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2015•arXiv (Cornell University)Open access

Peller's problem concerning Koplienko-Neidhardt trace formulae: the\n unitary case

Clément Coine, Christian Le Merdy, Denis Potapov, Fedor Sukochev, Anna Tomskova

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Abstract

We prove the existence of a complex valued $C^2$-function on the unit circle,\na unitary operator U and a self-adjoint operator Z in the Hilbert-Schmidt class\n$S^2$, such that the perturbated operator $$ f(e^{iZ}U)-f(U)\n-\\frac{d}{dt}\\bigl(f(e^{itZ}U)\\bigr)_{\\vert t=0} $$ does not belong to the\nspace $S^1$ of trace class operators. This resolves a problem of Peller\nconcerning the validity of the Koplienko-Neidhardt trace formula for unitaries.\n

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We prove the existence of a complex valued $C^2$-function on the unit circle,\na unitary operator U and a self-adjoint operator Z in the Hilbert-Schmidt class\n$S^2$, such that the perturbated operator $$ f(e^{iZ}U)-f(U)\n-\\frac{d}{dt}\\bigl(f(e^{itZ}U)\\bigr)_{\\vert t=0} $$ does not belong to the\nspace $S^1$ of trace class operators. This resolves a problem of Peller\nconcerning the validity of the Koplienko-Neidhardt trace formula for unitaries.\n

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Available abstract

We prove the existence of a complex valued $C^2$-function on the unit circle,\na unitary operator U and a self-adjoint operator Z in the Hilbert-Schmidt class\n$S^2$, such that the perturbated operator $$ f(e^{iZ}U)-f(U)\n-\\frac{d}{dt}\\bigl(f(e^{itZ}U)\\bigr)_{\\vert t=0} $$ does not belong to the\nspace $S^1$ of trace class operators. This resolves a problem of Peller\nconcerning the validity of the Koplienko-Neidhardt trace formula for unitaries.\n

Key concepts: Trace class, Mathematics, TRACE (psycholinguistics), Unitary operator, Unitary state, Hilbert space, Operator (biology), Nuclear operator

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