Lecture Notes in Functional Analysis

by William L. Paschke

edition 0.9

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Contents

Frontmatter

I. Hilbert Space

II. Bounded Operators

III. Compact Operators

IV. The Spectral Theorem

Index and References

B. Trace Class Operators

We again assume HHilbert space\( H \) is a separable, infinite-dimensional Hilbert space.

Proposition III.19

For Aself-adjoint operator≥greater than or equal to0zero \( A\geq 0 \) in ℒbounded linear operators⁡(HHilbert space) \( \mathcal{L}\mathopen{}\left( H\right)\mathclose{} \) and orthonormal bases (sequenceeunit vectorninteger)sequenceninteger=1one∞infinity \( \mathopen{}\left({e}_{n}\right)\mathclose{}_{n=1}^{\infty} \) and (sequencefunit vectorninteger)sequenceninteger=1one∞infinity \( \mathopen{}\left({f}_{n}\right)\mathclose{}_{n=1}^{\infty} \), we have ∑summationninteger=1one∞infinity 〈Aself-adjoint operator⁡(eunit vectorninteger), eunit vectorninteger〉 =equals∑summationninteger=1one∞infinity 〈Aself-adjoint operator⁡(funit vectorninteger), funit vectorninteger〉 \( \sum_{n=1}^{\infty}{} \mathopen{}\left\langle{}A\mathopen{}\left( {e}_{n}\right)\mathclose{}, {e}_{n}\right\rangle\mathclose{} = \sum_{n=1}^{\infty}{} \mathopen{}\left\langle{}A\mathopen{}\left( {f}_{n}\right)\mathclose{}, {f}_{n}\right\rangle\mathclose{} \).

Proof. Since we may interchange the order of summands in a double series with nonnegative terms, ∑summationninteger 〈Aself-adjoint operator⁡(eunit vectorninteger), eunit vectorninteger〉 =equals∑summationninteger 〈Aself-adjoint operator1one2two⁡(eunit vectorninteger), Aself-adjoint operator1one2two⁡(eunit vectorninteger)〉 =equals∑summationninteger ‖Aself-adjoint operator1one2two⁡(eunit vectorninteger)‖ 2two =equals∑summationninteger ∑summationminteger |modulus〈Aself-adjoint operator1one2two⁡(eunit vectorninteger), funit vectorminteger〉|modulus 2two =equals∑summationninteger ∑summationminteger |modulus〈Aself-adjoint operator1one2two⁡(funit vectorminteger), eunit vectorninteger〉|modulus 2two =equals∑summationminteger ∑summationninteger |modulus〈Aself-adjoint operator1one2two⁡(funit vectorminteger), eunit vectorninteger〉|modulus 2two =equals∑summationminteger 〈Aself-adjoint operator⁡(funit vectorminteger), funit vectorminteger〉 . \[ \sum_{n}{} \mathopen{}\left\langle{}A\mathopen{}\left( {e}_{n}\right)\mathclose{}, {e}_{n}\right\rangle\mathclose{} = \sum_{n}{} \mathopen{}\left\langle{}{A}^{\frac{1}{2}}\mathopen{}\left( {e}_{n}\right)\mathclose{}, {A}^{\frac{1}{2}}\mathopen{}\left( {e}_{n}\right)\mathclose{}\right\rangle\mathclose{} = \sum_{n}{} {\mathopen{}\left\lVert{}{A}^{\frac{1}{2}}\mathopen{}\left( {e}_{n}\right)\mathclose{}\right\rVert\mathclose{}}^{2} = \sum_{n}{} \sum_{m}{} {\mathopen{}\left\lvert{}\mathopen{}\left\langle{}{A}^{\frac{1}{2}}\mathopen{}\left( {e}_{n}\right)\mathclose{}, {f}_{m}\right\rangle\mathclose{}\right\rvert\mathclose{}}^{2} = \sum_{n}{} \sum_{m}{} {\mathopen{}\left\lvert{}\mathopen{}\left\langle{}{A}^{\frac{1}{2}}\mathopen{}\left( {f}_{m}\right)\mathclose{}, {e}_{n}\right\rangle\mathclose{}\right\rvert\mathclose{}}^{2} = \sum_{m}{} \sum_{n}{} {\mathopen{}\left\lvert{}\mathopen{}\left\langle{}{A}^{\frac{1}{2}}\mathopen{}\left( {f}_{m}\right)\mathclose{}, {e}_{n}\right\rangle\mathclose{}\right\rvert\mathclose{}}^{2} = \sum_{m}{} \mathopen{}\left\langle{}A\mathopen{}\left( {f}_{m}\right)\mathclose{}, {f}_{m}\right\rangle\mathclose{} \text{.} \]

Notation III.20

Let ℒbounded linear operators⁡(HHilbert space)+positive elements=equals{setAself-adjoint operator∈element ofℒbounded linear operators⁡(HHilbert space)|such that Aself-adjoint operator≥greater than or equal to0zero }set \( {\mathcal{L}\mathopen{}\left( H\right)\mathclose{}}^{+}= \mathopen{}\left\{\, A\in \mathcal{L}\mathopen{}\left( H\right)\mathclose{}\,\middle\vert\, , A\geq 0, \,\right\}\mathclose{} \). Define Trtrace:maps ℒbounded linear operators⁡(HHilbert space)+positive elements →to [interval0zero, +⁢∞infinity]interval \( \operatorname{Tr} : {\mathcal{L}\mathopen{}\left( H\right)\mathclose{}}^{+} \to \mathopen{}\left[0, {+}\infty\right]\mathclose{} \) by Trtrace⁡(Aself-adjoint operator)=equals∑summationninteger=1one∞infinity 〈Aself-adjoint operator⁡(eunit vectorninteger), eunit vectorninteger〉 \( \operatorname{Tr}\mathopen{}\left( A\right)\mathclose{}= \sum_{n=1}^{\infty}{} \mathopen{}\left\langle{}A\mathopen{}\left( {e}_{n}\right)\mathclose{}, {e}_{n}\right\rangle\mathclose{} \).

