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

• Published in: Journal of functional analysis Vol. 271; no. 11; pp. 3300 - 3322
• Main Authors: Aleksandrov, A.B., Peller, V.V.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.12.2016
• Subjects: Almost commuting operators; Haagerup-like tensor products; Helton–Howe trace formula; Triple operator integrals; Helton–Howe trace formula; Haagerup-like tensor products; Almost commuting operators; Triple operator integrals
• ISSN: 0022-1236; 1096-0783
• DOI: 10.1016/j.jfa.2016.09.004

Let A and B be almost commuting (i.e., the commutator AB−BA belongs to trace class) self-adjoint operators. We construct a functional calculus φ↦φ(A,B) for functions φ in the Besov class B∞,11(R2). This functional calculus is linear, the operators φ(A,B) and ψ(A,B) almost commute for φ,ψ∈B∞,11(R2), and φ(A,B)=u(A)v(B) whenever φ(s,t)=u(s)v(t). We extend the Helton–Howe trace formula for arbitrary functions in B∞,11(R2). The main tool is triple operator integrals with integrands in Haagerup-like tensor products of L∞ spaces.