Functions of almost commuting operators and an extension of the Helton-Howe trace formula

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• 作者：A.B. Aleksandrov^a ; ^b ; V.V. Peller^c ; ^{peller@math.msu.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Almost commuting operators ; Triple operator integrals ; Haagerup-like tensor products ; Helton&ndash ; Howe trace formula
• 刊名：Journal of Functional Analysis
• 出版年：2016
• 出版时间：1 December 2016
• 年：2016
• 卷：271
• 期：11
• 页码：3300-3322
• 全文大小：451 K

文摘

Let A and B   be almost commuting (i.e., the commutator AB−BA$\mathit{AB}-\mathit{BA}$ belongs to trace class) self-adjoint operators. We construct a functional calculus φ↦φ(A,B)$\phi \mapsto \phi (A,B)$ for functions φ   in the Besov class $B_{\infty ,1}^1({\mathbb{R}}^2)$. This functional calculus is linear, the operators φ(A,B)$\phi (A,B)$ and ψ(A,B)$\psi (A,B)$ almost commute for $\phi ,\psi \in B_{\infty ,1}^1({\mathbb{R}}^2)$, and φ(A,B)=u(A)v(B)$\phi (A,B)=u(A)v(B)$ whenever φ(s,t)=u(s)v(t)$\phi (s,t)=u(s)v(t)$. We extend the Helton–Howe trace formula for arbitrary functions in $B_{\infty ,1}^1({\mathbb{R}}^2)$. The main tool is triple operator integrals with integrands in Haagerup-like tensor products of L^∞$L^{\infty }$ spaces.