Proposition III.21

For Aself-adjoint operator∈element ofℒbounded linear operators⁡(HHilbert space)+positive elements \( A\in {\mathcal{L}\mathopen{}\left( H\right)\mathclose{}}^{+} \), Trtrace⁡(Aself-adjoint operator)<less than∞infinity \( \operatorname{Tr}\mathopen{}\left( A\right)\mathclose{}\lt \infty \) if and only if Aself-adjoint operator∈element of𝒦compact linear operators⁡(HHilbert space) \( A\in \mathcal{K}\mathopen{}\left( H\right)\mathclose{} \) and ∑summationλnonnegative real numberninteger<less than∞infinity \( \sum{}{λ}_{n}\lt \infty \) where (sequenceλnonnegative real numberninteger)sequence \( \mathopen{}\left({λ}_{n}\right)\mathclose{} \) are the eigenvalues of Aself-adjoint operator\( A \). In this case, Trtrace⁡(Aself-adjoint operator)=equals∑summationλnonnegative real numberninteger \( \operatorname{Tr}\mathopen{}\left( A\right)\mathclose{}= \sum{}{λ}_{n} \) where the eigenvalues λnonnegative real numberninteger \( {λ}_{n} \) are counted according to multiplicity (that is, if the dimension of Kerkernel⁡(λnonnegative real number-minusAself-adjoint operator) \( \operatorname{Ker}\mathopen{}\left( λ-A\right)\mathclose{} \) is dmetric\( d \), then λnonnegative real number\( λ \) appears dmetric\( d \) times in ∑summationλnonnegative real numberninteger \( \sum{}{λ}_{n} \)).

Proof. (⇒) Pick an orthonormal basis (sequenceeunit vectorninteger)sequenceninteger=1one∞infinity \( \mathopen{}\left({e}_{n}\right)\mathclose{}_{n=1}^{\infty} \). Then Trtrace⁡(Aself-adjoint operator)=equals∑summation ‖Aself-adjoint operator1one2two⁡(eunit vectorninteger)‖ 2two <less than∞infinity \( \operatorname{Tr}\mathopen{}\left( A\right)\mathclose{}= \sum{} {\mathopen{}\left\lVert{}{A}^{\frac{1}{2}}\mathopen{}\left( {e}_{n}\right)\mathclose{}\right\rVert\mathclose{}}^{2} \lt \infty \). For xvector∈element ofHHilbert space \( x\in H \), Aself-adjoint operator⁡(xvector)=equalsAself-adjoint operator1one2two⁡(Aself-adjoint operator1one2two⁡(xvector))=equalsAself-adjoint operator1one2two⁢times∑summationninteger 〈Aself-adjoint operator1one2two⁡(xvector), eunit vectorninteger〉⁢timeseunit vectorninteger =equals∑summationninteger 〈Aself-adjoint operator1one2two⁡(xvector), eunit vectorninteger〉⁢timesAself-adjoint operator1one2two⁡(eunit vectorninteger) =equals∑summationninteger (Aself-adjoint operator1one2two⁡(eunit vectorninteger)⊗Aself-adjoint operator1one2two⁡(eunit vectorninteger))⁡(xvector) . \[ A\mathopen{}\left( x\right)\mathclose{}= {A}^{\frac{1}{2}}\mathopen{}\left( {A}^{\frac{1}{2}}\mathopen{}\left( x\right)\mathclose{}\right)\mathclose{}= {A}^{\frac{1}{2}}\sum_{n}{} \mathopen{}\left\langle{}{A}^{\frac{1}{2}}\mathopen{}\left( x\right)\mathclose{}, {e}_{n}\right\rangle\mathclose{}{e}_{n} = \sum_{n}{} \mathopen{}\left\langle{}{A}^{\frac{1}{2}}\mathopen{}\left( x\right)\mathclose{}, {e}_{n}\right\rangle\mathclose{}{A}^{\frac{1}{2}}\mathopen{}\left( {e}_{n}\right)\mathclose{} = \sum_{n}{} \mathopen{}\left({A}^{\frac{1}{2}}\mathopen{}\left( {e}_{n}\right)\mathclose{}\otimes {A}^{\frac{1}{2}}\mathopen{}\left( {e}_{n}\right)\mathclose{}\right)\mathclose{}\mathopen{}\left( x\right)\mathclose{} \text{.} \] Further, ∑summationninteger ‖Aself-adjoint operator1one2two⁡(eunit vectorninteger)⊗Aself-adjoint operator1one2two⁡(eunit vectorninteger)‖ =equals∑summationninteger ‖Aself-adjoint operator1one2two⁡(eunit vectorninteger)‖ 2two <less than∞infinity . \[ \sum_{n}{} \mathopen{}\left\lVert{}{A}^{\frac{1}{2}}\mathopen{}\left( {e}_{n}\right)\mathclose{}\otimes {A}^{\frac{1}{2}}\mathopen{}\left( {e}_{n}\right)\mathclose{}\right\rVert\mathclose{} = \sum_{n}{} {\mathopen{}\left\lVert{}{A}^{\frac{1}{2}}\mathopen{}\left( {e}_{n}\right)\mathclose{}\right\rVert\mathclose{}}^{2} \lt \infty \text{.} \] The series ∑summationninteger Aself-adjoint operator1one2two⁡(eunit vectorninteger)⊗Aself-adjoint operator1one2two⁡(eunit vectorninteger) \( \sum_{n}{} {A}^{\frac{1}{2}}\mathopen{}\left( {e}_{n}\right)\mathclose{}\otimes {A}^{\frac{1}{2}}\mathopen{}\left( {e}_{n}\right)\mathclose{} \) converges in norm with sum Aself-adjoint operator\( A \). This puts Aself-adjoint operator∈element ofℱfinite rank operators⁡(HHilbert space)¯=equals𝒦compact linear operators⁡(HHilbert space) \( A\in \overline{\mathcal{F}\mathopen{}\left( H\right)\mathclose{}}= \mathcal{K}\mathopen{}\left( H\right)\mathclose{} \). So there exists an orthonormal basis (sequencefunit vectorminteger)sequenceminteger=1one∞infinity \( \mathopen{}\left({f}_{m}\right)\mathclose{}_{m=1}^{\infty} \) and a sequence of nonnegative real numbers (sequenceλnonnegative real numberiinteger)sequenceiinteger=1one∞infinity \( \mathopen{}\left({λ}_{i}\right)\mathclose{}_{i=1}^{\infty} \) such that λnonnegative real numberminteger→converges to0zero \( {λ}_{m} \to 0 \), Aself-adjoint operator=equals∑summationminteger λnonnegative real numberminteger⁢timesfunit vectorminteger⊗funit vectorminteger \( A= \sum_{m}{} {λ}_{m}{f}_{m}\otimes {f}_{m} \). Furthermore, ∑summationminteger 〈Aself-adjoint operator⁡(funit vectorminteger), funit vectorminteger〉 =equals∑summationminteger λnonnegative real numberminteger⁢times〈funit vectorminteger, funit vectorminteger〉 =equals∑summationmintegerλnonnegative real numberminteger . \[ \sum_{m}{} \mathopen{}\left\langle{}A\mathopen{}\left( {f}_{m}\right)\mathclose{}, {f}_{m}\right\rangle\mathclose{} = \sum_{m}{} {λ}_{m}\mathopen{}\left\langle{}{f}_{m}, {f}_{m}\right\rangle\mathclose{} = \sum_{m}{}{λ}_{m} \text{.} \]

