Operator Estimates for Fredholm Modules

Sukochev, F. A.

Canadian journal of mathematics, Tome 52 (2000) no. 4, pp. 849-896

Résumé

We study estimates of the type 1 $${{\left\| \phi (D)\,-\,\phi ({{D}_{0}}) \right\|}_{E(\mathcal{M},\,\tau )}}\,\le \,C\,\cdot \,{{\left\| D\,-\,{{D}_{0}} \right\|}^{\alpha }},\,\,\,\,\,\,\,\alpha \,=\,\frac{1}{2},\,1$$ where $\phi (t)\,=\,t{{(1\,+\,{{t}^{2}})}^{-1/2}},\,{{D}_{0}}\,=\,D_{0}^{*}$ is an unbounded linear operator affiliated with a semifinite von Neumann algebra $M,D-{{D}_{0}}$ is a bounded self-adjoint linear operator from $\mathcal{M}$ and ${{(1+D_{0}^{2})}^{-1/2}}\in E(M,\tau )$ , where $E(\mathcal{M},\tau )$ is a symmetric operator space associated with $\mathcal{M}$ . In particular, we prove that $\phi \left( D \right)-\phi \left( {{D}_{0}} \right)$ belongs to the non-commutative ${{L}_{p}}$ -space for some $p\in (1,\infty )$ , provided ${{(1+D_{0}^{2})}^{-1/2}}$ belongs to the noncommutative weak ${{L}_{r}}$ -space for some $r\in [1,p)$ . In the case $\mathcal{M}\,=\,\mathcal{B}\left( \mathcal{H} \right)$ and $1\,\le \,p\,\le \,2$ , we show that this result continues to hold under the weaker assumption ${{(1+D_{0}^{2})}^{-1/2}}\in {{C}_{p}}$ . This may be regarded as an odd counterpart of A. Connes’ result for the case of even Fredholm modules.

Détail

DOI : 10.4153/CJM-2000-037-8

Mots-clés : 46L50, 46E30, 46L87, 47A55, 58B15

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