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ORTHOGONALITY AND PARALLELISM OF OPERATORS ON VARIOUS BANACH SPACES

  • T. BOTTAZZI (a1), C. CONDE (a2) (a3), M. S. MOSLEHIAN (a4), P. WÓJCIK (a5) and A. ZAMANI (a6)...

Abstract

We present some properties of orthogonality and relate them with support disjoint and norm inequalities in $p$ -Schatten ideals. In addition, we investigate the problem of characterization of norm-parallelism for bounded linear operators. We consider the characterization of the norm-parallelism problem in $p$ -Schatten ideals and locally uniformly convex spaces. Later on, we study the case when an operator is norm-parallel to the identity operator. Finally, we give some equivalence assertions about the norm-parallelism of compact operators. Some applications and generalizations are discussed for certain operators.

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The third author was supported by a grant from Ferdowsi University of Mashhad (no. 1/43523).

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[1] Abatzoglou, T. J., ‘Norm derivatives on spaces of operators’, Math. Ann. 239(2) (1979), 129135.
[2] Alonso, J., Martini, H. and Wu, S., ‘On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces’, Aequationes Math. 83 (2012), 153189.
[3] Arazy, J., ‘The isometries of C p ’, Israel J. Math. 22(3–4) (1975), 247256.
[4] Benítez, C., Fernández, M. and Soriano, M. L., ‘Orthogonality of matrices’, Linear Algebra Appl. 422 (2007), 155163.
[5] Bhatia, R. and Šemrl, P., ‘Orthogonality of matrices and some distance problems’, Linear Algebra Appl. 287(1–3) (1999), 7785.
[6] Bhattacharyya, T. and Grover, P., ‘Characterization of Birkhoff–James orthogonality’, J. Math. Anal. Appl. 407(2) (2013), 350358.
[7] Birkhoff, G., ‘Orthogonality in linear metric spaces’, Duke Math. J. 1 (1935), 169172.
[8] Busch, P., ‘Stochastic isometries in quantum mechanics’, Math. Phys. Anal. Geom. 2(1) (1999), 83106.
[9] Chmieliński, J., ‘Operators reversing orthogonality in normed spaces’, Adv. Oper. Theory 1(1) (2016), 814.
[10] Clarkson, J. A., ‘Uniformly convex spaces’, Trans. Amer. Math. Soc. 40(3) (1936), 396414.
[11] Dragomir, S. S., Semi-Inner Products and Applications (Nova Science, Hauppauge, NY, 2004).
[12] Ghosh, P., Sain, D. and Paul, K., ‘On symmetry of Birkhoff–James orthogonality of linear operators’, Adv. Oper. Theory 2(4) (2017), 428434.
[13] Giles, J. R., ‘Classes of semi-inner-product spaces’, Trans. Amer. Math. Soc. 129 (1967), 436446.
[14] Gohberg, I. C. and Kreĭn, M. G., Introduction to the Theory of Linear Nonselfadjoint Operators, Translations of Mathematical Monographs, 18 (American Mathematical Society, Providence, RI, 1969), translated from the Russian by A. Feinstein.
[15] Grover, P., ‘Orthogonality of matrices in the Ky Fan k-norms’, Linear Multilinear Algebra 65(3) (2017), 496509.
[16] James, R. C., ‘Orthogonality in normed linear spaces’, Duke Math. J. 12 (1945), 291301.
[17] James, R. C., ‘Orthogonality and linear functionals in normed linear spaces’, Trans. Amer. Math. Soc. 61 (1947), 265292.
[18] Kittaneh, F., ‘On zero-trace matrices’, Linear Algebra Appl. 151 (1991), 119124.
[19] Li, Y. and Li, Y.-E., ‘Some characterizations of the trace norm triangle equality’, Linear Algebra Appl. 484 (2015), 396408.
[20] Lumer, G., ‘Semi-inner-product spaces’, Trans. Amer. Math. Soc. 100 (1961), 2943.
[21] Magajna, B., ‘On the distance to finite-dimensional subspaces in operator algebras’, J. Lond. Math. Soc. (2) 47(3) (1993), 516532.
[22] Maher, P. J., ‘Some operator inequalities concerning generalized inverses’, Illinois J. Math. 34(3) (1990), 503514.
[23] Maher, P. J., ‘Some norm inequalities concerning generalized inverses’, Linear Algebra Appl. 174 (1992), 99110.
[24] Maligranda, L., ‘Some remarks on the triangle inequality for norms’, Banach J. Math. Anal. 2(2) (2008), 3141.
[25] McCarthy, C. A., ‘ c p ’, Israel J. Math. 5 (1967), 249271.
[26] Moslehian, M. S. and Zamani, A., ‘Characterizations of operator Birkhoff–James orthogonality’, Canad. Math. Bull. 60(4) (2017), 816829.
[27] Paul, K., Sain, D. and Ghosh, P., ‘Birkhoff–James orthogonality and smoothness of bounded linear operators’, Linear Algebra Appl. 506 (2016), 551563.
[28] Sain, D., ‘On the norm attainment set of a bounded linear operator’, J. Math. Anal. Appl. 457(1) (2018), 6776.
[29] Sain, D., Paul, K. and Hait, S., ‘Operator norm attainment and Birkhoff–James orthogonality’, Linear Algebra Appl. 476 (2015), 8597.
[30] Seddik, A., ‘Rank one operators and norm of elementary operators’, Linear Algebra Appl. 424 (2007), 177183.
[31] Stampfli, J. G., ‘The norm of a derivation’, Pacific J. Math. 33 (1970), 737747.
[32] Werner, D., ‘An elementary approach to the Daugavet equation’, in: Interaction Between Functional Analysis, Harmonic Analysis, and Probability (Columbia, MO, 1994), Lecture Notes in Pure and Applied Mathematics, 175 (Marcel Dekker, New York, 1996), 449454.
[33] Wójcik, P., ‘The Birkhoff orthogonality in pre-Hilbert C -modules’, Oper. Matrices 10(3) (2016), 713729.
[34] Wójcik, P., ‘Orthogonality of compact operators’, Expo. Math. 35(1) (2017), 8694.
[35] Wójcik, P., ‘Norm-parallelism in classical M-ideals’, Indag. Math. (N.S.) 28(2) (2017), 287293.
[36] Zamani, A., ‘The operator-valued parallelism’, Linear Algebra Appl. 505 (2016), 282295.
[37] Zamani, A. and Moslehian, M. S., ‘Exact and approximate operator parallelism’, Canad. Math. Bull. 58(1) (2015), 207224.
[38] Zamani, A. and Moslehian, M. S., ‘Norm-parallelism in the geometry of Hilbert C -modules’, Indag. Math. (N.S.) 27(1) (2016), 266281.
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ORTHOGONALITY AND PARALLELISM OF OPERATORS ON VARIOUS BANACH SPACES

  • T. BOTTAZZI (a1), C. CONDE (a2) (a3), M. S. MOSLEHIAN (a4), P. WÓJCIK (a5) and A. ZAMANI (a6)...

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