# On the polynomial eigenvalue problem with positive operators and location of the spectral radius

Aplikace matematiky (1969)

• Volume: 14, Issue: 2, page 146-159
• ISSN: 0862-7940

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## Abstract

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The purpose of this article is to give some estimates for the spectral radius of the polynomial eigenvalue problem, i.e. to derive some estimates for the singularity of the operator-function $F$, $F\left(\lambda \right)={\lambda }^{m}{A}_{0}-{\sum }_{k=1}^{m}{\lambda }^{m-k}{A}_{k}$ with the maximal absolute value. It is assumed that ${A}_{1},...,{A}_{m},{A}_{0}^{-1}$ are bounded linear operators mapping a Banach space into itself. Further, it is assumed that the operators ${B}_{j}$, where ${B}_{j}={A}_{0}^{-1}{A}_{j},j=1,2,...,m$, leave a cone invariant.

## How to cite

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Marek, Ivo. "On the polynomial eigenvalue problem with positive operators and location of the spectral radius." Aplikace matematiky 14.2 (1969): 146-159. <http://eudml.org/doc/14586>.

@article{Marek1969,
abstract = {The purpose of this article is to give some estimates for the spectral radius of the polynomial eigenvalue problem, i.e. to derive some estimates for the singularity of the operator-function $F$, $F(\lambda )=\lambda ^mA_0-\sum ^m_\{k=1\} \lambda ^\{m-k\}A_k$ with the maximal absolute value. It is assumed that $A_1,\ldots ,A_m,A^\{-1\}_0$ are bounded linear operators mapping a Banach space into itself. Further, it is assumed that the operators $B_j$, where $B_j=A^\{-1\}_0 A_j, j=1,2,\ldots ,m$, leave a cone invariant.},
author = {Marek, Ivo},
journal = {Aplikace matematiky},
keywords = {functional analysis},
language = {eng},
number = {2},
pages = {146-159},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the polynomial eigenvalue problem with positive operators and location of the spectral radius},
url = {http://eudml.org/doc/14586},
volume = {14},
year = {1969},
}

TY - JOUR
AU - Marek, Ivo
TI - On the polynomial eigenvalue problem with positive operators and location of the spectral radius
JO - Aplikace matematiky
PY - 1969
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 14
IS - 2
SP - 146
EP - 159
AB - The purpose of this article is to give some estimates for the spectral radius of the polynomial eigenvalue problem, i.e. to derive some estimates for the singularity of the operator-function $F$, $F(\lambda )=\lambda ^mA_0-\sum ^m_{k=1} \lambda ^{m-k}A_k$ with the maximal absolute value. It is assumed that $A_1,\ldots ,A_m,A^{-1}_0$ are bounded linear operators mapping a Banach space into itself. Further, it is assumed that the operators $B_j$, where $B_j=A^{-1}_0 A_j, j=1,2,\ldots ,m$, leave a cone invariant.
LA - eng
KW - functional analysis
UR - http://eudml.org/doc/14586
ER -

## References

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1. K. P. Hadeler, Eigenwerte von Operatorpolynomen, Arch. Rational Mech. Anal. 20 (1965) 72-80. (1965) Zbl0136.12601MR0184419
2. M. A. Krasnoselski, Положительные решения операторных уравнений, (Positive Solutions of Operator Equations.) Gostechizdat, Moscow 1962. (1962) MR0145331
3. M. G. Krejn M. A. Rutman, Линейные операторы оставлающие инвариантным конус в пространстве Банаха, (Linear operators leaving a cone invariant in a Banach space.) Uspehi Mat. Nauk 3 (1948), 3-95. (1948)
4. I. Marek, Přibližné stanovení spektrálního poloměru kladného nerozložitelného zobrazení, (Spektralradius einer positiven unzerlegbaren Abbildung). Apl. mat. 12 (1967), 351 - 363. (1967) MR0225191
5. I. Marek, A note on $\chi$-positive operators, Commen. Math. Univ. Carolinae 4 (1963), 137-146. (1963) MR0167843
6. I. Marek, On the approximate construction of eigenvectors corresponding to a pair of complex conjugated eigenvalues, Mat.-Fyz. časopis Sloven. Akad. Vied 14 (1964), 277-288. (1964) MR0191081
7. I. Marek, Spektraleigenschaften der $\chi$-positiven Operatoren und Einschliessungssätze für den Spektralradius, Czechoslovak Math. J. 16 (1966), 493 - 517. (1966) MR0217622
8. I. Marek, Über einen speziellen Typus der linearen Gleichungen im Hilbertschen Raume, Časopis pěst. mat. 89 (1964), 155-172. (1964) Zbl0187.38202MR0185443
9. I. Marek, 10.1137/0115044, SIAM J. Appl. Math. 15 (1967), 484-494. (1967) MR0233176DOI10.1137/0115044
10. P. H. Müller, Eine neue Methode zur Behandlung nichtlinearer Eigenwertaufgaben, Math. Z. 70 (1959), 381-406. (1959) MR0105024
11. I. Sawashima, On spectral properties of some positive operators, Natur. Sci. Rep. Ochanomizu Univ. 15 (1964), 55-64. (1964) Zbl0138.07801MR0187096
12. H. Schaefer, 10.1007/BF01236904, Arch. Math. 11 (1960), 40-43. (1960) Zbl0093.12402MR0112059DOI10.1007/BF01236904
13. H. Schaefer, 10.2140/pjm.1960.10.1009, Pacific J. Math. 10 (1960), 1009-1019. (1960) MR0115090DOI10.2140/pjm.1960.10.1009
14. T. Yamamoto, 10.1007/BF02162977, Numer. Math. 8 (1966), 324-333. (1966) Zbl0163.38801MR0218011DOI10.1007/BF02162977

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