×

Approximate Birkhoff-James orthogonality in normed linear spaces and related topics. (English) Zbl 07678104

Aron, Richard M. (ed.) et al., Operator and norm inequalities and related topics. Cham: Birkhäuser. Trends Math., 303-320 (2022).
Summary: The classical Birkhoff-James orthogonality (BJ-orthogonality) in a real normed linear space is one of many possible, but arguably the most adequate, generalizations of the usual orthogonality relation in an inner product space. In this work, however, we are dealing not so much with the exact BJ-orthogonality as with its approximate version. In the first section of this chapter we introduce basic definitions connected with the notion of approximate BJ-orthogonality. Then we present a package of equivalent statements, defining in various ways the introduced concept. Some of these characterizations are known but some other are new. The second part of the paper is a survey on selected results depicting the areas where the approximate BJ-orthogonality can be applied or where it stimulates further studies.
For the entire collection see [Zbl 1504.47002].

MSC:

47A30 Norms (inequalities, more than one norm, etc.) of linear operators
47A50 Equations and inequalities involving linear operators, with vector unknowns
47A63 Linear operator inequalities
46B20 Geometry and structure of normed linear spaces
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] J. Alonso, C. Benitez, Orthogonality in normed linear spaces: a survey. Part I: main properties. Extracta Math. 3(1), 1-15 (1988). Part II: Relations between main orthogonalities. Extracta Math. 4(3), 121-131 (1989)
[2] J. Alonso, H. Martini, S. Wu, On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces. Aequationes Math. 83(1-2), 153-189 (2012) · Zbl 1241.46006 · doi:10.1007/s00010-011-0092-z
[3] C. Alsina, J. Sikorska, M. Santos Tomás, Norm Derivatives and Characterizations of Inner Product Spaces (World Scientific, Hackensack, 2010) · Zbl 1196.46001
[4] Lj. Arambašić, R. Rajić, The Birkhoff-James orthogonality in Hilbert \(C^∗\)-modules. Linear Algebra Appl. 437(7), 1913-1929 (2012) · Zbl 1257.46025
[5] Lj. Arambašić, R. Rajić, On three concepts of orthogonality in Hilbert \(C^∗\)-modules. Linear Multilinear Algebra 63(7), 1485-1500 (2015) · Zbl 1327.46050
[6] G. Birkhoff, Orthogonality in linear metric spaces. Duke Math. J. 1(2), 169-172 (1935) · JFM 61.0634.01
[7] C. Benítez, M. Fernández, M.L. Soriano, Orthogonality of matrices. Linear Algebra Appl. 422(1), 155-163 (2007) · Zbl 1125.15026 · doi:10.1016/j.laa.2006.09.018
[8] R. Bhatia, P. Šemrl, Orthogonality of matrices and some distance problems. Linear Algebra Appl. 287(1-3), 77-85 (1999) · Zbl 0937.15023 · doi:10.1016/S0024-3795(98)10134-9
[9] A. Blanco, A. Turnšek, On maps that preserve orthogonality in normed spaces. Proc. R. Soc. Edinburgh Sect. A 136(4), 709-716 (2006) · Zbl 1115.46016 · doi:10.1017/S0308210500004674
[10] J. Chmieliński, On an \(ε\)-Birkhoff orthogonality. J. Inequal. Pure Appl. Math. 6(3) (2005). Art. 79 · Zbl 1095.46011
[11] J. Chmieliński, Linear mappings approximately preserving orthogonality. J. Math. Anal. Appl. 304(1), 158-169 (2005) · Zbl 1090.46017 · doi:10.1016/j.jmaa.2004.09.011
[12] J. Chmieliński, Stability of the orthogonality preserving property in finite-dimensional inner product spaces. J. Math. Anal. Appl. 318(2), 433-443 (2006) · Zbl 1103.46016 · doi:10.1016/j.jmaa.2005.06.016
