Comptes Rendus
Analyse mathématique
Normes des extensions quaternionique d'opérateurs réels
[Norms of quaternionic extensions of real operators]
Comptes Rendus. Mathématique, Volume 340 (2005) no. 9, pp. 639-643.

We consider bounded linear operators defined on real normed spaces, and with range in quaternionic spaces. We study the norms of the quaternionic extensions of such operators.

Nous considérons des opérateurs linéaires bornés définis sur des espaces normés réels, et dont les images sont dans des espaces quaternioniques. Nous étudions les normes des extensions quaternioniques de ces opérateurs.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2005.03.030
Daniel Alpay 1; Maria-Elena Luna-Elizarrarás 2; Michael Shapiro 2

1 Department of Mathematics, Ben–Gurion University of the Negev, Beer-Sheva 84105, Israel
2 Departamento de Matemáticas, Escuela Superior de Física y Mathemáticas, Instituto Politécnico Nacional, 07300 México, D.F., México
@article{CRMATH_2005__340_9_639_0,
     author = {Daniel Alpay and Maria-Elena Luna-Elizarrar\'as and Michael Shapiro},
     title = {Normes des extensions quaternionique d'op\'erateurs r\'eels},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {639--643},
     publisher = {Elsevier},
     volume = {340},
     number = {9},
     year = {2005},
     doi = {10.1016/j.crma.2005.03.030},
     language = {fr},
}
TY  - JOUR
AU  - Daniel Alpay
AU  - Maria-Elena Luna-Elizarrarás
AU  - Michael Shapiro
TI  - Normes des extensions quaternionique d'opérateurs réels
JO  - Comptes Rendus. Mathématique
PY  - 2005
SP  - 639
EP  - 643
VL  - 340
IS  - 9
PB  - Elsevier
DO  - 10.1016/j.crma.2005.03.030
LA  - fr
ID  - CRMATH_2005__340_9_639_0
ER  - 
%0 Journal Article
%A Daniel Alpay
%A Maria-Elena Luna-Elizarrarás
%A Michael Shapiro
%T Normes des extensions quaternionique d'opérateurs réels
%J Comptes Rendus. Mathématique
%D 2005
%P 639-643
%V 340
%N 9
%I Elsevier
%R 10.1016/j.crma.2005.03.030
%G fr
%F CRMATH_2005__340_9_639_0
Daniel Alpay; Maria-Elena Luna-Elizarrarás; Michael Shapiro. Normes des extensions quaternionique d'opérateurs réels. Comptes Rendus. Mathématique, Volume 340 (2005) no. 9, pp. 639-643. doi : 10.1016/j.crma.2005.03.030. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.03.030/

[1] S.L. Adler Quaternionic quantum mechanics and non commutative dynamics (available at) | arXiv

[2] S.L. Adler Quaternionic Quantum Mechanics and Quantum Fields, Internat. Ser. Monographs Phys., vol. 88, Clarendon Press, Oxford University Press, New York, 1995

[3] D. Alpay; M. Shapiro Reproducing kernel quaternionic Pontryagin spaces, Integral Equations Operator Theory, Volume 50 (2004), pp. 431-476

[4] N. Bourbaki Espaces vectoriels topologiques. Chapitres 1 à 5, Masson, Paris, 1981 (Éléments de mathématique [Elements of mathematics])

[5] E.G. Effros; Z.J. Ruan Operator Spaces, London Math. Soc. Monographs, vol. 23, 2000

[6] I.M. Glazman; Ju.I. Ljubic Finite-Dimensional Linear Analysis: A Systematic Presentation in Problem Form, MIT Press, Cambridge, 1974

[7] A. Grothendieck Résumé des résultats essentiels dans la théorie des produits tensoriels topologiques et des espaces nucléaires, Ann. Inst. Fourier (Grenoble), Volume 4 (1954), pp. 73-112 (1952)

[8] J. Holbrook Concerning the Hahn–Banach theorem, Proc. Amer. Math. Soc., Volume 50 (1975), pp. 322-327

[9] M. Karow Self-adjoint operators and pairs of Hermitian forms over the quaternions, Linear Algebra Appl., Volume 299 (1999) no. 1–3, pp. 101-117

[10] M.E. Luna-Elizarrarás, M. Shapiro, On some properties of quaternionic inner product spaces, Proceedings of the Twenty-Fifth International Colloquium on Group Theoretical Methods in Physics, in press

[11] M.E. Luna-Elizarrarás, M. Shapiro, Preservation of the norms of linear operators acting on some quaternionic function spaces, Oper. Theory: Adv. Appl., in press

[12] M. Morimoto Analytic Functionals on the Sphere, Transl. Math. Monographs, vol. 178, American Mathematical Society, 1998

[13] M. Riesz Sur les maxima des formes bilinéaires et sur les fonctionelles linéaires, Acta Math., Volume 49 (1926), pp. 465-497

[14] C.S. Sharma Complex structure on a real Hilbert space and symplectic structure on a complex Hilbert space, J. Math. Phys., Volume 29 (1988), pp. 1069-1078

[15] C.S. Sharma; T.J. Coulson Spectral theory for unitary operators on a quaternionic Hilbert space, J. Math. Phys., Volume 28 (1987), pp. 1941-1946

[16] G.A. Soukhomlinoff Über Fortsetzung von linearen Funktionalen in linearen komplexen Räumen und linearen Quaternionräumen, Mat. Sb. (N.S.), Volume 3 (1938), pp. 353-358

[17] G.O. Thorin Convexity theorems generalizing those of M. Riesz and Hadamard with some applications, Comm. Sem. Math. Univ. Lund = Medd. Lunds Univ. Sem., Volume 9 (1948), pp. 1-58

[18] I.E. Verbitski Some relations between the norm of an operator and that of its complex extension, Mat. Issled., Volume 42 (1976), pp. 3-12

[19] K. Viswanath Normal operators on quaternionic Hilbert spaces, Trans. Amer. Math. Soc., Volume 162 (1971), pp. 337-350

[20] F. Zhang Quaternions and matrices of quaternions, Linear Algebra Appl., Volume 251 (1997), pp. 21-57

Cited by Sources:

Comments - Policy


Articles of potential interest

Problème de Gleason et interpolation pour les fonctions hyper-analytiques

Daniel Alpay; Michael Shapiro

C. R. Math (2002)


Fonctions rationnelles et théorie de la réalisation: le cas hyper-analytique

Daniel Alpay; Baruch Schneider; Michael Shapiro; ...

C. R. Math (2003)


Fonctions rationnelles et problème de Gleason associés à l'opérateur de Dirac

Daniel Alpay; Flor de María Correa-Romero; María Elena Luna-Elizarrarás; ...

C. R. Math (2006)