {Tr} A<\infty .} Therefore, a self-adjoint operator A {\displaystyle A} is trace-class <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> its positive part A + {\displaystyle A^{+}} and negative part A − {\displaystyle A^{-}} are both trace-class. (The positive and negative parts of a self-adjoint operator are obtained by the <a href="/facts/Continuous_functional_calculus/cJ2Nj9Bo">continuous functional calculus</a>.)</li> <li>The trace is a <a href="/facts/Linear_functional/oGh7Asrz">linear functional</a> over the space of trace-class operators, that is, Tr ( a A + b B ) = a Tr ( A ) + b Tr ( B ) . {\displaystyle \operatorname {Tr} (aA+bB)=a\operatorname {Tr} (A)+b\operatorname {Tr} (B).} The bilinear map ⟨ A , B ⟩ = Tr ( A ∗ B ) {\displaystyle \langle A,B\rangle =\operatorname {Tr} (A^{*}B)} is an <a href="/facts/Inner_product/HoyjElEy">inner product</a> on the trace class; the corresponding norm is called the <a href="/facts/Hilbert%25E2%2580%2593Schmidt_operator/Xhy17yn6">Hilbert–Schmidt</a> norm. The completion of the trace-class operators in the Hilbert–Schmidt norm are called the Hilbert–Schmidt operators.</li> <li> Tr : B 1 ( H ) → C {\displaystyle \operatorname {Tr} :B_{1}(H)\to \mathbb {C} } is a positive linear functional such that if T {\displaystyle T} is a trace class operator satisfying T ≥ 0 and Tr T = 0 , {\displaystyle T\geq 0{\text{ and }}\operatorname {Tr} T=0,} then T = 0. {\displaystyle T=0.} <a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a></li> <li>If T : H → H {\displaystyle T:H\to H} is trace-class then so is T ∗ {\displaystyle T^{*}} and ‖ T ‖ 1 = ‖ T ∗ ‖ 1 . {\displaystyle \|T\|_{1}=\left\|T^{*}\right\|_{1}.} <a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a></li> <li>If A : H → H {\displaystyle A:H\to H} is bounded, and T : H → H {\displaystyle T:H\to H} is trace-class, then A T {\displaystyle AT} and T A {\displaystyle TA} are also trace-class (i.e. the space of trace-class operators on <i>H</i> is a two-sided <a href="/facts/Ideal_(ring_theory)/vCvytduX">ideal</a> in the algebra of bounded linear operators on <i>H</i>), and<a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a><a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a> ‖ A T ‖ 1 = Tr ( | A T | ) ≤ ‖ A ‖ ‖ T ‖ 1 , ‖ T A ‖ 1 = Tr ( | T A | ) ≤ ‖ A ‖ ‖ T ‖ 1 . {\displaystyle \|AT\|_{1}=\operatorname {Tr} (|AT|)\leq \|A\|\|T\|_{1},\quad \|TA\|_{1}=\operatorname {Tr} (|TA|)\leq \|A\|\|T\|_{1}.} Furthermore, under the same hypothesis,<a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a> Tr ( A T ) = Tr ( T A ) {\displaystyle \operatorname {Tr} (AT)=\operatorname {Tr} (TA)} and | Tr ( A T ) | ≤ ‖ A ‖ ‖ T ‖ . {\displaystyle |\operatorname {Tr} (AT)|\leq \|A\|\|T\|.} The last assertion also holds under the weaker hypothesis that <i>A</i> and <i>T</i> are Hilbert–Schmidt.</li> <li>If ( e k ) k {\displaystyle \left(e_{k}\right)_{k}} and ( f k ) k {\displaystyle \left(f_{k}\right)_{k}} are two orthonormal bases of <i>H</i> and if <i>T</i> is trace class then ∑ k | ⟨ T e k , f k ⟩ | ≤ ‖ T ‖ 1 . {\textstyle \sum _{k}\left|\left\langle Te_{k},f_{k}\right\rangle \right|\leq \|T\|_{1}.} <a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a></li> <li>If <i>A</i> is trace-class, then one can define the <a href="/facts/Fredholm_determinant/9TvIBCvI">Fredholm determinant</a> of I + A {\displaystyle I+A} : det ( I + A ) := ∏ n ≥ 1 [ 1 + λ n ( A ) ] , {\displaystyle \det(I+A):=\prod _{n\geq 1}[1+\lambda _{n}(A)],} where { λ n ( A ) } n {\displaystyle \{\lambda _{n}(A)\}_{n}} is the spectrum of A . {\displaystyle A.