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Article

Keywords:
functional analysis
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.
References:
[1] K. P. Hadeler: Eigenwerte von Operatorpolynomen. Arch. Rational Mech. Anal. 20 (1965) 72-80. MR 0184419 | Zbl 0136.12601
[2] M. A. Krasnoselski: Положительные решения операторных уравнений. (Positive Solutions of Operator Equations.) Gostechizdat, Moscow 1962. MR 0145331
[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.
[4] I. Marek: Přibližné stanovení spektrálního poloměru kladného nerozložitelného zobrazení. (Spektralradius einer positiven unzerlegbaren Abbildung). Apl. mat. 12 (1967), 351 - 363. MR 0225191
[5] I. Marek: A note on $\chi$-positive operators. Commen. Math. Univ. Carolinae 4 (1963), 137-146. MR 0167843
[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
[7] I. Marek: Spektraleigenschaften der $\chi$-positiven Operatoren und Einschliessungssätze für den Spektralradius. Czechoslovak Math. J. 16 (1966), 493 - 517. MR 0217622
[8] I. Marek: Über einen speziellen Typus der linearen Gleichungen im Hilbertschen Raume. Časopis pěst. mat. 89 (1964), 155-172. MR 0185443 | Zbl 0187.38202
[9] I. Marek: $u_0$-positive operators and some of their applications. SIAM J. Appl. Math. 15 (1967), 484-494. DOI 10.1137/0115044 | MR 0233176
[10] P. H. Müller: Eine neue Methode zur Behandlung nichtlinearer Eigenwertaufgaben. Math. Z. 70 (1959), 381-406. MR 0105024
[11] I. Sawashima: On spectral properties of some positive operators. Natur. Sci. Rep. Ochanomizu Univ. 15 (1964), 55-64. MR 0187096 | Zbl 0138.07801
[12] H. Schaefer: On the singularities of an analytic function with values in a Banach space. Arch. Math. 11 (1960), 40-43. DOI 10.1007/BF01236904 | MR 0112059 | Zbl 0093.12402
[13] H. Schaefer: Spectral properties of positive linear transformations. Pacific J. Math. 10 (1960), 1009-1019. DOI 10.2140/pjm.1960.10.1009 | MR 0115090
[14] T. Yamamoto: A computational method for the dominant root of a nonnegative irreducible matrix. Numer. Math. 8 (1966), 324-333. DOI 10.1007/BF02162977 | MR 0218011 | Zbl 0163.38801
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