Singular Bilinear Integrals in Quantum Physics

Jefferies, Brian

doi:10.3390/math3030563

Cited by 2 publications

(2 citation statements)

References 44 publications

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“…The equivalence between (i) and (iv) is stated in [25,27], and also in [20,21] to which we refer for various proofs. It follows from this analysis that if condition (i) from Theorem 23 holds true, then the above factorization (iv) can be achieved with (a, b) satisfying…”

Section: Further It Is Easy To Deduce From the Above Definition Of In...mentioning

confidence: 95%

When do triple operator integrals take value in the trace class?

Coine¹,

Merdy²,

Sukochev³

2017

Preprint

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Consider three normal operators A, B, C on separable Hilbert space H as well as scalar-valued spectral measures λ A on σ(A), λ B on σ(B) and λ C on σ(C). For any φ ∈ L ∞ (λ A × λ B × λ C ) and any X, Y ∈ S 2 (H), the space of Hilbert-Schmidt operators on H, we provide a general definition of a triple operator integral Γ A,B,C (φ)(X, Y ) belonging to S 2 (H) in such a way that Γ A,B,C (φ) belongs to the space B 2 (S 2 (H) × S 2 (H), S 2 (H)) of bounded bilinear operators on S 2 (H), and the resulting mapping) has the property that Γ A,B,C (φ) maps S 2 (H)×S 2 (H) into S 1 (H), the space of trace class operators on H, if and only if it has the following factorization property: there exist a Hilbert space H and two functions a. This is a bilinear version of Peller's Theorem characterizing double operator integral mappings S 1 (H) → S 1 (H). In passing we show that for any separable Banach spaces E, F , any w *measurable esssentially bounded function valued in the Banach space Γ 2 (E, F * ) of operators from E into F * factoring through Hilbert space admits a w * -measurable Hilbert space factorization.

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Section: Further It Is Easy To Deduce From the Above Definition Of In...mentioning

confidence: 95%

When do triple operator integrals take value in the trace class?

Coine¹,

Merdy²,

Sukochev³

2017

Preprint

View full text Add to dashboard Cite

show abstract

“…The equivalence between (1) and ( 4) is stated in [28,30], and also in [22,23] to which we refer for various proofs. It follows from this analysis that if condition (1) holds true, then the above factorization (4) can be achieved with (a, b) satisfying…”

Section: Additional Commentsmentioning

confidence: 97%

When do triple operator integrals take value in the trace class?

Coine¹,

Merdy²,

Sukochev³

2022

Annales De l'Institut Fourier

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Consider three normal operators A, B, C on a separable Hilbert space H as well as scalar-valued spectral measures λ A on σ(A), λ B on σ(B) and λ C on σ(C). For any φ ∈ L ∞ (λ A × λ B × λ C ) and any X, Y ∈ S 2 (H), the space of Hilbert-Schmidt operators on H, we provide a general definition of a triple operator integral Γ A,B,C (φ)(X, Y ) belonging to S 2 (H) in such a way that Γ A,B,C (φ) belongs to the space B 2 (S 2 (H) × S 2 (H), S 2 (H)) of bounded bilinear operators on S 2 (H), and the resulting mappinginto S 1 (H), the space of trace class operators on H, if and only if it has the following factorization property: there exist a Hilbert space H and two func-. This is a bilinear version of Peller's Theorem characterizing double operator integral mappings S 1 (H) → S 1 (H). In passing we show that for any separable Banach spaces E, F , any w *measurable esssentially bounded function valued in the Banach space Γ 2 (E, F * ) of operators from E into F * factoring through Hilbert space admits a w * -measurable Hilbert space factorization.Résumé. -Considérons trois opérateurs normaux A, B, C sur un espace de Hilbert séparable H ainsi que des mesures spectrales scalairesH)) est une isométrie w * -continue. On montre alors qu'étant donnée une fonction φ ∈ L ∞ (λ A × λ B × λ C ), Γ A,B,C (φ) envoie S 2 (H) × S 2 (H) dans S 1 (H), l'espace des opérateurs à trace sur H, si et seulement si φ vérifie

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Singular Bilinear Integrals in Quantum Physics

Cited by 2 publications

References 44 publications

When do triple operator integrals take value in the trace class?

When do triple operator integrals take value in the trace class?

When do triple operator integrals take value in the trace class?

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