# Extensions of Perron–Frobenius theory

Extensions of Perron–Frobenius theory The classical Perron–Frobenius theory asserts that, for two matrices \$\$A\$\$ and \$\$B\$\$ , if \$\$0\le B \le A\$\$ and \$\$r(A)=r(B)\$\$ with \$\$A\$\$ being irreducible, then \$\$A=B\$\$ . It has been extended to infinite-dimensional Banach lattices under certain additional conditions, including that \$\$r(A)\$\$ is a pole of the resolvent of \$\$A\$\$ . In this paper, we prove that the same result holds if \$\$B\$\$ is irreducible and \$\$r(B)\$\$ is a pole of the resolvent for \$\$B\$\$ . We also prove some other interesting extensions of the theorem for infinite-dimensional Banach lattices. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Extensions of Perron–Frobenius theory

, Volume 17 (4) – Nov 28, 2012
13 pages

/lp/springer_journal/extensions-of-perron-frobenius-theory-Va40lxLfDE
Publisher
Springer Basel
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-012-0215-3
Publisher site
See Article on Publisher Site

### Abstract

The classical Perron–Frobenius theory asserts that, for two matrices \$\$A\$\$ and \$\$B\$\$ , if \$\$0\le B \le A\$\$ and \$\$r(A)=r(B)\$\$ with \$\$A\$\$ being irreducible, then \$\$A=B\$\$ . It has been extended to infinite-dimensional Banach lattices under certain additional conditions, including that \$\$r(A)\$\$ is a pole of the resolvent of \$\$A\$\$ . In this paper, we prove that the same result holds if \$\$B\$\$ is irreducible and \$\$r(B)\$\$ is a pole of the resolvent for \$\$B\$\$ . We also prove some other interesting extensions of the theorem for infinite-dimensional Banach lattices.

### Journal

PositivitySpringer Journals

Published: Nov 28, 2012

### References

• The irreducibility in ordered Banach algebras
Alekhno, EA

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