| Title:
|
On the polynomial eigenvalue problem with positive operators and location of the spectral radius (English) |
| Author:
|
Marek, Ivo |
| Language:
|
English |
| Journal:
|
Aplikace matematiky |
| ISSN:
|
0373-6725 |
| Volume:
|
14 |
| Issue:
|
2 |
| Year:
|
1969 |
| Pages:
|
146-159 |
| Summary lang:
|
English |
| Summary lang:
|
Czech |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
functional analysis |
| MSC:
|
47-30 |
| idZBL:
|
Zbl 0175.13701 |
| idMR:
|
MR0246148 |
| DOI:
|
10.21136/AM.1969.103217 |
| . |
| Date available:
|
2008-05-20T17:44:47Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/103217 |
| . |
| Reference:
|
[1] K. P. Hadeler: Eigenwerte von Operatorpolynomen.Arch. Rational Mech. Anal. 20 (1965) 72-80. Zbl 0136.12601, MR 0184419, 10.1007/BF00250191 |
| Reference:
|
[2] M. A. Krasnoselski: Положительные решения операторных уравнений.(Positive Solutions of Operator Equations.) Gostechizdat, Moscow 1962. MR 0145331 |
| Reference:
|
[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. |
| Reference:
|
[4] I. Marek: Přibližné stanovení spektrálního poloměru kladného nerozložitelného zobrazení.(Spektralradius einer positiven unzerlegbaren Abbildung). Apl. mat. 12 (1967), 351 - 363. MR 0225191 |
| Reference:
|
[5] I. Marek: A note on $\chi$-positive operators.Commen. Math. Univ. Carolinae 4 (1963), 137-146. MR 0167843 |
| Reference:
|
[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. MR 0191081 |
| Reference:
|
[7] I. Marek: Spektraleigenschaften der $\chi$-positiven Operatoren und Einschliessungssätze für den Spektralradius.Czechoslovak Math. J. 16 (1966), 493 - 517. MR 0217622 |
| Reference:
|
[8] I. Marek: Über einen speziellen Typus der linearen Gleichungen im Hilbertschen Raume.Časopis pěst. mat. 89 (1964), 155-172. Zbl 0187.38202, MR 0185443 |
| Reference:
|
[9] I. Marek: $u_0$-positive operators and some of their applications.SIAM J. Appl. Math. 15 (1967), 484-494. MR 0233176, 10.1137/0115044 |
| Reference:
|
[10] P. H. Müller: Eine neue Methode zur Behandlung nichtlinearer Eigenwertaufgaben.Math. Z. 70 (1959), 381-406. MR 0105024 |
| Reference:
|
[11] I. Sawashima: On spectral properties of some positive operators.Natur. Sci. Rep. Ochanomizu Univ. 15 (1964), 55-64. Zbl 0138.07801, MR 0187096 |
| Reference:
|
[12] H. Schaefer: On the singularities of an analytic function with values in a Banach space.Arch. Math. 11 (1960), 40-43. Zbl 0093.12402, MR 0112059, 10.1007/BF01236904 |
| Reference:
|
[13] H. Schaefer: Spectral properties of positive linear transformations.Pacific J. Math. 10 (1960), 1009-1019. MR 0115090, 10.2140/pjm.1960.10.1009 |
| Reference:
|
[14] T. Yamamoto: A computational method for the dominant root of a nonnegative irreducible matrix.Numer. Math. 8 (1966), 324-333. Zbl 0163.38801, MR 0218011, 10.1007/BF02162977 |
| . |
