Bounds for the spectral radius of positive operators

Drnovšek, Roman

Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000), p. 459-467 / Harvested from Czech Digital Mathematics Library

Résumé

Let $f$ be a non-zero positive vector of a Banach lattice $L$, and let $T$ be a positive linear operator on $L$ with the spectral radius $r(T)$. We find some groups of assumptions on $L$, $T$ and $f$ under which the inequalities

$$\sup \{c \geq 0 : T f \geq c \, f\} \leq r(T) \leq \inf \{c \geq 0 : T f \leq c \, f\}$$ hold. An application of our results gives simple upper and lower bounds for the spectral radius of a product of positive operators in terms of positive eigenvectors corresponding to the spectral radii of given operators. We thus extend the matrix result obtained by Johnson and Bru which was the motivation for this paper.

Détail

Publié le : 2000-01-01

MR 1795077 | Zbl 1040.46021

Classification:  46B42,  47A10,  47B65

Bibliographie

Aliprantis C.D.; Burkinshaw O. Positive Operators, Academic Press, Orlando, 1985. | MR 0809372 | Zbl 1098.47001

Förster K.-H.; Nagy B. On the Collatz-Wielandt numbers and the local spectral radius of a nonnegative operator, Linear Algebra Appl. 120 193-205 (1989). (1989) | MR 1010057

Förster K.-H.; Nagy B. On the local spectral radius of a nonnegative element with respect to an irreducible operator, Acta Sci. Math. 55 155-166 (1991). (1991) | MR 1124954

Grobler J.J. A note on the theorems of Jentzsch-Perron and Frobenius, Indagationes Math. 49 381-391 (1987). (1987) | MR 0922442 | Zbl 0634.47034

Johnson C.R.; Bru R. The spectral radius of a product of nonnegative matrices, Linear Algebra Appl. 141 227-240 (1990). (1990) | MR 1076115 | Zbl 0712.15013

Marek I.; Varga R.S. Nested bounds for the spectral radius, Numer. Math. 14 49-70 (1969). (1969) | MR 0257779 | Zbl 0221.65063

Marek I. Frobenius theory of positive operators: Comparison theorems and applications, SIAM J. Appl. Math. 19 607-628 (1970). (1970) | MR 0415405 | Zbl 0219.47022

Marek I. Collatz-Wielandt numbers in general partially ordered spaces, Linear Algebra Appl. 173 165-180 (1992). (1992) | MR 1170509 | Zbl 0777.47003

Schaefer H.H. Banach lattices and positive operators, (Grundlehren Math. Wiss. Bd. 215), Springer-Verlag, New York, 1974. | MR 0423039 | Zbl 0296.47023

Schaefer H.H. A minimax theorem for irreducible compact operators in $L^p$-spaces, Israel J. Math. 48 196-204 (1984). (1984) | MR 0770701

Wielandt H. Unzerlegbare, nicht-negative Matrizen,, Math. Z. 52 642-648 (1950). (1950) | MR 0035265 | Zbl 0035.29101

Zaanen A.C. Riesz Spaces II, North Holland, Amsterdam, 1983. | MR 0704021 | Zbl 0519.46001