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Core partial order for finite potent endomorphisms

Diego Alba Alonso

TL;DR

This work extends the Core Inverse and the associated core order from finite matrices to arbitrary vector spaces by employing finite potent endomorphisms. Central to the development is the CN-decomposition, which isolates a finite-potency

Abstract

The aim of this paper is to generalize the Core Inverse to arbitrary vector spaces using finite potent endomorphisms. As an application, the core partial order is studied in the set of finite potent endomorphisms (of index lesser or equal than one), thus generalizing the theory of this order to infinite dimensional vector spaces. Moreover, a pre-order is presented using the CN-decomposition of a finite potent endomorphism. Finally, some questions concerning this pre-order are posed. Throughout the paper, some remarks are also made in the framework of arbitrary Hilbert spaces using bounded finite potent endomorphisms.

Core partial order for finite potent endomorphisms

TL;DR

This work extends the Core Inverse and the associated core order from finite matrices to arbitrary vector spaces by employing finite potent endomorphisms. Central to the development is the CN-decomposition, which isolates a finite-potency

Abstract

The aim of this paper is to generalize the Core Inverse to arbitrary vector spaces using finite potent endomorphisms. As an application, the core partial order is studied in the set of finite potent endomorphisms (of index lesser or equal than one), thus generalizing the theory of this order to infinite dimensional vector spaces. Moreover, a pre-order is presented using the CN-decomposition of a finite potent endomorphism. Finally, some questions concerning this pre-order are posed. Throughout the paper, some remarks are also made in the framework of arbitrary Hilbert spaces using bounded finite potent endomorphisms.

Paper Structure

This paper contains 17 sections, 30 theorems, 97 equations.

Table of Contents

  1. Introduction.
  2. Preliminaries
  3. Finite Potent Endomorphisms
  4. CN Decomposition of a Finite Potent Endomorphism
  5. Bounded finite potent endomorphisms on Hilbert spaces.
  6. The adjoint operator of a bounded finite potent endomorphism.
  7. Generalized Inverses.
  8. Group Inverse of Finite Potent Endomorphisms
  9. Moore-Penrose inverse of a bounded linear map.
  10. Moore-Penrose inverse of a bounded linear map.
  11. Space Pre-Order
  12. Some remarks on the Moore-Penrose inverse of a bounded finite potent operator
  13. On the closedness of the image of a bounded finite potent operator.
  14. Core Inverse of Finite Potent Endomorphisms
  15. Some remarks on the Core Inverse for bounded finite potent endomorphisms over Hilbert spaces.
  16. ...and 2 more sections

Key Result

Theorem 2.1

Fpa-Boun(Characterization of bounded finite potent endomorphisms). Given a Hilbert space $\mathcal{H}$ and an endomorphism $\varphi \in \operatorname{End}_{\mathbb{C}}(\mathcal{H}),$ then the following conditions are equivalent:

Theorems & Definitions (63)

  • Theorem 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Definition 2.5
  • Lemma 3.2
  • Proposition 3.3
  • proof
  • Definition 4.1
  • Proposition 4.2
  • ...and 53 more