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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 $ξ, η$ the function $t → <e^{itD} x e^{−itD} ξ, η>$ 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] || ≤ (||h||_\infty + ||k||_1)( 4||x||+ 4||[D, x]|| + 2||[D, [D, x]]||)$  . 
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