Comparative index and Hörmander index in finite dimension and their connections-Reference-Cited by-同舟云学术

Comparative index and Hörmander index in finite dimension and their connections

Published:2023 Issue:16 Volume:37 Page:5243-5257
ISSN:0354-5180
Container-title:Filomat
language:en
Short-container-title:Filomat

Author:

Elyseeva Julia¹,Sepitka Peter²,Simon-Hilscher Roman²

Affiliation:

1. Department of Applied Mathematics, Moscow State University of Technology, Moscow, Russia

2. bDepartment of Mathematics and Statistics, Faculty of Science, Masaryk University, Brno, Czech Republic

Abstract

In this paper we prove new relations between the comparative index and the H?rmander index (and the Maslov index) in the finite dimensional case. As a main result we derive an algebraic formula for calculating the H?rmander index of four given Lagrangian planes as a difference of two comparative indices involving certain transformed Lagrangian planes, or as a combination of four comparative indices. This result is based on a generalization of the comparison theorem for the Maslov index involving three Lagrangian paths. In this way we contribute to the recent efforts in the literature (by Zhou, Wu, Zhu in 2018 and by Howard in 2021) devoted to an efficient calculation of the H?rmander index in this finite dimensional case.

Publisher

National Library of Serbia

Reference31 articles.

1. A. Ben-Israel, T. N. E. Greville, Generalized Inverses: Theory and Applications, Second Edition, Springer-Verlag, New York, NY, 2003.

2. B. Booss-Bavnbek, C. Zhu, The Maslov index in symplectic Banach spaces, Memoirs of the American Mathematical Society 252 (2018), no. 1201, x+118 pp.

3. S. L. Campbell, C. D. Meyer, Generalized Inverses of Linear Transformations, Reprint of the 1991 corrected reprint of the 1979 original, Classics in Applied Mathematics, Vol. 56, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2009.

4. S. E. Cappell, R. Lee, E. Y. Miller, On the Maslov index, Communications on Pure and Applied Mathematics 47 (1994), no. 2, 121-186.

5. W. A. Coppel, Disconjugacy, Lecture Notes in Mathematics, Vol. 220, Springer-Verlag, Berlin, 1971.