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Some applications of the matched projections of idempotents

Xiaofeng Zhang, Xiaoyi Tian, Qingxiang Xu

TL;DR

The paper analyzes idempotents on a Hilbert space through their matched projections $m(Q)$, enabling a precise distance theory between projections and a fixed idempotent, a complete block-structure characterization of the C$^*$-algebra $C^\{Q\}$ for non-projection $Q$, and the construction of universal $r$-idempotents. It also develops operator-valued numerical-range theory, including an operator version of the elliptical range theorem and explicit descriptions of numerical ranges for non-quadratic operators built from $Q$ and $m(Q)$. Collectively, these results extend classical finite-dimensional projections to a richer operator-algebraic setting, providing exact norm formulas, universal properties, and new geometric insights into numerical ranges with potential applications in operator theory and functional calculus.

Abstract

For every idempotent $Q$ on a Hilbert space $H$, the matched projection $m(Q)$ is a well-established concept. This paper explores several applications of the matched projections. The first application addresses the distances from projections on $H$ to a given idempotent $Q$. Using $m(Q)$, a complete characterization of these distances is established, covering the minimum, maximum, and intermediate values. The second application focuses on the $C^*$-algebra $C^*\{Q\}$ generated by a single non-projection idempotent $Q$. A new $4\times 4$ block matrix representation of $Q$, induced by $m(Q)$, yields novel formulas for $Q$, leading to a full characterization of all elements in $C^*\{Q\}$ via explicit $4\times 4$ block matrices. Furthermore, for each $r>1$, a family of universal $r$-idempotents is introduced. These idempotents possess a universal property distinct from known properties of projection pairs. Some necessary and sufficient conditions are provided for such universal $r$-idempotents. The third application presents new characterizations of the numerical ranges. An operator version of the elliptical range theorem is established. Using a general non-projection idempotent $Q$ and its matched projection $m(Q)$, a non-quadratic operator is constructed, and its numerical range is described in detail. Additionally, another operator is introduced whose numerical range closure is not an elliptical disk, and the numerical range itself is neither closed nor open.

Some applications of the matched projections of idempotents

TL;DR

The paper analyzes idempotents on a Hilbert space through their matched projections , enabling a precise distance theory between projections and a fixed idempotent, a complete block-structure characterization of the C-algebra for non-projection , and the construction of universal -idempotents. It also develops operator-valued numerical-range theory, including an operator version of the elliptical range theorem and explicit descriptions of numerical ranges for non-quadratic operators built from and . Collectively, these results extend classical finite-dimensional projections to a richer operator-algebraic setting, providing exact norm formulas, universal properties, and new geometric insights into numerical ranges with potential applications in operator theory and functional calculus.

Abstract

For every idempotent on a Hilbert space , the matched projection is a well-established concept. This paper explores several applications of the matched projections. The first application addresses the distances from projections on to a given idempotent . Using , a complete characterization of these distances is established, covering the minimum, maximum, and intermediate values. The second application focuses on the -algebra generated by a single non-projection idempotent . A new block matrix representation of , induced by , yields novel formulas for , leading to a full characterization of all elements in via explicit block matrices. Furthermore, for each , a family of universal -idempotents is introduced. These idempotents possess a universal property distinct from known properties of projection pairs. Some necessary and sufficient conditions are provided for such universal -idempotents. The third application presents new characterizations of the numerical ranges. An operator version of the elliptical range theorem is established. Using a general non-projection idempotent and its matched projection , a non-quadratic operator is constructed, and its numerical range is described in detail. Additionally, another operator is introduced whose numerical range closure is not an elliptical disk, and the numerical range itself is neither closed nor open.
Paper Structure (10 sections, 42 theorems, 349 equations)

This paper contains 10 sections, 42 theorems, 349 equations.

Table of Contents

  1. Introduction
  2. The operator distances from projections to an idempotent
  3. The minimum and the maximum operator distances
  4. The intermediate values of the operator distances
  5. $C^*$-algebras generated by non-projection idempotents
  6. The structure of the $C^*$-algebra generated by a single non-projection idempotent
  7. The universal $r$-idempotents
  8. New characterizations of the numerical ranges
  9. An operator version of the elliptical range theorem
  10. The numerical ranges of certain non-quadratic operators

Key Result

Lemma 2.1

TXF02 For any idempotent $Q\in \mathbb{B}(H)$,

Theorems & Definitions (100)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Corollary 2.5
  • Remark 2.1
  • Example 2.1
  • ...and 90 more