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Commutator inequalities via Schur products

Erik Christensen
(University of Copenhagen)
Torsdag, 4 februar, 2016, at 16:15-17:15, in Aud. D3 (1531-215)

Let $D$ be an unbounded self-adjoint operator on a Hilbert space $H$.

A bounded operator $x$ in $B(H)$ is $n$-times weakly $D$-differentiable and belongs to the algebra $C^n (D)$ if for any pair of vectors $\xi,\eta$ the function $t \to \langle e^{itD} x e^{-itD} \xi, \eta\rangle$ is $n$-times differentiable on $\mathbb{R}$.

During my studies in noncommutative geometry I have realized that weak differentiability may be expressed as a property of certain matrices in a linear space consisting of infinite matrices with bounded operators as entries. This makes it possible to apply the theory of Schur products of such matrices, i.e. the product $(a_{ij} ) * (b_{ij} ) := (a_{ij} b_{ij} )$.

In this way we can among other results prove the following theorem. Let $D$ be a self-adjoint operator, $x$ a bounded operator in $C^2 (D)$ and $g(t)$ a complex absolutely continuous function on $\mathbb{R}$. If $g(t)$ has a derivative $g'(t) = h(t) + k(t)$ such that $h(t)$ is essentially bounded and $k(t)$ is integrable, then $x$ is weakly $g(D)$-differentiable and

$ ||[g(D), x] || \leq (||h||_{\infty} + ||k||_1)( 4||x||+ 4||[D, x]|| + 2||[D, [D, x]]||)$

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