Bounds for the spectral radius of positive operators

Roman Drnovšek

Commentationes Mathematicae Universitatis Carolinae (2000)

  • Volume: 41, Issue: 3, page 459-467
  • ISSN: 0010-2628

Abstract

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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 ≥ 0 : T f ≥ c f } ≤ r ( T ) ≤ inf { c ≥ 0 : T f ≤ 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.

How to cite

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Drnovšek, Roman. "Bounds for the spectral radius of positive operators." Commentationes Mathematicae Universitatis Carolinae 41.3 (2000): 459-467. <http://eudml.org/doc/248602>.

@article{Drnovšek2000,
abstract = {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 \[ \sup \lbrace c \ge 0 : T f \ge c \, f\rbrace \le r(T) \le \inf \lbrace c \ge 0 : T f \le c \, f\rbrace \] 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.},
author = {Drnovšek, Roman},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Banach lattices; positive operators; spectral radius; Banach lattice; positive operator; spectral radius; Collatz-Wielandt bounds},
language = {eng},
number = {3},
pages = {459-467},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Bounds for the spectral radius of positive operators},
url = {http://eudml.org/doc/248602},
volume = {41},
year = {2000},
}

TY - JOUR
AU - Drnovšek, Roman
TI - Bounds for the spectral radius of positive operators
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2000
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 41
IS - 3
SP - 459
EP - 467
AB - 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 \[ \sup \lbrace c \ge 0 : T f \ge c \, f\rbrace \le r(T) \le \inf \lbrace c \ge 0 : T f \le c \, f\rbrace \] 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.
LA - eng
KW - Banach lattices; positive operators; spectral radius; Banach lattice; positive operator; spectral radius; Collatz-Wielandt bounds
UR - http://eudml.org/doc/248602
ER -

References

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