Random Series of Trace Class Operators
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abstract
In this lecture, we present some results on Gaussian (or Rademacher) random series of trace class operators, mainly due jointly with F. Lust-Piquard. We will emphasize the probabilistic reformulation of these results, as well as the open problems suggested by them. We start by a brief survey of what is known about the problem of characterizing a.s. convergent (Gaussian or Rademacher) series of random vectors in a Banach space. The main result presented here is that for certain pairs of Banach spaces $E,F$ that include Hilbert spaces (and type 2 spaces with the analytic UMD property), we have $$ R(Ehatotimes F) =R(E)hatotimes F + Ehatotimes R(F) $$ where $R(E)$ denotes the space of convergent Rademacher series with coefficients in $E$ and $Ehatotimes F$ denotes the projective tensor product.
