35 KAM Mathematical Colloquium

Prof. GILLES PISIER

Texas A&M  University and Universite Paris VI      

SIMILARITY PROBLEMS FOR HILBERT SPACE OPERATORS

AND RELATED ALGEBRAS

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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.