} The trace class condition on A {\displaystyle A} guarantees that the infinite product is finite: indeed, det ( I + A ) ≤ e ‖ A ‖ 1 . {\displaystyle \det(I+A)\leq e^{\|A\|_{1}}.} It also implies that det ( I + A ) ≠ 0 {\displaystyle \det(I+A)\neq 0} if and only if ( I + A ) {\displaystyle (I+A)} is invertible.</li> <li>If A : H → H {\displaystyle A:H\to H} is trace class then for any <a href="/facts/Orthonormal_basis/pzfKHlJG">orthonormal basis</a> ( e k ) k {\displaystyle \left(e_{k}\right)_{k}} of H , {\displaystyle H,} the sum of positive terms ∑ k | ⟨ A e k , e k ⟩ | {\textstyle \sum _{k}\left|\left\langle A\,e_{k},e_{k}\right\rangle \right|} is finite.<a class="footnote-ref" id="fnref:23" href="#fn:23"><sup>23</sup></a></li> <li>If A = B ∗ C {\displaystyle A=B^{*}C} for some <a href="/facts/Hilbert-Schmidt_operator/Xhy17yn6">Hilbert-Schmidt operators</a> B {\displaystyle B} and C {\displaystyle C} then for any normal vector e ∈ H , {\displaystyle e\in H,} | ⟨ A e , e ⟩ | = 1 2 ( ‖ B e ‖ 2 + ‖ C e ‖ 2 ) {\textstyle |\langle Ae,e\rangle |={\frac {1}{2}}\left(\|Be\|^{2}+\|Ce\|^{2}\right)} holds.<a class="footnote-ref" id="fnref:24" href="#fn:24"><sup>24</sup></a></li> </ol> <h3>Lidskii's theorem</h3> <p>Let A {\displaystyle A} be a trace-class operator in a separable Hilbert space H , {\displaystyle H,} and let { λ n ( A ) } n = 1 N ≤ ∞ {\displaystyle \{\lambda _{n}(A)\}_{n=1}^{N\leq \infty }} be the eigenvalues of A . {\displaystyle A.} Let us assume that λ n ( A ) {\displaystyle \lambda _{n}(A)} are enumerated with algebraic multiplicities taken into account (that is, if the algebraic multiplicity of λ {\displaystyle \lambda } is k , {\displaystyle k,} then λ {\displaystyle \lambda } is repeated k {\displaystyle k} times in the list λ 1 ( A ) , λ 2 ( A ) , … {\displaystyle \lambda _{1}(A),\lambda _{2}(A),\dots } ). Lidskii's theorem (named after <a href="/facts/Victor_Borisovich_Lidskii/JtGB8cqo">Victor Borisovich Lidskii</a>) states that Tr ( A ) = ∑ n = 1 N λ n ( A ) {\displaystyle \operatorname {Tr} (A)=\sum _{n=1}^{N}\lambda _{n}(A)} </p><p>Note that the series on the right converges absolutely due to <a href="/facts/Weyl%2527s_inequality/kFmWmSX8">Weyl's inequality</a> ∑ n = 1 N | λ n ( A ) | ≤ ∑ m = 1 M s m ( A ) {\displaystyle \sum _{n=1}^{N}\left|\lambda _{n}(A)\right|\leq \sum _{m=1}^{M}s_{m}(A)} between the eigenvalues { λ n ( A ) } n = 1 N {\displaystyle \{\lambda _{n}(A)\}_{n=1}^{N}} and the <a href="/facts/Singular_value/0QWHnqGG">singular values</a> { s m ( A ) } m = 1 M {\displaystyle \{s_{m}(A)\}_{m=1}^{M}} of the compact operator A . {\displaystyle A.} <a class="footnote-ref" id="fnref:25" href="#fn:25"><sup>25</sup></a> </p> <h3>Relationship between common classes of operators</h3> <p>One can view certain classes of bounded operators as noncommutative analogue of classical <a href="/facts/Sequence_space/2HkcM12K">sequence spaces</a>, with trace-class operators as the noncommutative analogue of the <a href="/facts/Sequence_space/2HkcM12K">sequence space</a> ℓ 1 ( N ) . {\displaystyle \ell ^{1}(\mathbb {N} ).} </p><p>Indeed, it is possible to apply the <a href="/facts/Spectral_theorem/Vqc04B21">spectral theorem</a> to show that every normal trace-class operator on a separable Hilbert space can be realized in a certain way as an ℓ 1 {\displaystyle \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 ℓ ∞ ( N ) , {\displaystyle \ell ^{\infty }(\mathbb {N} ),} the <a href="/facts/Compact_operator_on_Hilbert_space/xaL1NE6M">compact operators</a> that of c 0 {\displaystyle c_{0}} (the sequences convergent to 0), Hilbert–Schmidt operators correspond to ℓ 2 ( N ) , {\displaystyle \ell ^{2}(\mathbb {N} ),} and <a