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Analysis Seminar – Edward Timko, Indiana University
November 4, 2016 @ 10:30 am - 11:30 am
Title : On polynomial n-tuples of commuting isometries
Abstract : We extend some of the results of Agler, Knese, and McCarthy to n-tuples of commuting isometries for n>2. Let V=(V_1,…,V_n) be an n-tuple of a commuting isometries on a Hilbert space and let Ann(V) denote the set of all n-variable polynomials p such that p(V)=0. When Ann(V) defines an affine algebraic variety of dimension 1 and V is completely non-unitary, we show that V decomposes as a direct sum of n-tuples W=(W_1,…,W_n) with the property that, for each i=1,…,n, W_i is either a shift or a scalar multiple of the identity. If V is a cyclic n-tuple of commuting shifts, then we show that V is determined by Ann(V) up to ‘near unitary equivalence’, as defined by Agler, Knese, and McCarthy.