(⇐) If you know Aself-adjoint operator∈element of𝒦compact linear operators⁡(HHilbert space)+positive elements \( A\in {\mathcal{K}\mathopen{}\left( H\right)\mathclose{}}^{+} \), then Aself-adjoint operator=equals∑summation λnonnegative real numberminteger⁢timesfunit vectorminteger⊗funit vectorminteger \( A= \sum{} {λ}_{m}{f}_{m}\otimes {f}_{m} \) as above, and Trtrace⁡(Aself-adjoint operator)=equals∑summation 〈Aself-adjoint operator⁡(funit vectorminteger), funit vectorminteger〉 =equals∑summation λnonnegative real numberminteger <less than∞infinity . \[ \operatorname{Tr}\mathopen{}\left( A\right)\mathclose{}= \sum{} \mathopen{}\left\langle{}A\mathopen{}\left( {f}_{m}\right)\mathclose{}, {f}_{m}\right\rangle\mathclose{} = \sum{} {λ}_{m} \lt \infty \text{.} \]

Proposition III.22

For Tlinear map∈element ofℒbounded linear operators⁡(HHilbert space) \( T\in \mathcal{L}\mathopen{}\left( H\right)\mathclose{} \), the following are equivalent:

(a)
There exist {setxvector1onexvector2two…}set \( \mathopen{}\left\{\, {x}_{1}, {x}_{2}, \dotsc\,\right\}\mathclose{} \) and {setyvector1oneyvector2two…}set∈element ofHHilbert space \( \mathopen{}\left\{\, {y}_{1}, {y}_{2}, \dotsc\,\right\}\mathclose{}\in H \) such that ∑summationjinteger ‖xvectorjinteger‖⁢times‖yvectorjinteger‖ <less than∞infinity \[ \sum_{j}{} \mathopen{}\left\lVert{}{x}_{j}\right\rVert\mathclose{}\mathopen{}\left\lVert{}{y}_{j}\right\rVert\mathclose{} \lt \infty \] and Tlinear map=equals∑summationjinteger xvectorjinteger⊗yvectorjinteger \( T= \sum_{j}{} {x}_{j}\otimes {y}_{j} \).
(b)
For all orthonormal sets {seteunit vectorninteger}set \( \mathopen{}\left\{\, {e}_{n}\,\right\}\mathclose{} \) and {setfunit vectorninteger}set \( \mathopen{}\left\{\, {f}_{n}\,\right\}\mathclose{} \) in HHilbert space\( H \), ∑summationninteger |modulus〈Tlinear map⁡(eunit vectorninteger), funit vectorninteger〉|modulus <less than∞infinity \( \sum_{n}{} \mathopen{}\left\lvert{}\mathopen{}\left\langle{}T\mathopen{}\left( {e}_{n}\right)\mathclose{}, {f}_{n}\right\rangle\mathclose{}\right\rvert\mathclose{} \lt \infty \).
(c)
Trtrace⁡(|modulusTlinear map|modulus)<less than∞infinity \( \operatorname{Tr}\mathopen{}\left( \mathopen{}\left\lvert{}T\right\rvert\mathclose{}\right)\mathclose{}\lt \infty \).

Proof. ((a)⇒(b)) Using the Cauchy-Schwarz inequality in l2 \( \mathrm{l}^{0} \) and Bessel's inequality, we have ∑summationninteger |modulus〈Tlinear map⁡(eunit vectorninteger), funit vectorninteger〉|modulus =equals∑summationninteger |modulus∑summationjinteger 〈eunit vectorninteger, yvectorjinteger〉⁢times〈xvectorjinteger, funit vectorninteger〉 |modulus ≤less than or equal to∑summationninteger ∑summationjinteger |modulus〈eunit vectorninteger, yvectorjinteger〉|modulus⁢times|modulus〈xvectorjinteger, funit vectorninteger〉|modulus =equals∑summationjinteger ∑summationninteger |modulus〈eunit vectorninteger, yvectorjinteger〉|modulus⁢times|modulus〈xvectorjinteger, funit vectorninteger〉|modulus ≤less than or equal to∑summationjinteger (∑summationninteger |modulus〈eunit vectorninteger, yvectorjinteger〉|modulus 2two ) 1one2two ⁢times (∑summationninteger |modulus〈xvectorjinteger, funit vectorninteger〉|modulus 2two ) 1one2two ≤less than or equal to∑summationjinteger ‖yvectorjinteger‖⁢times‖xvectorjinteger‖ ≤less than or equal to∞infinity . \[ \sum_{n}{} \mathopen{}\left\lvert{}\mathopen{}\left\langle{}T\mathopen{}\left( {e}_{n}\right)\mathclose{}, {f}_{n}\right\rangle\mathclose{}\right\rvert\mathclose{} = \sum_{n}{} \mathopen{}\left\lvert{}\sum_{j}{} \mathopen{}\left\langle{}{e}_{n}, {y}_{j}\right\rangle\mathclose{}\mathopen{}\left\langle{}{x}_{j}, {f}_{n}\right\rangle\mathclose{} \right\rvert\mathclose{} \leq \sum_{n}{} \sum_{j}{} \mathopen{}\left\lvert{}\mathopen{}\left\langle{}{e}_{n}, {y}_{j}\right\rangle\mathclose{}\right\rvert\mathclose{}\mathopen{}\left\lvert{}\mathopen{}\left\langle{}{x}_{j}, {f}_{n}\right\rangle\mathclose{}\right\rvert\mathclose{} = \sum_{j}{} \sum_{n}{} \mathopen{}\left\lvert{}\mathopen{}\left\langle{}{e}_{n}, {y}_{j}\right\rangle\mathclose{}\right\rvert\mathclose{}\mathopen{}\left\lvert{}\mathopen{}\left\langle{}{x}_{j}, {f}_{n}\right\rangle\mathclose{}\right\rvert\mathclose{} \leq \sum_{j}{} {\mathopen{}\left(\sum_{n}{} {\mathopen{}\left\lvert{}\mathopen{}\left\langle{}{e}_{n}, {y}_{j}\right\rangle\mathclose{}\right\rvert\mathclose{}}^{2} \right)\mathclose{}}^{\frac{1}{2}}{\mathopen{}\left(\sum_{n}{} {\mathopen{}\left\lvert{}\mathopen{}\left\langle{}{x}_{j}, {f}_{n}\right\rangle\mathclose{}\right\rvert\mathclose{}}^{2} \right)\mathclose{}}^{\frac{1}{2}} \leq \sum_{j}{} \mathopen{}\left\lVert{}{y}_{j}\right\rVert\mathclose{}\mathopen{}\left\lVert{}{x}_{j}\right\rVert\mathclose{} \leq \infty \text{.} \]

