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APPROXIMATELY ANGLE PRESERVING MAPPINGS

Part of: Special classes of linear operators Linear spaces and algebras of operators Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  14 February 2019

MOHAMMAD SAL MOSLEHIAN Open the ORCID record for MOHAMMAD SAL MOSLEHIAN [Opens in a new window] ,
ALI ZAMANI Open the ORCID record for ALI ZAMANI [Opens in a new window]  and
PAWEŁ WÓJCIK Open the ORCID record for PAWEŁ WÓJCIK [Opens in a new window]
MOHAMMAD SAL MOSLEHIAN*
Affiliation:
Department of Pure Mathematics, Ferdowsi University of Mashhad, PO Box 1159, Mashhad 91775, Iran email moslehian@um.ac.ir
ALI ZAMANI
Affiliation:
Department of Mathematics, Farhangian University, Tehran, Iran email zamani.ali85@yahoo.com
PAWEŁ WÓJCIK
Affiliation:
Institute of Mathematics, Pedagogical University of Cracow, Podchora̧żych 2, 30-084 Kraków, Poland email pawel.wojcik@up.krakow.pl
*
✉moslehian@um.ac.ir
Rights & Permissions [Opens in a new window]

Abstract

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We study linear mappings which preserve vectors at a specific angle. We introduce the concept of $(\unicode[STIX]{x1D700},c)$-angle preserving mappings and define $\widehat{\unicode[STIX]{x1D700}}\,(T,c)$ as the ‘smallest’ number $\unicode[STIX]{x1D700}$ for which $T$ is an $(\unicode[STIX]{x1D700},c)$-angle preserving mapping. We derive an exact formula for $\widehat{\unicode[STIX]{x1D700}}\,(T,c)$ in terms of the norm $\Vert T\Vert$ and the minimum modulus $[T]$ of $T$. Finally, we characterise approximately angle preserving mappings.

Keywords

orthogonality preserving mappingapproximately angle preserving mappingisometryapproximate similarity

MSC classification

Primary: 47B49: Transformers, preservers (operators on spaces of operators)
Secondary: 46C05: Hilbert and pre-Hilbert spaces 47L05: Linear spaces of operators 39B82: Stability, separation, extension, and related topics
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 99 , Issue 3 , June 2019 , pp. 485 - 496
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

Footnotes

The first author is partially supported by a grant from Ferdowsi University of Mashhad (No. 2/47884).

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