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Chmieliński, Jacek; Wójcik, Paweł

Isosceles-orthogonality preserving property and its stability. (English) Zbl 1192.46013

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 3-4, 1445-1453 (2010).
Summary: For real normed spaces, we consider the class of linear operators, preserving approximately the relation of isosceles-orthogonality. We show some general properties of such mappings. Next, we examine whether an approximately orthogonality preserving mapping admits an approximation by an orthogonality preserving one. In regard to this, we generalize some results obtained earlier for inner product spaces with standard orthogonality relation.

Cited in 19 Documents

MSC:

46B20 Geometry and structure of normed linear spaces
47B99 Special classes of linear operators
39B82 Stability, separation, extension, and related topics for functional equations

Keywords:

isosceles-orthogonality; approximate orthogonality; orthogonality preserving mappings; linear preservers; stability
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References:

[1] Chmieliński, J., Linear mappings approximately preserving orthogonality, J. Math. Anal. Appl., 304, 158-169 (2005) · Zbl 1090.46017
[2] Chmieliński, J., Stability of the orthogonality preserving property in finite-dimensional inner product spaces, J. Math. Anal. Appl., 318, 433-443 (2006) · Zbl 1103.46016
[3] Turnšek, A., On mappings approximately preserving orthogonality, J. Math. Anal. Appl., 336, 625-631 (2007) · Zbl 1129.39011
[4] Ilišević, D.; Turnšek, A., Approximately orthogonality preserving mappings on \(C^\ast \)-modules, J. Math. Anal. Appl., 341, 298-308 (2008) · Zbl 1178.46055
[5] James, R. C., Orthogonality in normed linear linear spaces, Duke Math. J., 12, 291-301 (1945) · Zbl 0060.26202
[6] Amir, D., Characterization of Inner Product Spaces (1986), Birkhäuser Verlag: Birkhäuser Verlag Basel-Boston-Stuttgart · Zbl 0617.46030
[7] Hyers, D. H.; Ulam, S. M., On approximate isometries, Bull. Amer. Math. Soc., 51, 288-292 (1945) · Zbl 0060.26404
[8] Omladič, M.; Šemrl, P., On non linear perturbations of isometries, Math. Ann., 303, 617-628 (1995) · Zbl 0836.46014
[9] Dolinar, G., Generalized stability of isometries, J. Math. Anal. Appl., 242, 39-56 (2000) · Zbl 0956.46007
[10] Michael, E.; Pełczyński, A., Separable spaces which admit approximation, Israel J. Math., 4, 189-198 (1966) · Zbl 0151.17602
[11] Alspach, D. E., Small into isomorphism on \(L_p\) spaces, Ill. J. Math., 27, 300-314 (1983) · Zbl 0495.46011
[12] Benyamini, Y., Small into-isomorphisms between spaces of continuous functions, Proc. Amer. Math. Soc., 83, 479-485 (1981) · Zbl 0474.46012
[13] Benyamini, Y., Small into-isomorphisms between spaces of continuous functions II, Trans. Amer. Math. Soc., 277, 825-833 (1983) · Zbl 0515.46023
[14] Ding, G. G., The approximation problem of almost isometric operators by isometric operators, Acta Math. Sci. Engl. Ed., 8, 361-372 (1988) · Zbl 0664.41017
[15] Moszner, Z., Sur les définitions différentes de la stabilité des équations fonctionnelles, Aequationes Math., 68, 260-274 (2004) · Zbl 1060.39031
[16] Bellenot, S. F., Banach spaces with trivial isometries, Israel J. Math., 56, 89-96 (1986) · Zbl 0619.46012
[17] Jarosz, K., Small isomorphisms of \(C(X, E)\) spaces, Pacific J. Math., 138, 295-315 (1989) · Zbl 0698.46033
[18] V. Yu Protasov, On stability of isometries in Banach spaces, manuscript; V. Yu Protasov, On stability of isometries in Banach spaces, manuscript · Zbl 1252.46005
[19] Koldobsky, A., Operators preserving orthogonality are isometries, Proc. Roy. Soc. Edinburgh Sect. A, 123, 835-837 (1993) · Zbl 0806.46013
[20] Blanco, A.; Turnšek, A., On maps that preserve orthogonality in normed spaces, Proc. Roy. Soc. Edinburgh Sect. A, 136, 709-716 (2006) · Zbl 1115.46016
[21] Chmieliński, J., Remarks on orthogonality preserving mappings in normed spaces and some stability problems, Banach J. Math. Anal., 1, 1, 117-124 (2007) · Zbl 1135.46006
[22] Chmieliński, J., On an \(\varepsilon \)-Birkhoff orthogonality, J. Inequal. Pure Appl. Math., 6, 3 (2005), Art. 79 · Zbl 1095.46011
[23] Dragomir, S. S., On approximation of continuous linear functionals in normed linear spaces, An. Univ. Timişoara Ser. Ştiinţ. Mat., 29, 51-58 (1991) · Zbl 0786.46017
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