href="/facts/Finite-rank_operator/jA2kCmCA">finite-rank operators</a> to c 00 {\displaystyle 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. </p><p>Recall that every compact operator T {\displaystyle T} on a Hilbert space takes the following canonical form: there exist orthonormal bases ( u i ) i {\displaystyle (u_{i})_{i}} and ( v i ) i {\displaystyle (v_{i})_{i}} and a sequence ( α i ) i {\displaystyle \left(\alpha _{i}\right)_{i}} of non-negative numbers with α i → 0 {\displaystyle \alpha _{i}\to 0} such that T x = ∑ i α i ⟨ x , v i ⟩ u i for all x ∈ H . {\displaystyle Tx=\sum _{i}\alpha _{i}\langle x,v_{i}\rangle u_{i}\quad {\text{ for all }}x\in H.} Making the above heuristic comments more precise, we have that T {\displaystyle T} is trace-class iff the series ∑ i α i {\textstyle \sum _{i}\alpha _{i}} is convergent, T {\displaystyle T} is Hilbert–Schmidt iff ∑ i α i 2 {\textstyle \sum _{i}\alpha _{i}^{2}} is convergent, and T {\displaystyle T} is finite-rank iff the sequence ( α i ) i {\displaystyle \left(\alpha _{i}\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 {\displaystyle H} is infinite-dimensional: { finite rank } ⊆ { trace class } ⊆ { Hilbert--Schmidt } ⊆ { compact } . {\displaystyle \{{\text{ finite rank }}\}\subseteq \{{\text{ trace class }}\}\subseteq \{{\text{ Hilbert--Schmidt }}\}\subseteq \{{\text{ compact }}\}.} </p><p>The trace-class operators are given the trace norm ‖ T ‖ 1 = Tr [ ( T ∗ T ) 1 / 2 ] = ∑ i α i . {\textstyle \|T\|_{1}=\operatorname {Tr} \left[\left(T^{*}T\right)^{1/2}\right]=\sum _{i}\alpha _{i}.} The norm corresponding to the Hilbert–Schmidt inner product is ‖ T ‖ 2 = [ Tr ( T ∗ T ) ] 1 / 2 = ( ∑ i α i 2 ) 1 / 2 . {\displaystyle \|T\|_{2}=\left[\operatorname {Tr} \left(T^{*}T\right)\right]^{1/2}=\left(\sum _{i}\alpha _{i}^{2}\right)^{1/2}.} Also, the usual <a href="/facts/Operator_norm/a5l0TkFW">operator norm</a> is ‖ T ‖ = sup i ( α i ) . {\textstyle \|T\|=\sup _{i}\left(\alpha _{i}\right).} By classical inequalities regarding sequences, ‖ T ‖ ≤ ‖ T ‖ 2 ≤ ‖ T ‖ 1 {\displaystyle \|T\|\leq \|T\|_{2}\leq \|T\|_{1}} for appropriate T . {\displaystyle T.} </p><p>It is also clear that finite-rank operators are dense in both trace-class and Hilbert–Schmidt in their respective norms. </p> <h3>Trace class as the dual of compact operators</h3> <p>The <a href="/facts/Dual_space/4Uw4Knz1">dual space</a> of c 0 {\displaystyle c_{0}} is ℓ 1 ( N ) . {\displaystyle \ell ^{1}(\mathbb {N} ).} Similarly, we have that the dual of compact operators, denoted by K ( H ) ∗ , {\displaystyle K(H)^{*},} is the trace-class operators, denoted by B 1 . {\displaystyle B_{1}.} The argument, which we now sketch, is reminiscent of that for the corresponding sequence spaces. Let f ∈ K ( H ) ∗ , {\displaystyle f\in K(H)^{*},} we identify f {\displaystyle f} with the operator T f {\displaystyle T_{f}} defined by ⟨ T f x , y ⟩ = f ( S x , y ) , {\displaystyle \langle T_{f}x,y\rangle =f\left(S_{x,y}\right),} where S x , y {\displaystyle S_{x,y}} is the rank-one operator given by S x , y ( h ) = ⟨ h , y ⟩ x . {\displaystyle S_{x,y}(h)=\langle h,y\rangle x.} </p><p>This identification works because the finite-rank operators are norm-dense in K ( H ) . {\displaystyle K(H).} In the event that T f {\displaystyle T_{f}} is a positive operator, for any orthonormal basis u i , {\displaystyle u_{i},} one has ∑ i ⟨ T f u i , u i ⟩ = f ( I ) ≤ ‖ f ‖ , {\displaystyle \sum _{i}\langle T_{f}u_{i},u_{i}\rangle =f(I)\leq \|f\|,} where I {\displaystyle I} is the identity operator: I = ∑ i ⟨ ⋅ , u i ⟩ u i . {\displaystyle I=\sum _{i}\langle \cdot ,u_{i}\rangle u_{i}.