On von Neumann’s inequality for contractions on a Hilbert space

SpeakerSoumitra Ghara

Date: 13th November, 2018

Time: 9:15pm-10:15pm

In 1951, von Neumann proved that if T is a contraction on a Hilbert
space, i.e. the operator norm of T is less than or equal to 1, then for any polynomial p,
the operator norm of p(T) is less than or equal to the supremum of |p(z)| over
|z|<1. In this talk, I will start with a proof of this theorem when the Hilbert space

is finite dimensional, and then use a limiting argument to obtain a proof forthe
infinite dimensional case. Then we will see another proof of this theorem using
Sz.-Nazy’s dilation theorem (which will only be stated). If time permits, I will
also discuss some multivariate generalizations of this theorem.
Prerequisites: Linear algebra and elementary functional analysis.

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