Open Access
2007 Remarks on orthogonality preserving mappings in normed spaces and some stability problems
Jacek Chmielinski
Banach J. Math. Anal. 1(1): 117-124 (2007). DOI: 10.15352/bjma/1240321561
Abstract

We consider the Birkhoff-James orthogonality in normed spaces and classes of linear mappings exactly and approximately preserving this relation. Some related stability problems are posed.

References

1.

R. Badora, J. Chmieliński, Decomposition of mappings approximately inner product preserving, Nonlinear Anal. 62 (2005), 1015–1023. MR2152994R. Badora, J. Chmieliński, Decomposition of mappings approximately inner product preserving, Nonlinear Anal. 62 (2005), 1015–1023. MR2152994

2.

G. Birkhoff, Orthogonality in linear metric spaces, Duke Math. J. 1 (1935), 169-172. MR1545873 10.1215/S0012-7094-35-00115-6 euclid.dmj/1077488974  0012.30604G. Birkhoff, Orthogonality in linear metric spaces, Duke Math. J. 1 (1935), 169-172. MR1545873 10.1215/S0012-7094-35-00115-6 euclid.dmj/1077488974  0012.30604

3.

A. Blanco, A. Turnšek, On maps that preserve orthogonality in normed spaces, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006), 709–716. MR2250441 10.1017/S0308210500004674 1115.46016A. Blanco, A. Turnšek, On maps that preserve orthogonality in normed spaces, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006), 709–716. MR2250441 10.1017/S0308210500004674 1115.46016

4.

J. Chmieliński, On an $\varepsilon$-Birkhoff orthogonality, J. Inequal. Pure and Appl. Math., 6(3) (2005), Art. 79. MR2164320J. Chmieliński, On an $\varepsilon$-Birkhoff orthogonality, J. Inequal. Pure and Appl. Math., 6(3) (2005), Art. 79. MR2164320

5.

J. Chmieliński, Linear mappings approximately preserving orthogonality, J. Math. Anal. Appl., 304 (2005), 158–169. MR2124655 10.1016/j.jmaa.2004.09.011 1090.46017J. Chmieliński, Linear mappings approximately preserving orthogonality, J. Math. Anal. Appl., 304 (2005), 158–169. MR2124655 10.1016/j.jmaa.2004.09.011 1090.46017

6.

J. Chmieliński, Stability of the orthogonality preserving property in finite-dimensional inner product spaces, J. Math. Anal. Appl. 318 (2006), 433–443. MR2215159 10.1016/j.jmaa.2005.06.016 1103.46016J. Chmieliński, Stability of the orthogonality preserving property in finite-dimensional inner product spaces, J. Math. Anal. Appl. 318 (2006), 433–443. MR2215159 10.1016/j.jmaa.2005.06.016 1103.46016

7.

J. Chmieliński, Stability of the Wigner equation and related topics, Nonlinear Funct. Anal. Appl., 11 (2006), 859–879. MR2309591J. Chmieliński, Stability of the Wigner equation and related topics, Nonlinear Funct. Anal. Appl., 11 (2006), 859–879. MR2309591

8.

J. Chmielinski, M.S. Moslehian, Approximately $C^*$-inner product preserving mappings, Bull. Korean. Math. Soc. (to appear). MR2391464 1149.39019J. Chmielinski, M.S. Moslehian, Approximately $C^*$-inner product preserving mappings, Bull. Korean. Math. Soc. (to appear). MR2391464 1149.39019

9.

M.M. Day, Normed Linear Spaces, Springer-Verlag, Berlin – Heidelberg – New York, 1973. MR344849M.M. Day, Normed Linear Spaces, Springer-Verlag, Berlin – Heidelberg – New York, 1973. MR344849

10.

S.S. Dragomir, On approximation of continuous linear functionals in normed linear spaces, An. Univ. Timişoara Ser. Ştiinţ. Mat. 29 (1991), 51-58. MR1336199 0786.46017S.S. Dragomir, On approximation of continuous linear functionals in normed linear spaces, An. Univ. Timişoara Ser. Ştiinţ. Mat. 29 (1991), 51-58. MR1336199 0786.46017

11.

S.S. Dragomir, Semi-Inner Products and Applications, Nova Science Publishers, Inc., Hauppauge, NY, 2004. MR2047793S.S. Dragomir, Semi-Inner Products and Applications, Nova Science Publishers, Inc., Hauppauge, NY, 2004. MR2047793

12.

J.R. Giles, Classes of semi–inner–product spaces, Trans. Amer. Math. Soc. 129 (1967), 436-446. MR217574 10.2307/1994599 0157.20103J.R. Giles, Classes of semi–inner–product spaces, Trans. Amer. Math. Soc. 129 (1967), 436-446. MR217574 10.2307/1994599 0157.20103

13.

R.C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc. 61 (1947), 265-292. MR21241 10.2307/1990220 0037.08001R.C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc. 61 (1947), 265-292. MR21241 10.2307/1990220 0037.08001

14.

S.-M. Jung, Stability of the orthogonality equation on bounded domain, Nonlinear Anal. 47 (2001), 2655–2666. MR1972390S.-M. Jung, Stability of the orthogonality equation on bounded domain, Nonlinear Anal. 47 (2001), 2655–2666. MR1972390

15.

D. Koehler, P. Rosenthal, On isometries of normed linear spaces, Studia Math. 36 (1970), 213-216. MR275209D. Koehler, P. Rosenthal, On isometries of normed linear spaces, Studia Math. 36 (1970), 213-216. MR275209

16.

A. Koldobsky, Operators preserving orthogonality are isometries, Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), 835-837. MR1249689 0806.46013A. Koldobsky, Operators preserving orthogonality are isometries, Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), 835-837. MR1249689 0806.46013

17.

G. Lumer, Semi–inner–product spaces, Trans. Amer. Math. Soc. 100 (1961), 29-43. MR133024 10.2307/1993352 0102.32701G. Lumer, Semi–inner–product spaces, Trans. Amer. Math. Soc. 100 (1961), 29-43. MR133024 10.2307/1993352 0102.32701

18.

Th.M. Rassias, A new generalization of a theorem of Jung for the orthogonality equation, Applicable Analysis 81 (2002), 163–177. MR1926808 10.1080/0003681021000021132 1022.39025Th.M. Rassias, A new generalization of a theorem of Jung for the orthogonality equation, Applicable Analysis 81 (2002), 163–177. MR1926808 10.1080/0003681021000021132 1022.39025

19.

A. Tissier, A right-angle preserving mapping (a solution of a problem proposed in 1983 by H. Kestelman), Advanced Problem 6436, Amer. Math. Monthly 92 (1985), 291-292. MR1540626 10.2307/2323660A. Tissier, A right-angle preserving mapping (a solution of a problem proposed in 1983 by H. Kestelman), Advanced Problem 6436, Amer. Math. Monthly 92 (1985), 291-292. MR1540626 10.2307/2323660

20.

A. Turnšek, On mappings approximately preserving orthogonality, J. Math. Anal. Appl. 336 (2007), 625–631. MR2348530 10.1016/j.jmaa.2007.03.016A. Turnšek, On mappings approximately preserving orthogonality, J. Math. Anal. Appl. 336 (2007), 625–631. MR2348530 10.1016/j.jmaa.2007.03.016
Copyright © 2007 Tusi Mathematical Research Group
Jacek Chmielinski "Remarks on orthogonality preserving mappings in normed spaces and some stability problems," Banach Journal of Mathematical Analysis 1(1), 117-124, (2007). https://doi.org/10.15352/bjma/1240321561
Published: 2007
Vol.1 • No. 1 • 2007
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