[13] J. Chmieliński, Orthogonality Preserving Property and Its Ulam Stability. Functional Equations in Mathematical Analysis, Springer Optimization and Its Applications, vol. 52 (Springer, New York, 2012), pp. 33-58 · Zbl 1248.39023
[14] J. Chmieliński, D. Khurana, D. Sain, Local approximate symmetry of Birkhoff-James orthogonality in normed linear spaces. Results Math. 76(3) (2021). Paper No. 136 · Zbl 1471.46014
[15] J. Chmieliński, T. Stypuła, P. Wójcik, Approximate orthogonality in normed spaces and its applications. Linear Algebra Appl. 531, 305-317 (2017) · Zbl 1383.46013 · doi:10.1016/j.laa.2017.06.001
[16] J. Chmieliński, P. Wójcik, Isosceles-orthogonality preserving property and its stability. Nonlinear Anal. 72(3-4), 1445-1453 (2010) · Zbl 1192.46013 · doi:10.1016/j.na.2009.08.028
[17] J. Chmieliński, P. Wójcik, On a \(ρ\)-orthogonality. Aequationes Math. 80(1-2), 45-55 (2010) · Zbl 1208.46015 · doi:10.1007/s00010-010-0042-1
[18] J. Chmieliński, P. Wójcik, \(ρ\)-orthogonality and its preservation – revisited, in Recent Developments in Functional Equations and Inequalities, vol. 99 (Banach Center Publications, Warszawa, 2013), pp. 17-30 · Zbl 1292.46007
[19] J. Chmieliński, P. Wójcik, Approximate symmetry of Birkhoff orthogonality. J. Math. Anal. Appl. 461, 625-640 (2018) · Zbl 1402.46009 · doi:10.1016/j.jmaa.2018.01.031
[20] Ch. Chorianopoulos, P. Psarrakos, Birkhoff-James approximate orthogonality sets and numerical ranges. Linear Algebra Appl. 434(9), 2089-2108 (2011) · Zbl 1215.15024 · doi:10.1016/j.laa.2010.12.008
[21] Ch. Chorianopoulos, P. Psarrakos, On the continuity of Birkhoff-James \(ε\)-orthogonality sets. Linear Multilinear Algebra 61(11), 1447-1454 (2013) · Zbl 1309.15037 · doi:10.1080/03081087.2012.743024
[22] S.S. Dragomir, On approximation of continuous linear functionals in normed linear spaces. An. Univ. Timişoara Ser. Ştiinţ. Mat., 29(1), 51-58 (1991) · Zbl 0786.46017
[23] S.S. Dragomir, Semi-Inner Products and Applications (Nova Science Publishers, Inc., Hauppauge, NY, 2004) · Zbl 1060.46001
[24] W. Fechner, J. Sikorska, On the stability of orthogonal additivity. Bull. Pol. Acad. Sci. Math. 58(1), 23-30 (2010) · Zbl 1197.39016 · doi:10.4064/ba58-1-3
[25] J.R. Giles, Classes of semi-inner-product spaces. Trans. Am. Math. Soc., 129, 436-446 (1967) · Zbl 0157.20103 · doi:10.1090/S0002-9947-1967-0217574-1
[26] S. Gudder, D. Strawther, Orthogonally additive and orthogonally increasing functions on vector spaces. Pac. J. Math. 58(2), 427-436 (1975) · Zbl 0311.46015 · doi:10.2140/pjm.1975.58.427
[27] P. Grover, Orthogonality to matrix subspaces and a distance formula. Linear Algebra Appl. 445, 280-288 (2014) · Zbl 1286.15035 · doi:10.1016/j.laa.2013.11.040
[28] D. Ilišević, A. Turnšek, Approximately orthogonality preserving mappings on \(C^∗\)-modules. J. Math. Anal. Appl. 341(1), 298-308 (2008) · Zbl 1178.46055 · doi:10.1016/j.jmaa.2007.10.028
[29] R.C. James, Orthogonality in normed linear linear spaces. Duke Math. J. 12, 291-301 (1945) · Zbl 0060.26202 · doi:10.1215/S0012-7094-45-01223-3
[30] R.C. James, Orthogonality and linear functionals in normed linear spaces. Trans. Am. Math. Soc. 61, 265-292 (1947) · Zbl 0037.08001 · doi:10.1090/S0002-9947-1947-0021241-4
[31] R.C. James, Inner products in normed linear spaces. Bull. Am. Math. Soc. 53, 559-566 (1947) · Zbl 0041.43701 · doi:10.1090/S0002-9904-1947-08831-5
[32] M. Karamanlis, P.J. Psarrakos, Birkhoff-James \(ε\)-orthogonality sets in normed linear spaces, in The Natália Bebiano Anniversary, vol. 81-92. Textos Mat. Sér. B, vol. 44 (University of Coimbra, Coimbra, 2013) · Zbl 1303.46014
