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Cayley--Hamilton tuples: an interplay between algebraic varieties and joint spectra

B. Krishna Das, Poornendu Kumar, Haripada Sau

Abstract

We introduce the notion of Cayley--Hamilton tuples: these are commuting operator tuples that are annihilated by a non-zero polynomial and such that its Taylor joint spectrum coincides with the algebraic variety determined by its annihilating ideal. Commuting matrix tuples are Cayley--Hamilton tuples. We provide two families of Cayley--Hamilton tuples in the infinite dimensional setting with additional details. What arises as a by-product is a concrete characterization of distinguished varieties in the polydisk in terms of Taylor joint spectrum of commuting isometries. These varieties have been of interest in various fields of mathematics over the last two decades. The Taylor and Waelbroeck joint spectrum of a Cayley--Hamilton tuple are shown to be the same. It is also shown that the support of the annihilating ideal of a Cayley--Hamilton tuple is the same as its joint spectrum. As an application, we deduce an algebraic characterization of bi-variate polynomials whose zero set intersected with the closed bidisk is the joint spectrum of a commuting isometric pair.

Cayley--Hamilton tuples: an interplay between algebraic varieties and joint spectra

Abstract

We introduce the notion of Cayley--Hamilton tuples: these are commuting operator tuples that are annihilated by a non-zero polynomial and such that its Taylor joint spectrum coincides with the algebraic variety determined by its annihilating ideal. Commuting matrix tuples are Cayley--Hamilton tuples. We provide two families of Cayley--Hamilton tuples in the infinite dimensional setting with additional details. What arises as a by-product is a concrete characterization of distinguished varieties in the polydisk in terms of Taylor joint spectrum of commuting isometries. These varieties have been of interest in various fields of mathematics over the last two decades. The Taylor and Waelbroeck joint spectrum of a Cayley--Hamilton tuple are shown to be the same. It is also shown that the support of the annihilating ideal of a Cayley--Hamilton tuple is the same as its joint spectrum. As an application, we deduce an algebraic characterization of bi-variate polynomials whose zero set intersected with the closed bidisk is the joint spectrum of a commuting isometric pair.

Paper Structure

This paper contains 14 sections, 16 theorems, 161 equations.

Table of Contents

  1. Background and the main results
  2. Background
  3. The problem of interest
  4. The main results
  5. Continuity of joint eigenvalues and a Cayley--Hamilton Theorem
  6. A functional model for commuting isometries
  7. Joint Spectrum
  8. Continuity of joint eigenvalues and a Cayley--Hamilton Theorem
  9. The proofs of the main results
  10. Joint spectrum for commuting Toeplitz operators
  11. Joint spectrum for commuting isometries
  12. Varieties as joint spectrum of commuting isometries
  13. Joint spectra and annihilating ideals
  14. Further Problems

Key Result

Theorem 2.1

Suppose $(V_1, V_2, \dots, V_d)$ is a tuple of commuting isometries acting on a Hilbert space $\mathcal{H}$. The Wold decomposition of the product isometry $V := V_1 \cdots V_d$ decomposes each $V_j$ as BCLform for some unique orthogonal projections $P_j$, unitary operators $U_j$ in ${\mathcal{B}}({ for some orthogonal projection $P$, unitary operator $U$ in ${\mathcal{B}}({\mathcal{D}}_{V^*})$, a

Theorems & Definitions (47)

  • Definition 1.1
  • Definition 1.3
  • Definition 1.4
  • Example 1.5
  • Example 1.6
  • Definition 1.7
  • Theorem 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4: Taylor joint spectrum
  • ...and 37 more