((b)⇒(c)) Recall the polar decomposition (Proposition II.68): |modulusTlinear map|modulus=equals (Tlinear map*⁢timesTlinear map) 1one2two \( \mathopen{}\left\lvert{}T\right\rvert\mathclose{}= {\mathopen{}\left( T^{*}T\right)\mathclose{}}^{\frac{1}{2}} \) and Tlinear map=equalsVpartial isometry⁢times|modulusTlinear map|modulus \( T= V\mathopen{}\left\lvert{}T\right\rvert\mathclose{} \) where Vpartial isometry*⁢timesVpartial isometry \( V^{*}V \) is the projection on |modulusTlinear map|modulus⁡(HHilbert space)¯=equalsTlinear map*⁡(HHilbert space)¯ \( \overline{\mathopen{}\left\lvert{}T\right\rvert\mathclose{}\mathopen{}\left( H\right)\mathclose{}}= \overline{ T^{*}\mathopen{}\left( H\right)\mathclose{}} \). Let {setẽunit vectornintegerẽunit vectorninteger…}set \( \mathopen{}\left\{\, {ẽ}_{n}, {ẽ}_{n}, \dotsc\,\right\}\mathclose{} \) be an orthonormal basis for Kerkernel⁡(Tlinear map)=equalsKerkernel⁡(|modulusTlinear map|modulus) \( \operatorname{Ker}\mathopen{}\left( T\right)\mathclose{}= \operatorname{Ker}\mathopen{}\left( \mathopen{}\left\lvert{}T\right\rvert\mathclose{}\right)\mathclose{} \), and let {seteunit vector1oneeunit vector2two…}set \( \mathopen{}\left\{\, {e}_{1}, {e}_{2}, \dotsc\,\right\}\mathclose{} \) be an orthonormal basis for |modulusTlinear map|modulus⁡(HHilbert space)¯=equals (Kerkernel⁡(Tlinear map)) ⊥ \( \overline{\mathopen{}\left\lvert{}T\right\rvert\mathclose{}\mathopen{}\left( H\right)\mathclose{}}= { \mathopen{}\left(\operatorname{Ker}\mathopen{}\left( T\right)\mathclose{}\right)\mathclose{} }^{\perp} \). Then Trtrace⁡(|modulusTlinear map|modulus)=equals∑summationninteger 〈|modulusTlinear map|modulus⁡(eunit vectorninteger), eunit vectorninteger〉 +plus∑summationninteger 〈|modulusTlinear map|modulus⁡(ẽunit vectorninteger), ẽunit vectorninteger〉 =equals∑summationninteger 〈|modulusTlinear map|modulus⁡(eunit vectorninteger), Vpartial isometry*⁢timesVpartial isometry⁡(eunit vectorninteger)〉 =equals∑summationninteger 〈Vpartial isometry⁢times|modulusTlinear map|modulus⁡(eunit vectorninteger), Vpartial isometry⁡(eunit vectorninteger)〉 =equals∑summationninteger 〈Tlinear map⁡(eunit vectorninteger), Vpartial isometry⁡(eunit vectorninteger)〉 ≤less than or equal to∑summationninteger |modulus〈Tlinear map⁡(eunit vectorninteger), funit vectorninteger〉|modulus , \[ \operatorname{Tr}\mathopen{}\left( \mathopen{}\left\lvert{}T\right\rvert\mathclose{}\right)\mathclose{}= \sum_{n}{} \mathopen{}\left\langle{}\mathopen{}\left\lvert{}T\right\rvert\mathclose{}\mathopen{}\left( {e}_{n}\right)\mathclose{}, {e}_{n}\right\rangle\mathclose{} +\sum_{n}{} \mathopen{}\left\langle{}\mathopen{}\left\lvert{}T\right\rvert\mathclose{}\mathopen{}\left( {ẽ}_{n}\right)\mathclose{}, {ẽ}_{n}\right\rangle\mathclose{} = \sum_{n}{} \mathopen{}\left\langle{}\mathopen{}\left\lvert{}T\right\rvert\mathclose{}\mathopen{}\left( {e}_{n}\right)\mathclose{}, V^{*}V\mathopen{}\left( {e}_{n}\right)\mathclose{}\right\rangle\mathclose{} = \sum_{n}{} \mathopen{}\left\langle{}V\mathopen{}\left\lvert{}T\right\rvert\mathclose{}\mathopen{}\left( {e}_{n}\right)\mathclose{}, V\mathopen{}\left( {e}_{n}\right)\mathclose{}\right\rangle\mathclose{} = \sum_{n}{} \mathopen{}\left\langle{}T\mathopen{}\left( {e}_{n}\right)\mathclose{}, V\mathopen{}\left( {e}_{n}\right)\mathclose{}\right\rangle\mathclose{} \leq \sum_{n}{} \mathopen{}\left\lvert{}\mathopen{}\left\langle{}T\mathopen{}\left( {e}_{n}\right)\mathclose{}, {f}_{n}\right\rangle\mathclose{}\right\rvert\mathclose{} \text{,} \] where funit vectorninteger=equalsVpartial isometry⁡(eunit vectorninteger) \( {f}_{n}= V\mathopen{}\left( {e}_{n}\right)\mathclose{} \). Notice 〈funit vectorninteger, funit vectorjinteger〉=equals〈Vpartial isometry⁡(eunit vectorninteger), Vpartial isometry⁡(eunit vectorjinteger)〉=equals〈Vpartial isometry*⁢timesVpartial isometry⁡(eunit vectorninteger), eunit vectorjinteger〉=equals〈eunit vectorninteger, eunit vectorjinteger〉 \( \mathopen{}\left\langle{}{f}_{n}, {f}_{j}\right\rangle\mathclose{}= \mathopen{}\left\langle{}V\mathopen{}\left( {e}_{n}\right)\mathclose{}, V\mathopen{}\left( {e}_{j}\right)\mathclose{}\right\rangle\mathclose{}= \mathopen{}\left\langle{} V^{*}V\mathopen{}\left( {e}_{n}\right)\mathclose{}, {e}_{j}\right\rangle\mathclose{}= \mathopen{}\left\langle{}{e}_{n}, {e}_{j}\right\rangle\mathclose{} \), so the (sequencefunit vectorninteger)sequence \( \mathopen{}\left({f}_{n}\right)\mathclose{} \) are orthonormal. By (b), Trtrace⁡(|modulusTlinear map|modulus)<less than∞infinity \( \operatorname{Tr}\mathopen{}\left( \mathopen{}\left\lvert{}T\right\rvert\mathclose{}\right)\mathclose{}\lt \infty \).

