A “nonlinear” proof of Pitt’s compactness theorem

Fabian, M.; Zizler, V.

doi:10.1090/S0002-9939-03-07200-9

A “nonlinear” proof of Pitt’s compactness theorem
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by M. Fabian and V. Zizler PDF

Proc. Amer. Math. Soc. 131 (2003), 3693-3694 Request permission

Abstract:

Using Stegall’s variational principle, we present a simple proof of Pitt’s theorem that bounded linear operators from $\ell _q$ into $\ell _p$ are compact for $1\le p<q<+\infty$.

References

Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos Santalucía, Jan Pelant, and Václav Zizler, Functional analysis and infinite-dimensional geometry, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 8, Springer-Verlag, New York, 2001. MR 1831176, DOI 10.1007/978-1-4757-3480-5
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056, DOI 10.1007/978-3-642-66557-8
Robert R. Phelps, Convex functions, monotone operators and differentiability, 2nd ed., Lecture Notes in Mathematics, vol. 1364, Springer-Verlag, Berlin, 1993. MR 1238715
Charles Stegall, Optimization of functions on certain subsets of Banach spaces, Math. Ann. 236 (1978), no. 2, 171–176. MR 503448, DOI 10.1007/BF01351389
Charles Stegall, Optimization and differentiation in Banach spaces, Proceedings of the symposium on operator theory (Athens, 1985), 1986, pp. 191–211. MR 872283, DOI 10.1016/0024-3795(86)90314-9