Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

When is a CPD weighted shift similar to a subnormal operator?

Zenon Jan Jabłoński, Il Bong Jung, Jan Stochel

Abstract

We prove that a CPD unilateral weighted shift $W_λ$ of type III is a quasi-affine transform of the operator $M_z$ of multiplication by the independent variable on the $L^2(ρ)$-closure of analytic complex polynomials on the complex plane, where $ρ$ is a measure precisely determined by $W_λ$. By using this model, we provide necessary and sufficient conditions for similarity of $W_λ$ to $M_z$. Necessary conditions for a CPD operator to be similar to a subnormal one are given. A variety of concrete classes of non-subnormal CPD unilateral weighted shifts similar to subnormal operators are established.

When is a CPD weighted shift similar to a subnormal operator?

Abstract

We prove that a CPD unilateral weighted shift of type III is a quasi-affine transform of the operator of multiplication by the independent variable on the -closure of analytic complex polynomials on the complex plane, where is a measure precisely determined by . By using this model, we provide necessary and sufficient conditions for similarity of to . Necessary conditions for a CPD operator to be similar to a subnormal one are given. A variety of concrete classes of non-subnormal CPD unilateral weighted shifts similar to subnormal operators are established.

Paper Structure

This paper contains 14 sections, 28 theorems, 117 equations, 1 table.

Table of Contents

  1. Prerequisites
  2. Introduction
  3. A little harmonic analysis
  4. A little operator theory
  5. CPD weighted shifts
  6. Similarity
  7. Necessary conditions
  8. The proof of the main theorem
  9. Quasi-affine transform
  10. Criteria for type III
  11. Examples
  12. Examples via the measure $\nu$
  13. Examples via the scalar representing triplet $(\mathfrak{b},\mathfrak{c}, \nu)$
  14. Quasi-similarity versus similarity

Key Result

Theorem 1.1.2

Let $W_{\boldsymbol{\lambda}}\in \boldsymbol B(\ell^2)$ be a CPD unilateral weighted shift with weights $\boldsymbol{\lambda}=\{\lambda_n\}_{n=0}^{\infty}$ that is not of type I. Then there exists a nonzero, compactly supported, finite, rotation-invariant Borel measure $\rho$ on $\mathbb C$ and an o

Theorems & Definitions (53)

  • Theorem 1.1.2
  • Theorem 1.2.1: Ja-Ju-St22
  • Theorem 1.3.1: Ja-Ju-St22
  • Theorem 1.4.1: g-w70hal70; shi74
  • Theorem 1.4.2: Ja-Ju-Le-St23
  • Theorem 1.4.3: Ja-Ju-Le-St23
  • Corollary 1.4.4
  • Theorem 1.4.5: Ja-Ju-Le-St23
  • Proposition 1.4.6
  • proof
  • ...and 43 more