} </p><p>But this means that T f {\displaystyle T_{f}} is trace-class. An appeal to <a href="/facts/Polar_decomposition/XWQLdp9B">polar decomposition</a> extend this to the general case, where T f {\displaystyle T_{f}} need not be positive. </p><p>A limiting argument using finite-rank operators shows that ‖ T f ‖ 1 = ‖ f ‖ . {\displaystyle \|T_{f}\|_{1}=\|f\|.} Thus K ( H ) ∗ {\displaystyle K(H)^{*}} is <a href="/facts/Isometrically_isomorphic/K5wX8ah6">isometrically isomorphic</a> to B 1 . {\displaystyle B_{1}.} </p> <h3>As the predual of bounded operators</h3> <p>Recall that the dual of ℓ 1 ( N ) {\displaystyle \ell ^{1}(\mathbb {N} )} is ℓ ∞ ( N ) . {\displaystyle \ell ^{\infty }(\mathbb {N} ).} In the present context, the dual of trace-class operators B 1 {\displaystyle B_{1}} is the bounded operators B ( H ) . {\displaystyle B(H).} More precisely, the set B 1 {\displaystyle B_{1}} is a two-sided <a href="/facts/Ideal_(ring_theory)/vCvytduX">ideal</a> in B ( H ) . {\displaystyle B(H).} So given any operator T ∈ B ( H ) , {\displaystyle T\in B(H),} we may define a <a href="/facts/Continuous_linear_operator/KcqjDBrJ">continuous</a> <a href="/facts/Linear_functional/oGh7Asrz">linear functional</a> φ T {\displaystyle \varphi _{T}} on B 1 {\displaystyle B_{1}} by φ T ( A ) = Tr ( A T ) . {\displaystyle \varphi _{T}(A)=\operatorname {Tr} (AT).} This correspondence between bounded linear operators and elements φ T {\displaystyle \varphi _{T}} of the <a href="/facts/Dual_space/4Uw4Knz1">dual space</a> of B 1 {\displaystyle B_{1}} is an <a href="/facts/Isometric_isomorphism/K5wX8ah6">isometric isomorphism</a>. It follows that B ( H ) {\displaystyle B(H)} is the dual space of B 1 . {\displaystyle B_{1}.} This can be used to define the <a href="/facts/Weak-star_operator_topology/ySc1UXjd">weak-* topology</a> on B ( H ) . {\displaystyle B(H).} </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Nuclear_operators_between_Banach_spaces/rIhJTQ15">Nuclear operators between Banach spaces</a> – operators on Banach spaces with properties similar to finite-dimensional operatorsPages displaying wikidata descriptions as a fallback</li></ul> <h2 id="bibliography">Bibliography</h2> <ul><li>Conway, John B. (2000). <i>A Course in Operator Theory</i>. Providence (R.I.): American Mathematical Soc. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-2065-0.</li> <li>Conway, John B. (1990). <i>A course in functional analysis</i>. New York: Springer-Verlag. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-97245-9. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/21195908">21195908</a>.</li> <li>Dixmier, J. (1969). <i>Les Algebres d'Operateurs dans l'Espace Hilbertien</i>. Gauthier-Villars.</li> <li>Mittelstaedt, Peter (2009). "Mixed State". <i>Compendium of Quantum Physics</i>. Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 389–390. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-540-70626-7_120">10.1007/978-3-540-70626-7_120</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-70622-9.</li> <li><a href="/facts/Michael_C._Reed/PjUGTNwk">Reed, M.</a>; <a href="/facts/Barry_Simon/BImmcE2I">Simon, B.</a> (1980). <i>Methods of Modern Mathematical Physics: Vol 1: Functional analysis</i>. Academic Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-12-585050-6.</li> <li>Schaefer, Helmut H. (1999). <i>Topological Vector Spaces</i>. <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">GTM</a>. Vol. 3. New York, NY: Springer New York Imprint Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4612-7155-0. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/840278135">840278135</a>.