Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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MAPS PRESERVING m- ISOMETRIES ON HILBERT SPACE

  • Received : 2019.06.17
  • Accepted : 2019.08.01
  • Published : 2019.09.30
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Abstract

Let ${\mathcal{H}}$ be a complex Hilbert space and ${\mathcal{B}}({\mathcal{H}})$ the algebra of all bounded linear operators on ${\mathcal{H}}$. In this paper, we prove that if ${\varphi}:{\mathcal{B}}({\mathcal{H}}){\rightarrow}{\mathcal{B}}({\mathcal{H}})$ is a unital surjective bounded linear map, which preserves m- isometries m = 1, 2 in both directions, then there are unitary operators $U,V{\in}{\mathcal{B}}({\mathcal{H}})$ such that ${\varphi}(T)=UTV$ or ${\varphi}(T)=UT^{tr}V$ for all $T{\in}{\mathcal{B}}({\mathcal{H}})$, where $T^{tr}$ is the transpose of T with respect to an arbitrary but fixed orthonormal basis of ${\mathcal{H}}$.

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References

  1. F. Bayart, m-isometries on Banach spaces, Math. Nachr. 284 (17-18) (2011), 2141-2147. https://doi.org/10.1002/mana.200910029
  2. A. Chahbi and S. Kabbaj, Linear maps preserving G-unitary operators in Hilbert space, Arab J. Math. Sci. 21 (1) (2015), 109-117. https://doi.org/10.1016/j.ajmsc.2014.03.002
  3. I. N. Herstein, Topics in ring theory, University of Chicago Press, Chicago, 1969.
  4. A. Majidi and M. Amyari, Maps preserving quasi- isometries on Hilbert $C^*$-modules, Rocky Mountain J. Math. 48 (4) (2018).
  5. A. Majidi and M. Amyari, On maps that Preserve *-product of operators in $\mathcal{B}(\mathcal{H})$, Tamsui Oxford J. Math. Sci. 33 (1) (2019).
  6. L. Molnar, Selected preserver problems on algebraic structures of linear operators and on function spaces, Springer, 1895.
  7. G. J. Murphy, $C^*$-algebras and operator theory, Academic Press Inc, London, 1990.
  8. M. Rais, The unitary group preserving maps (the in finite-dimensional case), Linear Multilinear Algebra, 20 (1987), 337-345. https://doi.org/10.1080/03081088708817766
  9. Y. N. Wei and G. X. Ji, Maps preserving partial isometries of operator pencils, (Chinese) Acta Math. Sci. Ser. A Chin. Ed. 36 (3) (2016), 413-424.