Abstract
This paper introduces a new family of non-negative real-valued functions on the algebra of all bounded linear operators on a Hilbert space. These functions will be called semi-norms because they satisfy the norm axioms, except for the triangle inequality, which will be discussed for special cases. Many bounds and relations will be shown for this new family, with a connection to the existing literature.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Availability of data and materials
Not applicable.
References
Abu-Omar, A., Kittaneh, F.: A numerical radius inequality involving the generalized Aluthge transform. Stud. Math. 216, 69–75 (2013)
Bhatia, R.: The Riemannian mean of positive matrices. In: Nielsen, F., Bhatia, R. (eds.) Matrix Information Geometry. Springer, Berlin (2013). https://doi.org/10.1007/978-3-642-30232-9_2
Dragomir, S.S., Moslehian, M.S.: Some inequalities for \((\alpha , \beta )\)-normal operators in Hilbert spaces. Facta Univ. Math. Inform. 23, 39–47 (2008)
Furuichi, S., Moradi, H.R., Sababheh, M.: Some inequalities for interpolational operator means. J. Math. Inequal. 15(1), 107–116 (2021)
Gustafson, K.E., Rao, D.K.M.: Numerical Range: The Field of Values of Linear Operators and Matrices. Springer, New York (1997)
Heydarbeygi, Z., Sababheh, M., Moradi, H.R.: A convex treatment of numerical radius inequalities. Czechoslov. Math. J. 72(147), 601–614 (2022)
Holbrook, J., Bhatia, R.: Noncommutative geometric means. Math. Intell. 28, 32–39 (2006). https://doi.org/10.1007/BF02987000
Kato, T.: Notes on some inequalities for linear operators. Math. Ann. 125, 208–212 (1952)
Kittaneh, F.: Numerical radius inequalities for Hilbert space operators. Stud. Math. 168(1), 73–80 (2005)
Kittaneh, F.: Notes on some inequalities for Hilbert space operators. Publ. RIMS Kyoto Univ. 24, 283–293 (1988)
Kubo, F., Ando, T.: Means of positive linear operators. Math. Ann. 246, 205–224 (1980)
Moradi, H.R., Sababheh, M.: New estimates for the numerical radius. Filomat 35(14), 4957–4962 (2021)
Pečarić, J., Furuta, T., Mićić Hot, J., Seo, Y.: Mond-Pečarić Method in Operator Inequalities. Element, Zagreb (2005)
Pusz, W., Woronowicz, S.L.: Functional calculus for sesquilinear forms and the purification map. Rep. Math. Phys. 8, 159–170 (1975)
Yamazaki, T.: On upper and lower bounds of the numerical radius and an equality condition. Stud. Math. 178, 83–89 (2007)
Funding
Not available.
Author information
Authors and Affiliations
Contributions
Authors declare that they have contributed equally to this paper. All authors have read and approved this version.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Conde, C., Moradi, H.R. & Sababheh, M. A Family of Semi-norms Between the Numerical Radius and the Operator Norm. Results Math 79, 36 (2024). https://doi.org/10.1007/s00025-023-02059-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00025-023-02059-2