[33] D. Khurana, D. Sain, Norm derivatives and geometry of bilinear operators. Ann. Funct. Anal. 12(3) (2021). Paper No. 49 · Zbl 1481.46011
[34] A. Koldobsky, Operators preserving orthogonality are isometries. Proc. R. Soc. Edinburgh Sect. A 123(5), 835-837 (1993) · Zbl 0806.46013 · doi:10.1017/S0308210500029528
[35] G. Lumer, Semi-inner-product spaces. Trans. Am. Math. Soc. 100, 29-43 (1961) · Zbl 0102.32701 · doi:10.1090/S0002-9947-1961-0133024-2
[36] A. Mal, K. Paul, T.S.S.R.K. Rao, D. Sain, Approximate Birkhoff-James orthogonality and smoothness in the space of bounded linear operators. Monatsh. Math. 190(3), 549-558 (2019) · Zbl 1436.46015
[37] H. Martini, K.J. Swanepoel, Antinorms and Radon curves. Aequationes Math. 72(1-2), 110-138 (2006) · Zbl 1108.52005 · doi:10.1007/s00010-006-2825-y
[38] B. Mojškerc, A. Turnšek, Mappings approximately preserving orthogonality in normed spaces. Nonlinear Anal. 73(12), 3821-3831 (2010) · Zbl 1208.46016 · doi:10.1016/j.na.2010.08.007
[39] M.S. Moslehian, A. Zamani, Characterizations of operator Birkhoff-James orthogonality. Canad. Math. Bull. 60(4), 816-829 (2017) · Zbl 1387.46019 · doi:10.4153/CMB-2017-004-5
[40] V. Panagakou, P. Psarrakos, N. Yannakakis, Birkhoff-James \(ε\)-orthogonality sets of vectors and vector-valued polynomials. J. Math. Anal. Appl. 454(1), 59-78 (2017) · Zbl 1366.15017 · doi:10.1016/j.jmaa.2017.04.033
[41] K. Paul, Translatable radii of an operator in the direction of another operator. Sci. Math. 2(1), 119-122 (1999) · Zbl 0952.47032
[42] K. Paul, D. Sain Birkhoff-James Orthogonality and Its Application in the Study of Geometry of Banach Space. Advanced Topics in Mathematical Analysis (CRC Press, Boca Raton, FL, 2019), pp. 245-284 · Zbl 1430.46010
[43] K. Paul, D. Sain, A. Mal, Approximate Birkhoff-James orthogonality in the space of bounded linear operators. Linear Algebra Appl. 537, 348-357 (2018) · Zbl 1391.46014 · doi:10.1016/j.laa.2017.10.008
[44] J. Rätz, On orthogonally additive mappings. Aequationes Math. 28(1-2), 35-49 (1985) · Zbl 0569.39006 · doi:10.1007/BF02189390
[45] D. Sain, K. Paul, Operator norm attainment and inner product spaces. Linear Algebra Appl. 439(8), 2448-2452 (2013) · Zbl 1291.46024 · doi:10.1016/j.laa.2013.07.008
[46] D. Sain, K. Paul, S. Hait, Operator norm attainment and Birkhoff-James orthogonality. Linear Algebra Appl. 476, 85-97 (2015) · Zbl 1335.46008 · doi:10.1016/j.laa.2015.03.002
[47] D. Sain, K. Paul, A. Mal, On approximate Birkhoff-James orthogonality and normal cones in a normed space. J. Convex Anal. 26(1), 341-351 (2019) · Zbl 1420.46019
[48] J. Sen, D. Sain, K. Paul, On approximate orthogonality and symmetry of operators in semi-Hilbertian structure. Bull. Sci. Math. 170 (2021). Paper No. 102997 · Zbl 1479.46016
[49] A. Turnšek, On mappings approximately preserving orthogonality. J. Math. Anal. Appl. 336(1), 625-631 (2007) · Zbl 1129.39011 · doi:10.1016/j.jmaa.2007.03.016
[50] P. Wójcik, Characterization of smooth spaces by approximate orthogonalities. Aequationes Math. 89(4), 1189-1194 (2015) · Zbl 1331.46013 · doi:10.1007/s00010-014-0293-3
[51] P. Wójcik, Orthogonality of compact operators. Expo. Math. 35(1), 86-94 (2017) · Zbl 1385.46013 · doi:10.1016/j.exmath.2016.06.003
[52] A. Zamani, Birkhoff-James orthogonality of operators in semi-Hilbertian spaces and its applications. Ann. Funct. Anal. 10(3), 433-445 (2019) · Zbl 1436.46017 · doi:10.1215/20088752-2019-0001
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.