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35 KAM Mathematical Colloquium

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Prof. GILLES PISIER

Texas A&M University and Universite Paris VI
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SIMILARITY PROBLEMS FOR HILBERT SPACE OPERATORS

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AND RELATED ALGEBRAS

June 4, 1999
Lecture Room S6, Charles University, Malostranske nam. 25, Praha 1
10:30 AM

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Abstract

The Sz.-Nagy-Halmos problem asked whether a polynomially bounded operator
on a Hilbert space $H$ is necessarily similar to a contraction. A couterexample
has recently been found. The talk will discuss this as well as other related
but still open problems of the same nature, for homomorphisms from
an operator algebra $A$ (i.e. a closed subalgebra of $B(H)$) into
the algebra $B(H)$ of bounded operators on $H$. An analogous
property can be formulated for group algebras. We will
discuss mainly the cases when $A$ is a uniform algebra (for instance the
disk or the bidisk algebra) and a $C^*$-algebra. These questions
are closely related to a notion of ``length" for a Banach algebra generated
by a pair of subalgebras, analogous to the minimal length of a word
expressing a group element as a product of generators.