| KRATZ, Werner and Roman ŠIMON HILSCHER. A generalized index theorem for monotone matrix-valued functions with applications to discrete oscillation theory. SIAM Journal on Matrix Analysis and Applications. Philadelphia, PA, USA: SIAM, 2013, vol. 34, No 1, p. 228-243. ISSN 0895-4798. doi:10.1137/120873029. |
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@article{1079732,
author = {Kratz, Werner and Šimon Hilscher, Roman},
article_location = {Philadelphia, PA, USA},
article_number = {1},
doi = {http://dx.doi.org/10.1137/120873029},
keywords = {Index theorem; Rank theorem; Limit theorem; Oscillation theorem; Discrete symplectic system; Sturm--Liouville difference equation},
language = {eng},
issn = {0895-4798},
journal = {SIAM Journal on Matrix Analysis and Applications},
title = {A generalized index theorem for monotone matrix-valued functions with applications to discrete oscillation theory},
volume = {34},
year = {2013}
}
TY - JOUR
ID - 1079732
AU - Kratz, Werner - Šimon Hilscher, Roman
PY - 2013
TI - A generalized index theorem for monotone matrix-valued functions with applications to discrete oscillation theory
JF - SIAM Journal on Matrix Analysis and Applications
VL - 34
IS - 1
SP - 228-243
EP - 228-243
PB - SIAM
SN - 08954798
KW - Index theorem
KW - Rank theorem
KW - Limit theorem
KW - Oscillation theorem
KW - Discrete symplectic system
KW - Sturm--Liouville difference equation
N2 - An index theorem is a tool for computing the change of the index (i.e., the number of negative eigenvalues) of a symmetric monotone matrix-valued function when its variable passes through a singularity. In 1995, the first author proved an index theorem in which a certain critical matrix coefficient is constant. In this paper, we generalize the above index theorem to the case when this critical matrix may be varying, but its rank, as well as the rank of some additional matrix, are constant. This includes as a special case the situation when this matrix has a constant image. We also show that the index theorem does not hold when the main assumption on constant ranks is violated. Our investigation is motivated by the oscillation theory of discrete symplectic systems with nonlinear dependence on the spectral parameter, which was recently developed by the second author and for which we obtain new oscillation theorems.
ER -
KRATZ, Werner and Roman ŠIMON HILSCHER. A generalized index theorem for monotone matrix-valued functions with applications to discrete oscillation theory. \textit{SIAM Journal on Matrix Analysis and Applications}. Philadelphia, PA, USA: SIAM, 2013, vol.~34, No~1, p.~228-243. ISSN~0895-4798. doi:10.1137/120873029.
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