((c)⇒(a)) For an orthonormal basis (sequenceeunit vectorninteger)sequenceninteger=1one∞infinity \( \mathopen{}\left({e}_{n}\right)\mathclose{}_{n=1}^{\infty} \), Trtrace⁡(|modulusTlinear map|modulus)=equals∑summation ‖ |modulusTlinear map|modulus 1one2two ⁡(eunit vectorninteger)‖ 2two <less than∞infinity \( \operatorname{Tr}\mathopen{}\left( \mathopen{}\left\lvert{}T\right\rvert\mathclose{}\right)\mathclose{}= \sum{} {\mathopen{}\left\lVert{}{\mathopen{}\left\lvert{}T\right\rvert\mathclose{}}^{\frac{1}{2}}\mathopen{}\left( {e}_{n}\right)\mathclose{}\right\rVert\mathclose{}}^{2} \lt \infty \). So, |modulusTlinear map|modulus=equals∑summation |modulusTlinear map|modulus 1one2two ⁡(eunit vectorninteger)⊗ |modulusTlinear map|modulus 1one2two ⁡(eunit vectorninteger) \[ \mathopen{}\left\lvert{}T\right\rvert\mathclose{}= \sum{} {\mathopen{}\left\lvert{}T\right\rvert\mathclose{}}^{\frac{1}{2}}\mathopen{}\left( {e}_{n}\right)\mathclose{}\otimes {\mathopen{}\left\lvert{}T\right\rvert\mathclose{}}^{\frac{1}{2}}\mathopen{}\left( {e}_{n}\right)\mathclose{} \] (as in the proof of Proposition III.21) and Tlinear map=equalsVpartial isometry⁡(|modulusTlinear map|modulus)=equals∑summation Vpartial isometry⁢times |modulusTlinear map|modulus 1one2two ⁡(eunit vectorninteger)⊗ |modulusTlinear map|modulus 1one2two ⁡(eunit vectorninteger) . \[ T= V\mathopen{}\left( \mathopen{}\left\lvert{}T\right\rvert\mathclose{}\right)\mathclose{}= \sum{} V{\mathopen{}\left\lvert{}T\right\rvert\mathclose{}}^{\frac{1}{2}}\mathopen{}\left( {e}_{n}\right)\mathclose{}\otimes {\mathopen{}\left\lvert{}T\right\rvert\mathclose{}}^{\frac{1}{2}}\mathopen{}\left( {e}_{n}\right)\mathclose{} \text{.} \] Since ‖Vpartial isometry‖=equals1one \( \mathopen{}\left\lVert{}V\right\rVert\mathclose{}= 1 \) we can take xvectorninteger=equalsVpartial isometry⁢times |modulusTlinear map|modulus 1one2two ⁡(eunit vectorninteger) \( {x}_{n}= V{\mathopen{}\left\lvert{}T\right\rvert\mathclose{}}^{\frac{1}{2}}\mathopen{}\left( {e}_{n}\right)\mathclose{} \) and yvectorninteger=equals |modulusTlinear map|modulus 1one2two ⁡(eunit vectorninteger) \( {y}_{n}= {\mathopen{}\left\lvert{}T\right\rvert\mathclose{}}^{\frac{1}{2}}\mathopen{}\left( {e}_{n}\right)\mathclose{} \) to get (a).

Definition III.23

Call Tlinear map∈element ofℒbounded linear operators⁡(HHilbert space) \( T\in \mathcal{L}\mathopen{}\left( H\right)\mathclose{} \) a trace-class operator if Trtrace⁡(|modulusTlinear map|modulus)<less than∞infinity \( \operatorname{Tr}\mathopen{}\left( \mathopen{}\left\lvert{}T\right\rvert\mathclose{}\right)\mathclose{}\lt \infty \). Write 𝒯trace-class operators⁡(HHilbert space) \( \mathcal{T}\mathopen{}\left( H\right)\mathclose{} \) for the set of all trace-class operators.

Proposition III.24

𝒯trace-class operators⁡(HHilbert space) \( \mathcal{T}\mathopen{}\left( H\right)\mathclose{} \) is an ideal of ℒbounded linear operators⁡(HHilbert space) \( \mathcal{L}\mathopen{}\left( H\right)\mathclose{} \) with ℱfinite rank operators⁡(HHilbert space)⊆subset𝒯trace-class operators⁡(HHilbert space)⊆subset𝒦compact linear operators⁡(HHilbert space) , \( \mathcal{F}\mathopen{}\left( H\right)\mathclose{}\subseteq \mathcal{T}\mathopen{}\left( H\right)\mathclose{}\subseteq \mathcal{K}\mathopen{}\left( H\right)\mathclose{} \text{,} \) and Ttrace-class operator∈element of𝒯trace-class operators⁡(HHilbert space) \( T\in \mathcal{T}\mathopen{}\left( H\right)\mathclose{} \) implies Ttrace-class operator*∈element of𝒯trace-class operators⁡(HHilbert space) \( T^{*}\in \mathcal{T}\mathopen{}\left( H\right)\mathclose{} \).

Proof. By Proposition III.22, the generic trace-class operator is Ttrace-class operator=equals∑summation xvectorjinteger⊗yvectorjinteger \( T= \sum{} {x}_{j}\otimes {y}_{j} \), where ∑summation ‖xvectorjinteger‖⁢times‖yvectorjinteger‖ <less than∞infinity \( \sum{} \mathopen{}\left\lVert{}{x}_{j}\right\rVert\mathclose{}\mathopen{}\left\lVert{}{y}_{j}\right\rVert\mathclose{} \lt \infty \). The partial sums are in ℱfinite rank operators⁡(HHilbert space) \( \mathcal{F}\mathopen{}\left( H\right)\mathclose{} \) and converge in norm to Ttrace-class operator\( T \), so Ttrace-class operator∈element of𝒦compact linear operators⁡(HHilbert space) \( T\in \mathcal{K}\mathopen{}\left( H\right)\mathclose{} \). The sum of two such operators is another operator of the same type, and for Sbounded linear operator∈element ofℒbounded linear operators⁡(HHilbert space) \( S\in \mathcal{L}\mathopen{}\left( H\right)\mathclose{} \) we have Sbounded linear operator⁡(Ttrace-class operator)=equals∑summation Sbounded linear operator⁡(xvectorjinteger)⊗yvectorjinteger \( S\mathopen{}\left( T\right)\mathclose{}= \sum{} S\mathopen{}\left( {x}_{j}\right)\mathclose{}\otimes {y}_{j} \) and Ttrace-class operator⁡(Sbounded linear operator)=equals∑summation xvectorjinteger⊗Sbounded linear operator*⁡(yvectorjinteger) \( T\mathopen{}\left( S\right)\mathclose{}= \sum{} {x}_{j}\otimes S^{*}\mathopen{}\left( {y}_{j}\right)\mathclose{} \). Finally, Ttrace-class operator*=equals∑summation yvectorjinteger⊗xvectorjinteger \( T^{*}= \sum{} {y}_{j}\otimes {x}_{j} \).

