Event Title
On polynomial n-tuples of commuting isometries
Location
Cupples I Room 207
Start Date
7-18-2016 4:00 PM
End Date
18-7-2016 4:20 PM
Description
We extend some of the results of Agler, Knese, and McCarthy concerning pairs of commuting shifts to the case of n-tuples of commuting isometries, where n>2. Let $V=(V_1,\dots,V_n)$ be an $n$-tuple of 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 of $n$-tuples $(W_1,\dots,W_n)$ with the property that, for each $i$, $W_i$ 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.
On polynomial n-tuples of commuting isometries
Cupples I Room 207
We extend some of the results of Agler, Knese, and McCarthy concerning pairs of commuting shifts to the case of n-tuples of commuting isometries, where n>2. Let $V=(V_1,\dots,V_n)$ be an $n$-tuple of 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 of $n$-tuples $(W_1,\dots,W_n)$ with the property that, for each $i$, $W_i$ 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.