Geodesic
On the polynomial eigenvalue problem with positive operators and location of the spectral radius
Applications of Mathematics, Tome 14 (1969) no. 2, pp. 146-159

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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 $F$, $F(\lambda)=\lambda^mA_0-\sum^m_{k=1} \lambda^{m-k}A_k$ 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.
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.
DOI : 10.21136/AM.1969.103217
Classification : 47-30
Keywords: functional analysis 
Marek, Ivo. On the polynomial eigenvalue problem with positive operators and location of the spectral radius. Applications of Mathematics, Tome 14 (1969) no. 2, pp. 146-159. doi: 10.21136/AM.1969.103217
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