Example III.25

The Volterra operator TVolterra operator\( T \) is not trace-class. Look at efunctionninteger⁡(treal number)=equals2two⁢timescoscosine⁡(ninteger⁢timesπ⁢timestreal number) \( {e}_{n}\mathopen{}\left( t\right)\mathclose{}= \sqrt{2}\cos\mathopen{}\left( nπt\right)\mathclose{} \). Then TVolterra operator⁡(efunctionninteger⁡(treal number))=equals 2two ninteger⁢timesπ ⁢timessinsine⁡(ninteger⁢timesπ⁢timestreal number)=equals ffunctionninteger⁡(treal number) ninteger⁢timesπ , \[ T\mathopen{}\left( {e}_{n}\mathopen{}\left( t\right)\mathclose{}\right)\mathclose{}= \frac{\sqrt{2}}{nπ}\sin\mathopen{}\left( nπt\right)\mathclose{}= \frac{{f}_{n}\mathopen{}\left( t\right)\mathclose{}}{nπ} \text{,} \] where ffunctionninteger⁡(treal number)=equals2two⁢timessinsine⁡(ninteger⁢timesπ⁢timestreal number) \( {f}_{n}\mathopen{}\left( t\right)\mathclose{}= \sqrt{2}\sin\mathopen{}\left( nπt\right)\mathclose{} \). 〈TVolterra operator⁡(efunctionninteger), ffunctionninteger〉=equals 1one ninteger⁢timesπ \( \mathopen{}\left\langle{}T\mathopen{}\left( {e}_{n}\right)\mathclose{}, {f}_{n}\right\rangle\mathclose{}= \frac{1}{nπ} \), so TVolterra operator∉not an element of𝒯trace-class operators⁡(HHilbert space) \( T\notin \mathcal{T}\mathopen{}\left( H\right)\mathclose{} \) by Proposition III.22. On the other hand, TVolterra operator2two \( {T}^{2} \) is trace-class because, as we have seen above, TVolterra operator2two=equalsξfunction1one⊗(ξfunction0zero-minusξfunction1one)-minus1oneπ2two⁢times∑summationninteger=1one∞infinity 1oneninteger2two⁢timesefunctionninteger⊗efunctionninteger , \[ {T}^{2}= {ξ}_{1}\otimes \mathopen{}\left({ξ}_{0}-{ξ}_{1}\right)\mathclose{}-\frac{1}{{π}^{2}}\sum_{n=1}^{\infty}{} \frac{1}{{n}^{2}}{e}_{n}\otimes {e}_{n} \text{,} \] where ξfunctionjinteger⁡(treal number)=equalstreal numberjinteger \( {ξ}_{j}\mathopen{}\left( t\right)\mathclose{}= {t}^{j} \) and efunctionninteger⁡(treal number)=equals2two⁢timessinsine⁡(ninteger⁢timesπ⁢timestreal number) \( {e}_{n}\mathopen{}\left( t\right)\mathclose{}= \sqrt{2}\sin\mathopen{}\left( nπt\right)\mathclose{} \).

Proposition III.26

For Ttrace-class operator∈element of𝒯trace-class operators⁡(HHilbert space) \( T\in \mathcal{T}\mathopen{}\left( H\right)\mathclose{} \), written Ttrace-class operator=equals∑summationjinteger xvectorjinteger⊗yvectorjinteger \( T= \sum_{j}{} {x}_{j}\otimes {y}_{j} \) with ∑summationjinteger ‖xvectorjinteger‖⁢times‖yvectorjinteger‖ <less than∞infinity \( \sum_{j}{} \mathopen{}\left\lVert{}{x}_{j}\right\rVert\mathclose{}\mathopen{}\left\lVert{}{y}_{j}\right\rVert\mathclose{} \lt \infty \), we have

  1. ∑summationjinteger 〈xvectorjinteger, yvectorjinteger〉 \( \sum_{j}{} \mathopen{}\left\langle{}{x}_{j}, {y}_{j}\right\rangle\mathclose{} \) converges;
  2. ∑summationninteger 〈Ttrace-class operator⁡(eunit vectorninteger), eunit vectorninteger〉 \( \sum_{n}{} \mathopen{}\left\langle{}T\mathopen{}\left( {e}_{n}\right)\mathclose{}, {e}_{n}\right\rangle\mathclose{} \) converges for any orthonormal basis (sequenceeunit vectorninteger)sequenceninteger=1one∞infinity \( \mathopen{}\left({e}_{n}\right)\mathclose{}_{n=1}^{\infty} \), and ∑summationninteger 〈Ttrace-class operator⁡(eunit vectorninteger), eunit vectorninteger〉 =equals∑summationjinteger 〈xvectorjinteger, yvectorjinteger〉 . \[ \sum_{n}{} \mathopen{}\left\langle{}T\mathopen{}\left( {e}_{n}\right)\mathclose{}, {e}_{n}\right\rangle\mathclose{} = \sum_{j}{} \mathopen{}\left\langle{}{x}_{j}, {y}_{j}\right\rangle\mathclose{} \text{.} \]

Proof.

