Skip to content
Research Article Open access CC BY 4.0

The Solution of the Invariant Subspace Problem in Complex Hilbert Space: External non-Archimedean field *Rc by Cauchy Completion of the internal Non-Archimedean field *R (Part I)

Jaykov Foukzon

Journal of Advances in Mathematics and Computer Science · pp. 51–89 · Published 23 Nov 2022

10.9734/jamcs/2022/v37i101717

Abstract

The incompleteness of set theory ZFC leads one to look for natural extensions of ZFC in which one can prove statements wich appear as independent of ZFC but which look to be \true". In this paper we deal with set theory NC\(^{\#}{_\infty}_{\#}\) based on hyper infinitary logic with Restricted Modus Ponens Rule. Set theory NC\(^{\#}{_\infty}_{\#}\)contains Aczel's anti-foundation axiom. We present a new approach to the invariant subspace problem for complex Hilbert spaces.This approach based on nonconservative extension of the model theoretical NSA. Our main result will be that: if T is a bounded linear operator on an infinite-dimensional separable complex Hilbert space H;it follow that T has a non-trivial closed invariant subspace.

Set theory ZFC non-conservative extension of ZFC internal set theory IST external set theory HST A. Robinson model theoretical NSA bivalent gyper innitary logic modus ponens rule logic with restricted modus ponens rule internal non-archimedean eld invariant subspce problem

References (22)

  1. 1 Invariant Subspaces of Completely Continuous Operators [DOI]
  2. 2 Applied non-standard analysis [DOI]
  3. 3 On the invariant subspace problem for Banach spaces [DOI]
  4. 4 A Solution to the Invariant Subspace Problem [DOI]
  5. 5 Invariant subspaces of polynomially compact operators [DOI]
  6. 6 Tarski Grothendieck Set Theory
  7. 7 NONSTANDARD METHODS IN STOCHASTIC ANALYSIS AND MATHEMATICAL PHYSICS [DOI]
  8. 8 Set Theory INC# Based on Intuitionistic Logic with Restricted Modus Ponens Rule (Part. I) [DOI]
  9. 9 Set Theory INC# ∞# Based on Infinitary Intuitionistic Logic with Restricted Modus Ponens Rule (Part.II) Hyper Inductive Definitions [DOI]
  10. 10 Set theory with a filter quantifier [DOI]
  11. 11 Intelligent Computer Mathematics [DOI]
  12. 12 Internal Set Theory IST# Based on Hyper Infinitary Logic with Restricted Modus Ponens Rule: Nonconservative Extension of the Model Theoretical NSA [DOI]
  13. 13 Relevant first-order logic LP# and Curry’s paradox resolution
  14. 14 Set Theory INC# ∞# Based on Innitary Intuitionistic Logic with Restricted Modus Ponens Rule. Hyper Inductive Denitions. Application in Transcendental Number Theory. Generalized Lindemann-Weierstrass Theorem [DOI]
  15. 15 Realism, nonstandard set theory, and large cardinals [DOI]
  16. 16 Inconsistent Countable Set in Second Order ZFC and Nonexistence of the Strongly Inaccessible Cardinals [DOI]
  17. 17 There Is No Standard Model of ZFC and ZFC2 with Henkin Semantics [DOI]
  18. 18 Advances in Mathematics and Computer Science Vol. 1 [DOI]
  19. 19 Consistency Results in Topology and Homotopy Theory [DOI]
  20. 20 A note on invariant subspaces [DOI]
  21. 21 The Solution of the Invariant Subspace Problem. Part I. Complex Hilbert Space [DOI]
  22. 22 Set Theory NC_{∞}^{} Based on Bivalent Infinitary Logic with Restricted Modus Ponens Rule.Basic Real Analysis on External Non-Archimedean Field ℝ_{c}^{}. Basic Complecs Analysis on External Field ℂ_{c}^{}=ℝ_{c}^{}[i]. [DOI]

Cited by 4

Showing 2 of 4 known citations — external sources report more than can currently be individually listed.

Article metrics

Real usage data collected on this platform.

38

Page views

0

PDF downloads

4

Outbound clicks

4

Citations

Views over time

Views by country

Approximate, from request IP at view time — not citizenship or institution. Countries with fewer than 5 views are grouped as "Other".

Traffic sources

Referring site, by host.

Views and downloads exclude known bots/crawlers. Citations combines this platform's own DOI-resolved index with each external source's own reported total — see Cited by above for individually listed citing works. Last refreshed 0 seconds ago.