On polynomial $n$-tuples of commuting isometries

Abstract

We extend some of the results of Agler, Knese, and McCarthy [1] to n-tuples of commuting isometries for n > 2. Let V = (V1, . . . , Vn) be an ntuple 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 nonunitary, we show that V decomposes as a direct sum of n-tuples W = (W1, . . . , Wn) with the property that, for each i = 1, . . . , n, Wi 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 in [1].Theorem 3.4. Suppose V is completely non-unitary, and let I ⊆ Ann(V) be an ideal such that dim Z(I) = 1.(1) I = Ann(V) if and only if I is radical with prime factors I 1 , . . . , I m such that i =j I j Ann(V) for j = 1, . . . , m.(2) If I = Ann(V), then there exist non-zero mutually orthogonal V-invariant subspaces H 1 , . . . , H m of H such that I j = Ann(V|H j ) for each j ∈ {1, . . . , m}.An algebraic variety W in C k is said to be a distinguished variety ifwhere D is the open unit disc in C, T is the unit circle, E is the exterior of the closed unit disc, and exponents indicate Cartesian powers. In the special case that Ann(V) is a prime ideal, the n-tuple V has a particularly simple structure.Theorem 4.4. Suppose V is completely non-unitary. If Ann(V) is a prime ideal and Z(Ann(V)) has dimension 1, then, after a permutation of coordinates, there exists an s ∈ {1, . . . , n} such that (1) V = (V 1 , . . . , V s , λ s+1 I, . . . , λ n I) where V 1 , . . . , V s are shifts and λ s+1 , . . . , λ n are scalars of absolute value 1; and(2) Z(Ann(V)) = W × {(λ s+1 , . . . , λ n )} for some 1-dimensional distinguished variety W ⊆ C s .When Ann(V) is not prime, we have the following result.

A Classification of $${\varvec{n}}$$-Tuples of Commuting Shifts of Finite Multiplicity

“…The core of the proof can be found in [3,Thm.1.12] and [17,Prop. 6.3], but we provide details here for completeness.…”

“…, x n ] are finitely generated, the n-tuple V|H i has a finite cyclic set. By [17,Thm 6.2], each element of V|H i has finite multiplicity.…”

“…Observe that V|H j is an n-tuple of shifts, and therefore Z(V|H i ) has no 0dimensional components [17,Lemma 3.1]. Because Z(I i ) is an irreducible variety of dimension 1, we have that Z(Ann(V|H j )) = I j , and (4) now follows from (5).…”

“…For k = 1, the answer to Question A is 'yes', as shown for n = 2 in [2] and n > 2 in [17] Proof. By Lemma 4.9, there is a β ∈ Hom(G, U(C k )) such that V is virtually similar to M η |H 2 β (D, C k ).…”

“…. , x n ] : p(V) = 0}, It is known from [17,Prop. 6.3] that Ann(V) is a non-trivial ideal and that it determines a variety of pure dimension 1.…”