Banach Journal of Mathematical Analysis

On some geometric properties of operator spaces

Arpita Mal, Debmalya Sain, and Kallol Paul

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In this article, we study some geometric properties like parallelism, orthogonality, and semirotundity in the space of bounded linear operators. We completely characterize parallelism of two compact linear operators between normed linear spaces X and Y, assuming X to be reflexive. We also characterize parallelism of two bounded linear operators between normed linear spaces X and Y. We investigate parallelism and approximate parallelism in the space of bounded linear operators defined on a Hilbert space. Using the characterization of operator parallelism, we study Birkhoff–James orthogonality in the space of compact linear operators as well as bounded linear operators. Finally, we introduce the concept of semirotund points (semirotund spaces) which generalizes the notion of exposed points (strictly convex spaces). We further study semirotund operators and prove that B(X,Y) is a semirotund space which is not strictly convex if X,Y are finite-dimensional Banach spaces and Y is strictly convex.

Article information

Banach J. Math. Anal., Volume 13, Number 1 (2019), 174-191.

Received: 15 February 2018
Accepted: 22 June 2018
First available in Project Euclid: 4 December 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46B20: Geometry and structure of normed linear spaces
Secondary: 47L05: Linear spaces of operators [See also 46A32 and 46B28]

norm-parallelism orthogonality semirotund norm attainment


Mal, Arpita; Sain, Debmalya; Paul, Kallol. On some geometric properties of operator spaces. Banach J. Math. Anal. 13 (2019), no. 1, 174--191. doi:10.1215/17358787-2018-0021.

Export citation


  • [1] L. Arambašić and R. Rajić, The Birkhoff–James orthogonality in Hilbert $C^{\ast}$-modules, Linear Algebra Appl. 437 (2012), no. 7, 1913–1929.
  • [2] C. Benítez, M. Fernández, and M. L. Soriano, Orthogonality of matrices, Linear Algebra Appl. 422 (2007), no. 1, 155–163.
  • [3] R. Bhatia and P. Šemrl, Orthogonality of matrices and distance problems, Linear Algebra Appl. 287 (1999), no 1–3, 77–85.
  • [4] T. Bhattacharyya and P. Grover, Characterization of Birkhoff–James orthogonality, J. Math. Anal. Appl. 407 (2013), no. 2, 350–358.
  • [5] G. Birkhoff, Orthogonality in linear metric spaces, Duke Math. J. 1 (1935), no. 2, 169–172.
  • [6] T. Bottazzi, C. Conde, M. S. Moslehian, P. Wójcik, and A. Zamani, Orthogonality and parallelism of operators on various Banach spaces, to appear in J. Aust. Math. Soc., preprint, arXiv:1609.06615v3 [math.FA].
  • [7] J. Chmieliński, Linear mappings approximately preserving orthogonality, J. Math. Anal. Appl. 304 (2005), no. 1, 158–169.
  • [8] J. B. Conway, A Course in Functional Analysis, 2nd ed., Grad. Texts in Math. 96, Springer, New York, 1990.
  • [9] S. S. Dragomir, On approximation of continuous linear functionals in normed linear spaces, An. Univ. Timişoara Ser. Ştiinţ. Mat. 29 (1991), no. 1, 51–58.
  • [10] P. Ghosh, D. Sain, and K. Paul, On symmetry of Birkhoff–James orthogonality of linear operators, Adv. Oper. Theory 2 (2017), no. 4, 428–434.
  • [11] R. C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc. 61 (1947), 265–292.
  • [12] C. K. Li and H. Schneider, Orthogonality of matrices, Linear Algebra Appl. 347 (2002), no. 1–3, 115–122.
  • [13] K. Paul, D. Sain, and P. Ghosh, Birkhoff–James orthogonality and smoothness of bounded linear operators, Linear Algebra Appl. 506 (2016), 551–563.
  • [14] K. Paul, D. Sain, and K. Jha, On strong orthogonality and strictly convex normed linear spaces, J. Inequal. Appl. 2013, no. 242.
  • [15] K. Paul, D. Sain, A. Mal, and K. Mandal, Orthogonality of bounded linear operators on complex Banach spaces, Adv. Oper. Theory 3 (2018), no. 3, 699–709.
  • [16] D. Sain and K. Paul, Operator norm attainment and inner product spaces, Linear Algebra Appl. 439 (2013), no. 8, 2448–2452.
  • [17] D. Sain, K. Paul, and S. Hait, Operator norm attainment and Birkhoff–James orthogonality, Linear Algebra Appl. 476 (2015), 85–97.
  • [18] A. Seddik, Rank one operators and norm of elementary operators, Linear Algebra Appl. 424 (2007), no. 1, 177–183.
  • [19] P. Wójcik, Orthogonality of compact operators, Expo. Math. 35 (2017), no. 1, 86–94.
  • [20] A. Zamani and M. S. Moslehian, Exact and approximate operator parallelism, Canad. Math. Bull. 58 (2015), no. 1, 207–224.
  • [21] A. Zamani and M. S. Moslehian, Norm-parallelism in the geometry of Hilbert $C^{*}$-modules, Indag. Math. (N.S.) 27 (2016), no. 1, 266–281.