×

The \(\mathfrak j\)-eigenfunctions and \(s\)-numbers. (English) Zbl 1189.47060

Authors’ abstract: It is a truth universally acknowledged, that a compact linear map between Hilbert spaces has an excellent structure that can be described by projections on eigenmanifolds. However, until comparatively recently there were no similar results when the action takes place between Banach spaces. The focus of this paper is on these new developments.
Let \(X\) and \(Y\) be uniformly convex and uniformly smooth Banach spaces, and let \(T: X\to Y\) be a compact linear map. Denote by \(J_X \) and \(J_y\) normalized duality mappings on \(X\) and \(Y\), respectively. We describe a geometric approach for obtaining a “new” class of eigenfunctions and eigenvalues for nonlinear equations of the form \[ S^* J_Y Sx = \lambda J_X x, \] where \(S\) denotes the restriction of \(T\) to subspaces generated by James orthogonality. Our method, which seems to be more direct than the Lusternik-Schnirelmann method, is based on a procedure developed recently by Evans, Harris, and one of the authors together with the use of James (otherwise called Birkhoff) orthogonality as a decomposition tool. Using the Hardy operator, for which we prove that “classical” eigenvalues and eigenvalues obtained by the Lusternik-Schnirelmann method and all “strict” \(s\)-numbers are same, we give numerical computations indicating that these new eigenvalues lie outside the family of \(s\)-numbers and that the eigenfunctions are different from classical eigenfunctions

MSC:

47J10 Nonlinear spectral theory, nonlinear eigenvalue problems
46B50 Compactness in Banach (or normed) spaces
47B40 Spectral operators, decomposable operators, well-bounded operators, etc.
47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alber, James orthogonality and orthogonal decompositions in Banach spaces, J. Math. Anal. Appl. 312 pp 330– (2005) · Zbl 1095.46007
[2] Bennewitz, Approximation numbers = singular eigenvalues, J. Comput. Appl. Math. 208 pp 102– (2007) · Zbl 1126.47019
[3] Böttcher, Weighted Markov-type inequalities, norms of Volterra operators, and zeros of Bessel functions, Math. Nachr. 283 (1) pp 40– (2010) · Zbl 1186.41004
[4] Drábek, On the closed solution to some nonhomogeneous eigenvalue problems with p -Laplacian, Differential Integral Equations 12 (6) pp 773– (1999) · Zbl 1015.34071
[5] Edmunds, Representations of compact linear operators in Banach spaces and nonlinear eigenvalue problems, J. London Math. Soc. (2) 78 pp 65– (2008) · Zbl 1169.47047
[6] D. E. Edmunds W. D. Evans D. J. Harris A spectral analysis of compact linear operators in Banach spaces, Preprint (2009).
[7] Edmunds, Behaviour of the approximation numbers of a Sobolev embedding in the one-dimensional case, J. Funct. Anal. 206 (1) pp 149– (2004) · Zbl 1058.47016
[8] P. Halmos A Hilbert Space Problem Book (D. van Nostrand, Princeton, 1967).
[9] James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc. 61 pp 265– (1947) · Zbl 0037.08001 · doi:10.1090/S0002-9947-1947-0021241-4
[10] Lê, Eigenvalue problems for the p -Laplacian, Nonlinear Anal. 64 pp 1057– (2006)
[11] J. Lindenstrauss L. Tzafriri Classical Banach spaces. II. Function spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and related Areas] Bd. 97 (Springer-Verlag, Berlin-New York, 1979).
[12] Pietsch, s -numbers of operators in Banach spaces, Studia Math. 51 pp 201– (1974)
[13] A. Pietsch History of Banach Spaces and Linear Operators (Birkhäuser, Boston, 2007).
[14] Pinkus, n -widths of Sobolev Spaces in Lp, Constr. Approx. 1 pp 15– (1985)
[15] A. Pinkus n -widths in Approximation Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (Springer-Verlag, Berlin, 1985).
[16] Schmidt, Über die nebst ihren Ableitungen orthogonalen Polynomensysteme und das zugehörige Extremum, Math. Ann. 119 pp 165– (1944) · Zbl 0028.39402
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.