Abstract
The cosine function is a classical tool for measuring angles in inner product spaces, and it has various extensions to normed linear spaces. In this paper, we investigate a cosine function for the convex angle formed by two nonzero elements of a complex normed linear space, in connection with recent results on the Birkhoff-James approximate orthogonality sets.
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References
Amir, D.: Characterizations of inner product spaces. In: Operator Theory: Advances and Applications. Vol. 20, Birkhäuser Verlag (1986)
Balestro, V., Horváth, Á.G., Martini, H., Teixeira, R.: Angles in normed spaces. Aequationes Mathematicae 91, 201–236 (2017)
Bhatia, R., Šemrl, P.: Orthogonality of matrices and some distance problems. Linear Algebra Appl. 287, 77–86 (1999)
Birkhoff, G.: Orthogonality in linear metric spaces. Duke Math. J. 1, 169–172 (1935)
Bonsall, F.F., Duncan, J.: Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras. London Mathematical Society Lecture Note Series. Cambridge University Press, New York (1971)
Bonsall, F.F., Duncan, J.: Numerical Ranges II. London Mathematical Society Lecture Note SeriesLondon Mathematical Society Lecture Note Series. Cambridge University Press, New York (1973)
Chmieliński, J.: On an \(\varepsilon \)-Birkhoff orthogonality. J. Inequal. Pure Appl. Math. 79, 6 (2005)
Chorianopoulos, Ch., Karanasios, S., Psarrakos, P.: A definition of numerical range of rectangular matrices. Linear Multilinear Algebra 57, 459–475 (2009)
Chorianopoulos, Ch., Psarrakos, P.: Birkhoff-James approximate orthogonality sets and numerical ranges. Linear Algebra Appl. 434, 2089–2108 (2011)
Chorianopoulos, Ch., Psarrakos, P.: On the continuity of Birkhoff-James epsilon-orthogonality sets. Linear Multilinear Algebra 61, 1447–1454 (2013)
Day, M.M.: Some characterizations of inner product spaces. Trans. Am. Math. Soc. 62, 320–337 (1947)
Dragomir, S.S.: On approximation of continuous linear functionals in normed linear spaces. Analese Universităţii din Timişoara Seria Ştiinţe Matematice-Fizice 29, 51–58 (1991)
Dragomir, S.S.: Semi-Inner Products and Applications. Nova Science Publishers, New York (2004)
Giles, J.R.: Classes of semi-inner-product spaces. Trans. Am. Math. Soc. 129, 436–446 (1967)
Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)
James, R.C.: Orthogonality and linear functionals in normed linear spaces. Trans. Am. Math. Soc. 61, 265–292 (1947)
James, R.C.: Inner products in normed linear spaces. Bull. Am. Math. Soc. 53, 559–566 (1947)
Karamanlis, M., Psarrakos, P.J.: Birkhoff-James epsilon-orthogonality sets in normed linear spaces. Textos de Matematica, University of Coimbra 44, 81–92 (2013)
Lumer, G.: Semi-inner-product spaces. Trans. Am. Math. Soc. 100, 29–43 (1961)
Megginson, R.E.: An Introduction to Banach Space Theory, Graduate Texts in Mathematics, vol. 183. Springer, New York (1998)
Mojškerc, B., Turnšek, A.: Mappings approximately preserving orthogonality in normed spaces. Nonlinear Anal. 73, 3821–3831 (2010)
Ohira, K.: On some characterizations of abstract Euclidean spaces by properties of orthogonality. Kumamoto J. Sci. Ser. A 1, 23–26 (1952)
Panagakou, V., Psarrakos, P., Yannakakis, N.: Birkhoff-James epsilon-orthogonality sets of vectors and vector-valued polynomials. J. Math. Anal. Appl. 454, 59–78 (2017)
Sain, D., Paul, K., Mal, A.: A complete characterization of Birkhoff-James orthogonality in infinite dimensional normed space. J. Oper. Theory 80, 399–413 (2018)
Stampfli, J.G., Williams, J.P.: Growth conditions and the numerical range in a Banach algebra. T\({\hat{o}}\)hoku Math. J. 20, 417–424 (1968)
Szostok, T.: On a generalization of the sine function. Glasnik Matematički 38(58), 29–44 (2003)
Wilson, W.A.: A relation between metric and Euclidean spaces. Am. J. Math. 54(3), 505–517 (1932)
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Panagakou, V., Psarrakos, P. & Yannakakis, N. A Birkhoff-James cosine function for normed linear spaces. Aequat. Math. 95, 889–914 (2021). https://doi.org/10.1007/s00010-021-00791-0
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DOI: https://doi.org/10.1007/s00010-021-00791-0
Keywords
- Norm
- Birkhoff-James orthogonality
- Birkhoff-James \(\varepsilon \)-orthogonality
- Linear functional
- Cosine
- Sine