arXiv:2602.10962

A generalization of Frenkel's formula

Shmuel Friedland

math.FAmath-phquant-phSubmitted Feb 11, 2026• Announced Feb 12, 2026• Revised Feb 15, 2026 (v2)

Abstract

We generalize Frenkel's integral formula for traces of operators to operators. The resulting formula holds for bounded self-adjoint positive operators and $p$-Schatten class of compact positive operators.

Comments: Revised version, 14 pages

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arXiv:2602.10962

A generalization of Frenkel's formula

Shmuel Friedland

math.FAmath-phquant-phSubmitted Feb 11, 2026• Announced Feb 12, 2026• Revised Feb 15, 2026 (v2)

arXiv HTML PDF

Abstract

We generalize Frenkel's integral formula for traces of operators to operators. The resulting formula holds for bounded self-adjoint positive operators and

p

-Schatten class of compact positive operators.

Comments: Revised version, 14 pages

Related papers

A Note on the p->q norms of Completely Positive Maps

Koenraad M. R. Audenaert

King and Ruskai asked whether the $p\to q$ norm of a completely positive map $Φ$ , acting between Schatten $p$ and $q$ classes of self-adjoint operators, $||Φ||_{p\to q} = \max_{A=A^*} \frac{||Φ(A)||_q}{||A||_p}$ , is equal to the $p\to q$ norm of that map when acting between Schatten classes of general, not necessarily self-adjoint, operators. The first proof has been given by Watrous. We give an alternative proof of this statement.King and Ruskai asked whether the $p\to q$ norm of a completely positive map $Φ$, acting between Schatten $p$ and $q$ classes of self-adjoint operators, $||Φ||_{p\to q} = \max_{A=A^*} \frac{||Φ(A)||_q}{||A||_p}$, is equal to the $p\to q$ norm of that map when acting between Schatten classes of general, not necessarily self-adjoint, operators. The first proof has been given by Watrous. We give an alternative proof of this statement.

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We generalize Frenkel's integral formula for traces of operators to operators. The resulting formula holds for bounded self-adjoint positive operators and

p

-Schatten class of compact positive operators.

King and Ruskai asked whether the

p\to q

norm of a completely positive map

Φ

, acting between Schatten

p

and

q

classes of self-adjoint operators,

||Φ||_{p\to q} = \max_{A=A^*} \frac{||Φ(A)||_q}{||A||_p}

, is equal to the

p\to q

norm of that map when acting between Schatten classes of general, not necessarily self-adjoint, operators. The first proof has been given by Watrous. We give an alternative proof of this statement.