</li> <li><a href="/facts/Barry_Simon/BImmcE2I">Simon, Barry</a> (2010). <i>Szegő's theorem and its descendants: spectral theory for L² perturbations of orthogonal polynomials</i>. Princeton University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-691-14704-8.</li> <li><a href="/facts/Fran%25C3%25A7ois_Tr%25C3%25A8ves/cpjQbAov">Trèves, François</a> (2006) [1967]. <i>Topological Vector Spaces, Distributions and Kernels</i>. Mineola, N.Y.: Dover Publications. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-45352-1. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/853623322">853623322</a>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Mittelstaedt 2009, pp. 389–390. - Mittelstaedt, Peter (2009). "Mixed State". Compendium of Quantum Physics. Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 389–390. doi:10.1007/978-3-540-70626-7_120. ISBN 978-3-540-70622-9. <a href="https://doi.org/10.1007%2F978-3-540-70626-7_120" target="_blank">https://doi.org/10.1007%2F978-3-540-70626-7_120</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Conway 2000, p. 86. - Conway, John B. (2000). A Course in Operator Theory. Providence (R.I.): American Mathematical Soc. ISBN 978-0-8218-2065-0. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Reed & Simon 1980, p. 206. - Reed, M.; Simon, B. (1980). Methods of Modern Mathematical Physics: Vol 1: Functional analysis. Academic Press. ISBN 978-0-12-585050-6. <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Reed & Simon 1980, p. 196. - Reed, M.; Simon, B. (1980). Methods of Modern Mathematical Physics: Vol 1: Functional analysis. Academic Press. ISBN 978-0-12-585050-6. <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Reed & Simon 1980, p. 195. - Reed, M.; Simon, B. (1980). Methods of Modern Mathematical Physics: Vol 1: Functional analysis. Academic Press. ISBN 978-0-12-585050-6. <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Conway 2000, p. 86. - Conway, John B. (2000). A Course in Operator Theory. Providence (R.I.): American Mathematical Soc. ISBN 978-0-8218-2065-0. <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Trèves 2006, p. 494. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Conway 2000, p. 89. - Conway, John B. (2000). A Course in Operator Theory. Providence (R.I.): American Mathematical Soc. ISBN 978-0-8218-2065-0. <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Reed & Simon 1980, pp. 203–204, 209. - Reed, M.; Simon, B. (1980). Methods of Modern Mathematical Physics: Vol 1: Functional analysis. Academic Press. ISBN 978-0-12-585050-6. <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Conway 1990, p. 268. - Conway, John B. (1990). A course in functional analysis. New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908. <a href="https://search.worldcat.org/oclc/21195908" target="_blank">https://search.worldcat.org/oclc/21195908</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Trèves 2006, pp. 502–508. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. <a href="https://search.worldcat.org/oclc/853623322" target="_blank">https://search.worldcat.org/oclc/853623322</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Conway 1990, p. 267. - Conway, John B. (1990). A course in functional analysis. New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908. <a href="https://search.worldcat.org/oclc/21195908" target="_blank">https://search.worldcat.org/oclc/21195908</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Conway 1990, p. 267. - Conway, John B. (1990). A course in functional analysis. New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908. <a