Dimension of joint kernel of pure p-isopairs in terms of unitaries representing p for p(z,w) = z n − w2

Abstract

An inner toral polynomial is a polynomial in two complex variables whose zero set is contained in D2 ∪ T 2 ∪ E 2 , where D,T and E represent the open unit disk, the unit circle and the exterior of the closed unit disk, respectively. Given an inner toral polynomial, there exists a minimal version p where p is minimal in the sense that p divides any polynomial with the same zero set as itself. A zero of a polynomial p(z, w) is called a regular point for p, if p has a non-zero gradient evaluation. For a minimal inner toral polynomial p(z, w) of bidegree (n, m), there exists a unitary matrix U = ( A B C D ) of size (m + n) × (m + n) representing p such that p(z, w) is a constant multiple of det (A − wI zB C zD − I). A pure isometry is an isometry defined on a Hilbert space that behaves like a shift operator. Given a pair of pure isometries (S, T), its bimultiplicity is defined as (dim (ker(s ∗ )) , dim (ker (T ∗ ))), where ∗ denotes the adjoint operator. A pair of pure isometries (S, T) satisfying the algebraic relationship p(S, T) = 0, where p is a polynomial, is called a pure p-isopair. It was proven that, for a fixed minimal inner toral polynomial p with bidegree (n, m), and a unitary matrix U representing p, there exists a Hilbert space H such that the pair (Mz , Mw) on H is a pure p-isopair. Further, for a pure p-isopair (S, T) with finite bimultiplicity (m, n), dim [ker(S − λI) ∗ ∩ ker(T − μI) ∗ ] = 1 whenever (λ, μ) ∈ D2 is a regular point for p. It can be observed that at a non-regular point (λ, μ), the above dimension of the joint kernel could be one or higher. In this work, we proved that for minimal inner toral polynomials of the form p(z, w) = z n − w 2 where n ≥ 2, if the unitary matrix representing p has a zero diagonal and a non-zero block matrix A, then there exists a pure p-isopair (S, T) such that dimension of ker((S − λI) ∗ ) ∩ ker((T − μI) ∗ ) at the non-regular point (0,0) is one.