Approximation in reflexive Banach spaces and applications to the invariant subspace problem

Chalendar, Isabelle; Partington, Jonathan; Smith, Martin

doi:10.1090/S0002-9939-03-07152-1

Approximation in reflexive Banach spaces and applications to the invariant subspace problem
HTML articles powered by AMS MathViewer

by Isabelle Chalendar, Jonathan R. Partington and Martin Smith

Proc. Amer. Math. Soc. 132 (2004), 1133-1142

DOI: https://doi.org/10.1090/S0002-9939-03-07152-1

Published electronically: June 23, 2003

PDF | Request permission

Abstract:

We formulate a general approximation problem involving reflexive and smooth Banach spaces, and give its explicit solution. Two applications are presented—the first is to the Bounded Completion Problem involving approximation of Hardy class functions, while the second involves the construction of minimal vectors and hyperinvariant subspaces of linear operators, generalizing the Hilbert space technique of Ansari and Enflo.

References

R. F. Curtain, A. Bensoussan, and J.-L. Lions (eds.), Analysis and optimization of systems: state and frequency domain approaches for infinite-dimensional systems, Lecture Notes in Control and Information Sciences, vol. 185, Springer-Verlag, Berlin, 1993. MR 1208262, DOI 10.1007/BFb0115017
G. Androulakis. A note on the method of minimal vectors. Trends in Banach spaces and operator theory, Contemporary Mathematics, Ed. A. Kaminska, to appear.
Shamim Ansari and Per Enflo, Extremal vectors and invariant subspaces, Trans. Amer. Math. Soc. 350 (1998), no. 2, 539–558. MR 1407476, DOI 10.1090/S0002-9947-98-01865-0
L. Baratchart and J. Leblond, Hardy approximation to $L^p$ functions on subsets of the circle with $1\leq p<\infty$, Constr. Approx. 14 (1998), no. 1, 41–56. MR 1486389, DOI 10.1007/s003659900062
L. Baratchart, J. Leblond, and J. R. Partington, Problems of Adamjan-Arov-Krein type on subsets of the circle and minimal norm extensions, Constr. Approx. 16 (2000), no. 3, 333–357. MR 1759893, DOI 10.1007/s003659910015
Bernard Beauzamy, Introduction to Banach spaces and their geometry, Notas de Matemática [Mathematical Notes], vol. 86, North-Holland Publishing Co., Amsterdam-New York, 1982. MR 670943
Isabelle Chalendar and Jean Esterle, Le problème du sous-espace invariant, Development of mathematics 1950–2000, Birkhäuser, Basel, 2000, pp. 235–267 (French). MR 1796843
Alexander A. Borichev and Nikolai K. Nikolski (eds.), Systems, approximation, singular integral operators, and related topics, Operator Theory: Advances and Applications, vol. 129, Birkhäuser Verlag, Basel, 2001. Papers from the 11th International Workshop on Operator Theory and Applications (IWOTA 2000) held at the Université Bordeaux I, Bordeaux, June 13–16, 2000. MR 1882688, DOI 10.1007/978-3-0348-8362-7
I. Chalendar and J. R. Partington. Constrained approximation and invariant subspaces. J. Math. Anal. Appl., 280:176–187, 2003.
Robert Deville, Gilles Godefroy, and Václav Zizler, Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. MR 1211634
M. G. Kreĭn and P. Ja. Nudel′man, Approximation of functions in $L_{2}(\omega _{1},\omega _{2})$ by transmission functions of linear systems with minimal energy, Problemy Peredači Informacii 11 (1975), no. 2, 37–60 (Russian). MR 0481850
Juliette Leblond and Jonathan R. Partington, Constrained approximation and interpolation in Hilbert function spaces, J. Math. Anal. Appl. 234 (1999), no. 2, 500–513. MR 1689403, DOI 10.1006/jmaa.1999.6358
V. I. Lomonosov. Invariant subspaces for operators commuting with compact operators. Funct. Anal. Appl., 7:213–214, 1973.
Robert E. Megginson, An introduction to Banach space theory, Graduate Texts in Mathematics, vol. 183, Springer-Verlag, New York, 1998. MR 1650235, DOI 10.1007/978-1-4612-0603-3
Heydar Radjavi and Peter Rosenthal, Invariant subspaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 77, Springer-Verlag, New York-Heidelberg, 1973. MR 0367682, DOI 10.1007/978-3-642-65574-6
Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. MR 1157815
M. Smith. Constrained approximation in Banach spaces. Constr. Approx., to appear.
V. G. Troitsky. Minimal vectors in arbitrary Banach spaces. Preprint, 2002.