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Decomposition of Linear Operators on Pre-Euclidean Spaces by Means of Graphs

Author

Listed:
  • Hani Abdelwahab

    (Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt)

  • Elisabete Barreiro

    (University of Coimbra, CMUC, Department of Mathematics, FCTUC, Largo D. Dinis, 3000-143 Coimbra, Portugal)

  • Antonio J. Calderón

    (Department of Mathematics, University of Cádiz, 11510 Puerto Real, Spain)

  • José M. Sánchez

    (Department of Mathematics, University of Cádiz, 11510 Puerto Real, Spain)

Abstract

In this work, we study a linear operator f on a pre-Euclidean space V by using properties of a corresponding graph. Given a basis B of V , we present a decomposition of V as an orthogonal direct sum of certain linear subspaces { U i } i ∈ I , each one admitting a basis inherited from B , in such way that f = ∑ i ∈ I f i . Each f i is a linear operator satisfying certain conditions with respect to U i . Considering this new hypothesis, we assure the existence of an isomorphism between the graphs of f relative to two different bases. We also study the minimality of V by using the graph of f relative to B .

Suggested Citation

  • Hani Abdelwahab & Elisabete Barreiro & Antonio J. Calderón & José M. Sánchez, 2023. "Decomposition of Linear Operators on Pre-Euclidean Spaces by Means of Graphs," Mathematics, MDPI, vol. 11(3), pages 1-12, February.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:3:p:725-:d:1053544
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