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

top
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

top

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

top
  1. Aliprantis C.D., Burkinshaw O., Positive Operators, Academic Press, Orlando, 1985. Zbl1098.47001MR0809372
  2. 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) MR1010057
  3. 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) MR1124954
  4. Grobler J.J., A note on the theorems of Jentzsch-Perron and Frobenius, Indagationes Math. 49 381-391 (1987). (1987) Zbl0634.47034MR0922442
  5. Johnson C.R., Bru R., The spectral radius of a product of nonnegative matrices, Linear Algebra Appl. 141 227-240 (1990). (1990) Zbl0712.15013MR1076115
  6. Marek I., Varga R.S., Nested bounds for the spectral radius, Numer. Math. 14 49-70 (1969). (1969) Zbl0221.65063MR0257779
  7. Marek I., Frobenius theory of positive operators: Comparison theorems and applications, SIAM J. Appl. Math. 19 607-628 (1970). (1970) Zbl0219.47022MR0415405
  8. Marek I., Collatz-Wielandt numbers in general partially ordered spaces, Linear Algebra Appl. 173 165-180 (1992). (1992) Zbl0777.47003MR1170509
  9. Schaefer H.H., Banach lattices and positive operators, (Grundlehren Math. Wiss. Bd. 215), Springer-Verlag, New York, 1974. Zbl0296.47023MR0423039
  10. Schaefer H.H., A minimax theorem for irreducible compact operators in L p -spaces, Israel J. Math. 48 196-204 (1984). (1984) MR0770701
  11. Wielandt H., Unzerlegbare, nicht-negative Matrizen,, Math. Z. 52 642-648 (1950). (1950) Zbl0035.29101MR0035265
  12. Zaanen A.C., Riesz Spaces II, North Holland, Amsterdam, 1983. Zbl0519.46001MR0704021

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.