2020/04/05 by Martijn Caspers, Fedor Sukochev, Caspers, Martijn +3 · 3 citations
Mathematics · #Advanced Harmonic Analysis Research #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Operator Algebras (math.OA) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2004.02145
openalex publication_date 2020/04/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider the generalized absolute value function defined by \a(t) =
vert t\n
vert tn-1,
qquad t
in
mathbbR, n
in
mathbbN
geq 1. Further,\nconsider the n-th order divided difference function a[n]:\n\ℝn+1 \→ \ℂ and let 1 < p1, \…, pn <\n\∞ be such that \∑l=1n pl-1 = 1. Let \Spl\ndenote the Schatten-von Neumann ideals and let \S1,\∞ denote\nthe weak trace class ideal. We show that for any (n+1)-tuple bf A of\nbounded self-adjoint operators the multiple operator integral\nTa[n]^ bf A maps \Sp1 \× \… \×\n\Spn to \S1, \∞ boundedly with uniform bound in\n bf A. The same is true for the class of Cn+1-functions that outside\nthe interval [-1, 1] equal a. In [CLPST16] it was proved that for a\nfunction f in this class such boundedness of T^ bf A f[n] from\n\Sp1 \× \… \× \Spn to \S1\nmay fail, resolving a problem by V. Peller. This shows that the estimates in\nthe current paper are optimal. The proof is based on a new reduction method for\narbitrary multiple operator integrals of divided differences.\n