href="https://search.worldcat.org/oclc/21195908" target="_blank">https://search.worldcat.org/oclc/21195908</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Simon 2010, p. 21. - Simon, Barry (2010). Szegő's theorem and its descendants: spectral theory for L² perturbations of orthogonal polynomials. Princeton University Press. ISBN 978-0-691-14704-8. <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Conway 1990, p. 268. - Conway, John B. (1990). A course in functional analysis. New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908. <a href="https://search.worldcat.org/oclc/21195908" target="_blank">https://search.worldcat.org/oclc/21195908</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>Conway 1990, p. 268. - Conway, John B. (1990). A course in functional analysis. New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908. <a href="https://search.worldcat.org/oclc/21195908" target="_blank">https://search.worldcat.org/oclc/21195908</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>Conway 1990, p. 267. - Conway, John B. (1990). A course in functional analysis. New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908. <a href="https://search.worldcat.org/oclc/21195908" target="_blank">https://search.worldcat.org/oclc/21195908</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>Conway 1990, p. 267. - Conway, John B. (1990). A course in functional analysis. New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908. <a href="https://search.worldcat.org/oclc/21195908" target="_blank">https://search.worldcat.org/oclc/21195908</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> <li id="fn:19"><p>Conway 1990, p. 267. - Conway, John B. (1990). A course in functional analysis. New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908. <a href="https://search.worldcat.org/oclc/21195908" target="_blank">https://search.worldcat.org/oclc/21195908</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li> <li id="fn:20"><p>Reed & Simon 1980, p. 218. - Reed, M.; Simon, B. (1980). Methods of Modern Mathematical Physics: Vol 1: Functional analysis. Academic Press. ISBN 978-0-12-585050-6. <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li> <li id="fn:21"><p>Conway 1990, p. 267. - Conway, John B. (1990). A course in functional analysis. New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908. <a href="https://search.worldcat.org/oclc/21195908" target="_blank">https://search.worldcat.org/oclc/21195908</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></p></li> <li id="fn:22"><p>Conway 1990, p. 268. - Conway, John B. (1990). A course in functional analysis. New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908. <a href="https://search.worldcat.org/oclc/21195908" target="_blank">https://search.worldcat.org/oclc/21195908</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></p></li> <li id="fn:23"><p>Conway 1990, p. 267. - Conway, John B. (1990). A course in functional analysis. New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908. <a href="https://search.worldcat.org/oclc/21195908" target="_blank">https://search.worldcat.org/oclc/21195908</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></p></li> <li id="fn:24"><p>Conway 1990, p. 267. - Conway, John B. (1990). A course in functional analysis. New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908. <a href="https://search.worldcat.org/oclc/21195908" target="_blank">https://search.worldcat.org/oclc/21195908</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></p></li> <li id="fn:25"><p>Simon, B. (2005) Trace ideals and their applications, Second Edition, American Mathematical Society. <a href="#fnref:25" class="footnote-back-ref">↩</a></p></li> </ol>