Random Series of Trace Class Operators

Details Random Series of Trace Class Operators Random Series of Trace Class Operators

2011-03-10

arxiv.org/abs/1103.2090v1

Proceedings Cuarto CLAPEM Mexico 1990. Contribuciones en probabilidad y estadistica matematica (1992) 29-42

Mathematics

Functional Analysis

Proceedings Cuarto CLAPEM Mexico 1990. Text not widely available, so we decided to upload here

Scientific paper

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 $$ R(E\hat\otimes F) =R(E)\hat\otimes F + E\hat\otimes R(F) $$ where $R(E)$ denotes the space of convergent Rademacher series with coefficients in $E$ and $E\hat\otimes F$ denotes the projective tensor product.

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