**Speaker**: Soumitra Ghara

**Date: **13th November, 2018

**Time: **9:15pm-10:15pm

**Abstract:**

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

|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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