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Dixmier traces of operators on Banach and Hilbert spaces
Authors:Albrecht Pietsch
Institution:Biberweg 7, 07749 Jena, Germany. Phone: +49 3641 946110, Fax: +49 3641 946102
Abstract:We present a simple approach to the theory of traces on the Marcinkiewicz operator ideal equation image Particular attention is paid to the Dixmier traces, which may be obtained from both dilation‐invariant functionals and shift‐invariant functionals on $\mathfrak l_\infty$. Since the concept of positivity does not make sense for traces defined on operator ideals over Banach spaces, traces are not assumed to be positive in the Hilbert space setting, as well. This standpoint raises some new problems. The present studies will be continued in 35 , 36 .
Keywords:Traces  shift‐invariant functionals  dilation‐invariant functionals  operators  MSC (2010) 47B10  46L30
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