Monotonicity and limit results for certain symmetric
matrix-valued functions with applications in singular Sturmian
theory

J 2025

Monotonicity and limit results for certain symmetric matrix-valued functions with applications in singular Sturmian theory

ŠEPITKA, Peter and Roman ŠIMON HILSCHER

Basic information

Original name

Monotonicity and limit results for certain symmetric matrix-valued functions with applications in singular Sturmian theory

Authors

ŠEPITKA, Peter (703 Slovakia, belonging to the institution) and Roman ŠIMON HILSCHER (203 Czech Republic, guarantor, belonging to the institution)

Edition

Applied Mathematics in Science and Engineering, Taylor & Francis Ltd, 2025, 2769-0911

Other information

Language

English

Type of outcome

Article in a journal

Field of Study

10101 Pure mathematics

Country of publisher

United Kingdom of Great Britain and Northern Ireland

Confidentiality degree

is not subject to a state or trade secret

References:

URL

Impact factor

Impact factor: 1.900 in 2023

Organization unit

Faculty of Science

DOI

http://dx.doi.org/10.1080/27690911.2025.2494577

UT WoS

001484872600001

EID Scopus

2-s2.0-105004794444

Keywords in English

Symmetric matrix-valued function; limit theorem; Moore–Penrose pseudoinverse; linear hamiltonian system; Sturmian theory; Wronskian; Lidskii angles; principal solution at infinity

Abstract

V originále

In this paper we study the monotonicity and limit properties at infinity of certain symmetric matrix-valued functions arising in the singular Sturmian theory of canonical linear differential systems. We develop a new method for studying such matrices on an unbounded interval, where we employ the limit properties of Wronskians with the minimal principal solution at infinity to represent the value of the given symmetric matrix at infinity. Moreover, we use the Moore–Penrose pseudoinverse matrices to consider possibly noninvertible solutions of the system. We apply this knowledge for deriving singular Sturmian-type separation theorems on unbounded intervals, which are formulated in terms of the limit properties of the Lidskii angles of the symplectic fundamental matrix of the system. In this way we also extend to the unbounded intervals our results on this subject [Šepitka P, Šimon Hilscher R. Lidskii angles and Sturmian theory for linear Hamiltonian systems on compact interval. J Differ Equ. 2021;298:1–29. doi: 10.1016/j.jde.2021.06.037] regarding the Sturmian separation theorems on a compact interval.

Links

GA23-05242S, research and development project

Name: Oscilační teorie na hybridních časových doménách s aplikacemi ve spektrální teorii a maticové analýze

Investor: Czech Science Foundation, Oscillation theory on hybrid time domains with applications in spectral and matrix analysis

Cite

ŠEPITKA, Peter and Roman ŠIMON HILSCHER. Monotonicity and limit results for certain symmetric matrix-valued functions with applications in singular Sturmian theory. Applied Mathematics in Science and Engineering. Taylor & Francis Ltd, 2025, vol. 33, No 1, p. 1-39. ISSN 2769-0911. Available from: https://dx.doi.org/10.1080/27690911.2025.2494577.

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   author = {Šepitka, Peter and Šimon Hilscher, Roman},
   article_number = {1},
   doi = {http://dx.doi.org/10.1080/27690911.2025.2494577},
   keywords = {Symmetric matrix-valued function; limit theorem; Moore–Penrose pseudoinverse; linear hamiltonian system; Sturmian theory; Wronskian; Lidskii angles; principal solution at infinity},
   language = {eng},
   issn = {2769-0911},
   journal = {Applied Mathematics in Science and Engineering},
   title = {Monotonicity and limit results for certain symmetric matrix-valued functions with applications in singular Sturmian theory},
   url = {https://doi.org/10.1080/27690911.2025.2494577},
   volume = {33},
   year = {2025}
}

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AU  - Šepitka, Peter - Šimon Hilscher, Roman
PY  - 2025
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JF  - Applied Mathematics in Science and Engineering
VL  - 33
IS  - 1
SP  - 1-39
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PB  - Taylor & Francis Ltd
SN  - 27690911
KW  - Symmetric matrix-valued function
KW  - limit theorem
KW  - Moore–Penrose pseudoinverse
KW  - linear hamiltonian system
KW  - Sturmian theory
KW  - Wronskian
KW  - Lidskii angles
KW  - principal solution at infinity
UR  - https://doi.org/10.1080/27690911.2025.2494577
N2  - In this paper we study the monotonicity and limit properties at infinity of certain symmetric matrix-valued functions arising in the singular Sturmian theory of canonical linear differential systems. We develop a new method for studying such matrices on an unbounded interval, where we employ the limit properties of Wronskians with the minimal principal solution at infinity to represent the value of the given symmetric matrix at infinity. Moreover, we use the Moore–Penrose pseudoinverse matrices to consider possibly noninvertible solutions of the system. We apply this knowledge for deriving singular Sturmian-type separation theorems on unbounded intervals, which are formulated in terms of the limit properties of the Lidskii angles of the symplectic fundamental matrix of the system. In this way we also extend to the unbounded intervals our results on this subject [Šepitka P, Šimon Hilscher R. Lidskii angles and Sturmian theory for linear Hamiltonian systems on compact interval. J Differ Equ. 2021;298:1–29. doi: 10.1016/j.jde.2021.06.037] regarding the Sturmian separation theorems on a compact interval.
ER  -

ŠEPITKA, Peter and Roman ŠIMON HILSCHER. Monotonicity and limit results for certain symmetric matrix-valued functions with applications in singular Sturmian theory. \textit{Applied Mathematics in Science and Engineering}. Taylor \&{} Francis Ltd, 2025, vol.~33, No~1, p.~1-39. ISSN~2769-0911. Available from: https://dx.doi.org/10.1080/27690911.2025.2494577.

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