  1. Follows from |modulus〈xvectorjinteger, yvectorjinteger〉|modulus≤less than or equal to‖xvectorjinteger‖⁢times‖yvectorjinteger‖ \( \mathopen{}\left\lvert{}\mathopen{}\left\langle{}{x}_{j}, {y}_{j}\right\rangle\mathclose{}\right\rvert\mathclose{}\leq \mathopen{}\left\lVert{}{x}_{j}\right\rVert\mathclose{}\mathopen{}\left\lVert{}{y}_{j}\right\rVert\mathclose{} \) and absolute convergence implies convergence.
  2. ∑summationjinteger ∑summationninteger |modulus〈eunit vectorninteger, yvectorjinteger〉|modulus⁢times|modulus〈xvectorjinteger, eunit vectorninteger〉|modulus ≤less than or equal to∑summationjinteger (∑summationninteger |modulus〈eunit vectorninteger, yvectorjinteger〉|modulus 2two ) 1one2two ⁢times (∑summationninteger |modulus〈xvectorjinteger, eunit vectorninteger〉|modulus 2two ) 1one2two =equals∑summationjinteger ( ‖yvectoriinteger‖ 2two ) 1one2two ⁢times ( ‖xvectorjinteger‖ 2two ) 1one2two ≤less than or equal to∞infinity . \[ \sum_{j}{} \sum_{n}{} \mathopen{}\left\lvert{}\mathopen{}\left\langle{}{e}_{n}, {y}_{j}\right\rangle\mathclose{}\right\rvert\mathclose{}\mathopen{}\left\lvert{}\mathopen{}\left\langle{}{x}_{j}, {e}_{n}\right\rangle\mathclose{}\right\rvert\mathclose{} \leq \sum_{j}{} {\mathopen{}\left(\sum_{n}{} {\mathopen{}\left\lvert{}\mathopen{}\left\langle{}{e}_{n}, {y}_{j}\right\rangle\mathclose{}\right\rvert\mathclose{}}^{2} \right)\mathclose{}}^{\frac{1}{2}}{\mathopen{}\left(\sum_{n}{} {\mathopen{}\left\lvert{}\mathopen{}\left\langle{}{x}_{j}, {e}_{n}\right\rangle\mathclose{}\right\rvert\mathclose{}}^{2} \right)\mathclose{}}^{\frac{1}{2}} = \sum_{j}{} {\mathopen{}\left({\mathopen{}\left\lVert{}{y}_{i}\right\rVert\mathclose{}}^{2}\right)\mathclose{}}^{\frac{1}{2}}{\mathopen{}\left({\mathopen{}\left\lVert{}{x}_{j}\right\rVert\mathclose{}}^{2}\right)\mathclose{}}^{\frac{1}{2}} \leq \infty \text{.} \] So, both double series ∑summationjinteger ∑summationninteger 〈eunit vectorninteger, yvectorjinteger〉⁢times〈xvectorjinteger, eunit vectorninteger〉 =equals∑summationninteger ∑summationjinteger 〈eunit vectorninteger, yvectorjinteger〉⁢times〈xvectorjinteger, eunit vectorninteger〉 \[ \sum_{j}{} \sum_{n}{} \mathopen{}\left\langle{}{e}_{n}, {y}_{j}\right\rangle\mathclose{}\mathopen{}\left\langle{}{x}_{j}, {e}_{n}\right\rangle\mathclose{} = \sum_{n}{} \sum_{j}{} \mathopen{}\left\langle{}{e}_{n}, {y}_{j}\right\rangle\mathclose{}\mathopen{}\left\langle{}{x}_{j}, {e}_{n}\right\rangle\mathclose{} \] converge. Then ∑summationjinteger ∑summationninteger 〈eunit vectorninteger, yvectorjinteger〉⁢times〈xvectorjinteger, eunit vectorninteger〉 =equals∑summationjinteger 〈xvectorjinteger, yvectorjinteger〉 \[ \sum_{j}{} \sum_{n}{} \mathopen{}\left\langle{}{e}_{n}, {y}_{j}\right\rangle\mathclose{}\mathopen{}\left\langle{}{x}_{j}, {e}_{n}\right\rangle\mathclose{} = \sum_{j}{} \mathopen{}\left\langle{}{x}_{j}, {y}_{j}\right\rangle\mathclose{} \] and ∑summationninteger ∑summationjinteger 〈eunit vectorninteger, yvectorjinteger〉⁢times〈xvectorjinteger, eunit vectorninteger〉 =equals∑summationninteger ∑summationjinteger 〈〈eunit vectorninteger, yvectorjinteger〉⁢timesxvectorjinteger, eunit vectorninteger〉 =equals∑summationninteger 〈(∑summationjinteger xvectorjinteger⊗yvectorjinteger )⁢timeseunit vectorninteger, eunit vectorninteger〉 =equals∑summationninteger 〈Ttrace-class operator⁡(eunit vectorninteger), eunit vectorninteger〉 . \[ \sum_{n}{} \sum_{j}{} \mathopen{}\left\langle{}{e}_{n}, {y}_{j}\right\rangle\mathclose{}\mathopen{}\left\langle{}{x}_{j}, {e}_{n}\right\rangle\mathclose{} = \sum_{n}{} \sum_{j}{} \mathopen{}\left\langle{}\mathopen{}\left\langle{}{e}_{n}, {y}_{j}\right\rangle\mathclose{}{x}_{j}, {e}_{n}\right\rangle\mathclose{} = \sum_{n}{} \mathopen{}\left\langle{}\mathopen{}\left(\sum_{j}{} {x}_{j}\otimes {y}_{j} \right)\mathclose{}{e}_{n}, {e}_{n}\right\rangle\mathclose{} = \sum_{n}{} \mathopen{}\left\langle{}T\mathopen{}\left( {e}_{n}\right)\mathclose{}, {e}_{n}\right\rangle\mathclose{} \text{.} \]

Definition III.27

Let {seteunit vectorninteger}set \( \mathopen{}\left\{\, {e}_{n}\,\right\}\mathclose{} \) be an orthonormal basis of HHilbert space\( H \). For Ttrace-class operator∈element of𝒯trace-class operators⁡(HHilbert space) \( T\in \mathcal{T}\mathopen{}\left( H\right)\mathclose{} \), we define Trtrace⁡(Ttrace-class operator)=equals∑summation 〈Ttrace-class operator⁡(eunit vectorninteger), eunit vectorninteger〉 \( \operatorname{Tr}\mathopen{}\left( T\right)\mathclose{}= \sum{} \mathopen{}\left\langle{}T\mathopen{}\left( {e}_{n}\right)\mathclose{}, {e}_{n}\right\rangle\mathclose{} \).

Remark III.28

  1. It doesn't matter which orthonormal basis you use.
  2. If ∑summation ‖xvectorjinteger‖⁢times‖yvectorjinteger‖ <less than∞infinity \( \sum{} \mathopen{}\left\lVert{}{x}_{j}\right\rVert\mathclose{}\mathopen{}\left\lVert{}{y}_{j}\right\rVert\mathclose{} \lt \infty \), Trtrace⁡(∑summation xvectorjinteger⊗yvectorjinteger )=equals∑summation 〈xvectorjinteger, yvectorjinteger〉 \( \operatorname{Tr}\mathopen{}\left( \sum{} {x}_{j}\otimes {y}_{j} \right)\mathclose{}= \sum{} \mathopen{}\left\langle{}{x}_{j}, {y}_{j}\right\rangle\mathclose{} \).
  3. Trtrace:maps𝒯trace-class operators⁡(HHilbert space)→toCcomplex numbers \( \operatorname{Tr} : \mathcal{T}\mathopen{}\left( H\right)\mathclose{} \to \mathbb{C} \) is linear.

Proposition III.29

For Sbounded linear operator∈element ofℒbounded linear operators⁡(HHilbert space) \( S\in \mathcal{L}\mathopen{}\left( H\right)\mathclose{} \) and Ttrace-class operator∈element of𝒯trace-class operators⁡(HHilbert space) \( T\in \mathcal{T}\mathopen{}\left( H\right)\mathclose{} \),

  1. Trtrace⁡(Sbounded linear operator⁢timesTtrace-class operator)=equalsTrtrace⁡(Ttrace-class operator⁢timesSbounded linear operator) \( \operatorname{Tr}\mathopen{}\left( ST\right)\mathclose{}= \operatorname{Tr}\mathopen{}\left( TS\right)\mathclose{} \).
  2. |modulusTrtrace⁡(Sbounded linear operator⁢timesTtrace-class operator)|modulus≤less than or equal to‖Sbounded linear operator‖⁢timesTrtrace⁡(|modulusTtrace-class operator|modulus) \( \mathopen{}\left\lvert{}\operatorname{Tr}\mathopen{}\left( ST\right)\mathclose{}\right\rvert\mathclose{}\leq \mathopen{}\left\lVert{}S\right\rVert\mathclose{}\operatorname{Tr}\mathopen{}\left( \mathopen{}\left\lvert{}T\right\rvert\mathclose{}\right)\mathclose{} \).

