Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Inequivalent measures of noncompactness and the radius of the essential spectrum

Authors: John Mallet-Paret and Roger D. Nussbaum
Journal: Proc. Amer. Math. Soc. 139 (2011), 917-930
MSC (2010): Primary 47H08, 46B20; Secondary 46B25, 46B45, 47A10, 47H10
Published electronically: October 29, 2010
MathSciNet review: 2745644
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The Kuratowski measure of noncompactness $ \alpha$ on an infinite dimensional Banach space $ (X,\Vert\cdot\Vert)$ assigns to each bounded set $ S$ in $ X$ a nonnegative real number $ \alpha (S)$ by the formula

\begin{equation*} \begin{aligned} \alpha (S)= & \inf \{\delta >0 \mid S=\textsty... ...(S_i)\leq \delta,\hbox{ for }1\le i\le n<\infty \}. \end{aligned}\end{equation*}

In general a map $ \beta$ which assigns to each bounded set $ S$ in $ X$ a nonnegative real number and which shares most of the properties of $ \alpha$ is called a homogeneous measure of noncompactness or homogeneous MNC. Two homogeneous MNC's $ \beta$ and $ \gamma$ on $ X$ are called equivalent if there exist positive constants $ b$ and $ c$ with $ b\beta (S)\leq \gamma (S)\leq c\beta (S)$ for all bounded sets $ S\subset X$. There are many results which prove the equivalence of various homogeneous MNC's. Working with $ X=\ell^p (\mathbb{N})$ where $ 1\leq p\leq \infty$, we give the first examples of homogeneous MNC's which are not equivalent.

Further, if $ X$ is any complex, infinite dimensional Banach space and $ L:X\rightarrow X$ is a bounded linear map, one can define $ \rho (L)=\sup \{\vert\lambda\vert \mid \lambda \in \textrm{ess}(L)\}$, where $ \textrm{ess}(L)$ denotes the essential spectrum of $ L$. One can also define

$\displaystyle \beta (L)=\inf\{\lambda>0 \mid \beta(LS) \le\lambda\beta(S)\hbox{ for every }S\in{\mathcal{B}(X)}\}. $

The formula $ \rho (L)=\displaystyle{\lim_{m\rightarrow \infty}} \beta (L^m)^{1/m}$ is known to be true if $ \beta$ is equivalent to $ \alpha$, the Kuratowski MNC; however, as we show, it is in general false for MNC's which are not equivalent to $ \alpha$. On the other hand, if $ B$ denotes the unit ball in $ X$ and $ \beta$ is any homogeneous MNC, we prove that

$\displaystyle \rho (L)=\limsup_{m\to\infty}\beta(L^mB)^{1/m} =\inf \{\lambda>0 \mid \lim_{m\to \infty} \lambda^{-m} \beta (L^mB)=0\}. $

Our motivation for this study comes from questions concerning eigenvectors of linear and nonlinear cone-preserving maps.

References [Enhancements On Off] (What's this?)

  • 1. R.R. Akhmerov, M.I. Kamenskij, A.S. Potapov, A.E. Rodkina, and B.N. Sadovskij, Measures of Noncompactness and Condensing Operators (in Russian), Nauka, Novosibirsk, 1986; English translation: Birkhäuser Verlag, Basel, 1992. MR 1153247 (92k:47104)
  • 2. J. Appell, Measures of noncompactness, condensing operators and fixed points: An application-oriented survey, Fixed Point Theory 6 (2005), pp. 157-229. MR 2196709 (2006h:47121)
  • 3. J.M. Ayerbe Toledano, T. Dominguez Benavides, and G. López Acedo, Measures of Noncompactness in Metric Fixed Point Theory, Birkhäuser Verlag, Basel, 1997. MR 1483889 (99e:47070)
  • 4. J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, Marcel Dekker, New York, 1980. MR 591679 (82f:47066)
  • 5. F.E. Browder, On the spectral theory of elliptic differential operators, Math. Ann. 142 (1961), pp. 22-130. MR 0209909 (35:804)
  • 6. G. Darbo, Punti uniti in trasformazioni a condominio non compatto, Rend. Sem. Mat. Univ. Padova 24 (1955), pp. 84-92. MR 0070164 (16:1140f)
  • 7. M. Furi and A. Vignoli, On a property of the unit sphere in a linear normed space, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 18 (1970), pp. 333-334. MR 0264373 (41:8969)
  • 8. I. Gohberg and M.G. Krein, The basic propositions on defect numbers, root numbers and indices of linear operators, Amer. Math. Soc. Translations, Series 2, vol. 13 (1960), pp. 185-264. MR 0113146 (22:3984)
  • 9. T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966. MR 0203473 (34:3324)
  • 10. K. Kuratowski, Sur les espaces complets, Fund. Math. 15 (1930), pp. 301-309.
  • 11. J. Mallet-Paret and R.D. Nussbaum, Eigenvalues for a class of homogeneous cone maps arising from max-plus operators, Discrete and Continuous Dynamical Systems 8 (2002), pp. 519-562. MR 1897866 (2003c:47088)
  • 12. J. Mallet-Paret and R.D. Nussbaum, Inequivalent measures of noncompactness, Ann. Mat. Pura Appl., to appear.
  • 13. J. Mallet-Paret and R.D. Nussbaum, Generalizing the Krein-Rutman theorem, measures of noncompactness and the fixed point index, J. Fixed Point Theory and Appl. 7 (2010), pp. 103-143.
  • 14. R.D. Nussbaum, The radius of the essential spectrum, Duke Math. J. 37 (1970), pp. 473-478. MR 0264434 (41:9028)
  • 15. R.D. Nussbaum, A generalization of the Ascoli theorem and an application to functional differential equations, J. Math. Anal. Appl. 35 (1971), pp. 600-610. MR 0289898 (44:7085)
  • 16. R.D. Nussbaum, The fixed point index for local condensing maps, Ann. Mat. Pura Appl. 89 (1971), pp. 217-258. MR 0312341 (47:903)
  • 17. R.D. Nussbaum, Eigenvalues of nonlinear operators and the linear Krein-Rutman theorem, in Fixed Point Theory, Springer Lecture Notes in Math., vol. 886, Springer-Verlag, Berlin, 1981, pp. 309-331. MR 643014 (83b:47068)
  • 18. B.N. Sadovskij, Limit-compact and condensing operators, Uspekhi Mat. Nauk 27 (1972), pp. 81-146 (in Russian). MR 0428132 (55:1161)
  • 19. F. Wolf, On the essential spectrum of partial differential boundary problems, Comm. Pure Appl. Math. 12 (1959), pp. 211-228. MR 0107750 (21:6472)
  • 20. B. Yood, Properties of linear transformations preserved under addition of a completely continuous transformation, Duke Math. J. 18 (1951), pp. 599-612. MR 0044020 (13:355f)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 47H08, 46B20, 46B25, 46B45, 47A10, 47H10

Retrieve articles in all journals with MSC (2010): 47H08, 46B20, 46B25, 46B45, 47A10, 47H10

Additional Information

John Mallet-Paret
Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912

Roger D. Nussbaum
Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854

Keywords: Measure of noncompactness, essential spectral radius, cone map.
Received by editor(s): September 21, 2009
Received by editor(s) in revised form: January 16, 2010
Published electronically: October 29, 2010
Additional Notes: The first author was partially supported by NSF Grant DMS-0500674
The second author was partially supported by NSF Grant DMS-0701171
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society