×

A new sufficient condition for the denseness of norm attaining operators. (English) Zbl 0865.46005

Summary: We give a new sufficient condition for a Banach space \(Y\) to satisfy Lindenstrauss’s property \(B\), namely the set of norm-attaining operators from any other Banach space \(X\) into \(Y\) is dense. Even in the finite-dimensional case, our result gives new examples of Banach spaces with property \(B\).

MSC:

46B20 Geometry and structure of normed linear spaces
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] M.D. Acosta, Every real Banach space can be renormed to satisfy the denseness of numerical radius attaining operators , Israel J. Math. 81 (1993), 273-280. · Zbl 0795.47001 · doi:10.1007/BF02764831
[2] M.D. Acosta, F.J. Aguirre and R. Payá, A space by W. Gowers and new results on norm and numerical radius attaining operators , Acta Univ. Carolin. Math. Phys. 33 (1992), 5-14. · Zbl 0786.47002
[3] M.D. Acosta and R. Payá, Numerical radius attaining operators and the Radon-Nikodym property , Bull. London Math. Soc. 25 (1993), 67-73. · Zbl 0739.47004 · doi:10.1112/blms/25.1.67
[4] E.M. Alfsen, Compact convex sets and boundary integrals , Springer-Verlag, New York, 1971. · Zbl 0209.42601
[5] E. Bishop and R.R. Phelps, A proof that every Banach space is subreflexive , Bull. Amer. Math. Soc. 67 (1961), 97-98. · Zbl 0098.07905 · doi:10.1090/S0002-9904-1961-10514-4
[6] ——–, The support functionals of a convex set , in Convexity , Proc. Sympos. Pure Math. 7 (1963), 27-35. · Zbl 0149.08601
[7] J. Bourgain, On dentability and the Bishop-Phelps property , Israel J. Math. 28 (1977), 265-271. · Zbl 0365.46021 · doi:10.1007/BF02760634
[8] C. Cardassi, Numerical radius attaining operators on \(C(K)\) , Proc. Amer. Math. Soc. 95 (1985), 537-543. JSTOR: · Zbl 0602.47020 · doi:10.2307/2045839
[9] W.J. Davis, Positive bases in Banach spaces , Rev. Roumaine Math. Pures Appl. 16 (1971), 487-492. · Zbl 0228.46011
[10] W.T. Gowers, Symmetric block bases of sequences with large average growth , Israel J. Math. 69 (1990), 129-151. · Zbl 0721.46010 · doi:10.1007/BF02937300
[11] R. Huff, On non-density of norm attaining operators , Rev. Roumaine Math. Pures Appl. 25 (1980), 239-241. · Zbl 0434.46004
[12] R.B. Holmes, Geometric functional analysis and its applications , Springer-Verlag, New York, 1975. · Zbl 0336.46001
[13] J. Johnson and J. Wolfe, Norm attaining operators , Studia Math. 65 (1979), 7-19. · Zbl 0432.47024
[14] A. Lima, Intersection properties of balls in spaces of compact operators , Ann. Inst. Fourier Grenoble 28 (1978), 35-65. · Zbl 0347.46018 · doi:10.5802/aif.700
[15] J. Lindenstrauss, On operators which attain their norm , Israel J. Math. 1 (1963), 139-148. · Zbl 0127.06704 · doi:10.1007/BF02759700
[16] J.R. Partington, Norm attaining operators , Israel J. Math. 43 (1982), 273-276. · Zbl 0507.46008 · doi:10.1007/BF02761947
[17] R. Payá, A counterexample on numerical radius attaining operators , Israel J. Math. 79 (1992), 83-101. · Zbl 0784.47005 · doi:10.1007/BF02764803
[18] W. Schachermayer, Norm attaining operators and renormings of Banach spaces , Israel J. Math. 44 (1983), 201-212. · Zbl 0542.46013 · doi:10.1007/BF02760971
[19] C. Stegall, Optimization and differentiation in Banach spaces , Linear Algebra Appl. 84 (1986), 191-211. · Zbl 0633.46042 · doi:10.1016/0024-3795(86)90314-9
[20] V. Zizler, On some extremal problems in Banach spaces , Math. Scand. 32 (1973), 214-224. · Zbl 0269.46014
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.