Proof.

  1. Write Ttrace-class operator=equals∑summation xvectorjinteger⊗yvectorjinteger \( T= \sum{} {x}_{j}\otimes {y}_{j} \) with ∑summation ‖xvectorjinteger‖⁢times‖yvectorjinteger‖ <less than∞infinity \( \sum{} \mathopen{}\left\lVert{}{x}_{j}\right\rVert\mathclose{}\mathopen{}\left\lVert{}{y}_{j}\right\rVert\mathclose{} \lt \infty \). Then Trtrace⁡(Sbounded linear operator⁢timesTtrace-class operator)=equalsTrtrace⁡(∑summation Sbounded linear operator⁡(xvectorjinteger)⊗yvectorjinteger )=equals∑summationjinteger 〈Sbounded linear operator⁡(xvectorjinteger), yvectorjinteger〉 =equals∑summationjinteger 〈xvectorjinteger, Sbounded linear operator*⁡(yvectorjinteger)〉 =equalsTrtrace⁡(∑summation xvectorjinteger⊗Sbounded linear operator*⁡(yvectorjinteger) )=equalsTrtrace⁡(Ttrace-class operator⁢timesSbounded linear operator) . \[ \operatorname{Tr}\mathopen{}\left( ST\right)\mathclose{}= \operatorname{Tr}\mathopen{}\left( \sum{} S\mathopen{}\left( {x}_{j}\right)\mathclose{}\otimes {y}_{j} \right)\mathclose{}= \sum_{j}{} \mathopen{}\left\langle{}S\mathopen{}\left( {x}_{j}\right)\mathclose{}, {y}_{j}\right\rangle\mathclose{} = \sum_{j}{} \mathopen{}\left\langle{}{x}_{j}, S^{*}\mathopen{}\left( {y}_{j}\right)\mathclose{}\right\rangle\mathclose{} = \operatorname{Tr}\mathopen{}\left( \sum{} {x}_{j}\otimes S^{*}\mathopen{}\left( {y}_{j}\right)\mathclose{} \right)\mathclose{}= \operatorname{Tr}\mathopen{}\left( TS\right)\mathclose{} \text{.} \]
  2. By the spectral theorem, get an orthonormal basis (sequenceeunit vectorninteger)sequence \( \mathopen{}\left({e}_{n}\right)\mathclose{} \) for HHilbert space\( H \) and (non-negative) λnonnegative real number1one \( {λ}_{1} \), λnonnegative real number2two \( {λ}_{2} \), … such that |modulusTtrace-class operator|modulus=equals∑summation λnonnegative real numberninteger⁢times(eunit vectorninteger⊗eunit vectorninteger) \( \mathopen{}\left\lvert{}T\right\rvert\mathclose{}= \sum{} {λ}_{n}\mathopen{}\left({e}_{n}\otimes {e}_{n}\right)\mathclose{} \). Use the polar decomposition Ttrace-class operator=equalsVpartial isometry⁢times|modulusTtrace-class operator|modulus \( T= V\mathopen{}\left\lvert{}T\right\rvert\mathclose{} \). Then |modulusTrtrace⁡(Sbounded linear operator⁡(Ttrace-class operator))|modulus=equals|modulusTrtrace⁡(Sbounded linear operator⁢timesVpartial isometry⁢times|modulusTtrace-class operator|modulus)|modulus=equals|modulus∑summation 〈Sbounded linear operator⁡(Vpartial isometry⁡(|modulusTtrace-class operator|modulus⁡(eunit vectorninteger))), eunit vectorninteger〉 |modulus=equals|modulus∑summation λnonnegative real numberninteger⁢times〈Sbounded linear operator⁡(Vpartial isometry⁡(eunit vectorninteger)), eunit vectorninteger〉 |modulus≤less than or equal to∑summation λnonnegative real numberninteger⁢times|modulus〈Sbounded linear operator⁡(Vpartial isometry⁡(eunit vectorninteger)), eunit vectorninteger〉|modulus ≤less than or equal to‖Sbounded linear operator⁢timesVpartial isometry‖⁢times∑summation λnonnegative real numberninteger ≤less than or equal to‖Sbounded linear operator‖⁢timesTrtrace⁡(|modulusTtrace-class operator|modulus) \[ \mathopen{}\left\lvert{}\operatorname{Tr}\mathopen{}\left( S\mathopen{}\left( T\right)\mathclose{}\right)\mathclose{}\right\rvert\mathclose{}= \mathopen{}\left\lvert{}\operatorname{Tr}\mathopen{}\left( SV\mathopen{}\left\lvert{}T\right\rvert\mathclose{}\right)\mathclose{}\right\rvert\mathclose{}= \mathopen{}\left\lvert{}\sum{} \mathopen{}\left\langle{}S\mathopen{}\left( V\mathopen{}\left( \mathopen{}\left\lvert{}T\right\rvert\mathclose{}\mathopen{}\left( {e}_{n}\right)\mathclose{}\right)\mathclose{}\right)\mathclose{}, {e}_{n}\right\rangle\mathclose{} \right\rvert\mathclose{}= \mathopen{}\left\lvert{}\sum{} {λ}_{n}\mathopen{}\left\langle{}S\mathopen{}\left( V\mathopen{}\left( {e}_{n}\right)\mathclose{}\right)\mathclose{}, {e}_{n}\right\rangle\mathclose{} \right\rvert\mathclose{}\leq \sum{} {λ}_{n}\mathopen{}\left\lvert{}\mathopen{}\left\langle{}S\mathopen{}\left( V\mathopen{}\left( {e}_{n}\right)\mathclose{}\right)\mathclose{}, {e}_{n}\right\rangle\mathclose{}\right\rvert\mathclose{} \leq \mathopen{}\left\lVert{}SV\right\rVert\mathclose{}\sum{} {λ}_{n} \leq \mathopen{}\left\lVert{}S\right\rVert\mathclose{}\operatorname{Tr}\mathopen{}\left( \mathopen{}\left\lvert{}T\right\rvert\mathclose{}\right)\mathclose{} \] since ‖Vpartial isometry‖≤less than or equal to1one \( \mathopen{}\left\lVert{}V\right\rVert\mathclose{}\leq 1 \) and ∑summation λnonnegative real numberninteger =equalsTrtrace⁡(|modulusTtrace-class operator|modulus) \( \sum{} {λ}_{n} = \operatorname{Tr}\mathopen{}\left( \mathopen{}\left\lvert{}T\right\rvert\mathclose{}\right)\mathclose{} \).

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