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Numerical Radius Parallelism of Hilbert Space Operators

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Abstract

In this paper, we study the numerical radius parallelism for bounded linear operators on a Hilbert space \(\big ({\mathscr {H}}, \langle \cdot ,\cdot \rangle \big )\). More precisely, we consider bounded linear operators T and S which satisfy \(\omega (T + \lambda S) = \omega (T)+\omega (S)\) for some complex unit \(\lambda \), and is denoted by \(T \parallel _{\omega } S\). We show that \(T \parallel _{\omega } S\) if and only if there exists a sequence of unit vectors \(\{x_n\}\) in \({\mathscr {H}}\) such that

$$\begin{aligned} \displaystyle {\lim _{n\rightarrow \infty }}\big |\langle Tx_n, x_n\rangle \langle Sx_n, x_n\rangle \big | = \omega (T)\omega (S). \end{aligned}$$

We then apply it to give some applications.

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References

  1. Bottazzi, T., Conde, C., Moslehian, M.S., Wójcik, P., Zamani, A.: Orthogonality and parallelism of operators on various Banach spaces. J. Aust. Math. Soc. 106(2), 160–183 (2019)

    Article  MathSciNet  Google Scholar 

  2. Dragomir, S.S.: A survey of some recent inequalities for the norm and numerical radius of operators in Hilbert spaces. Banach J. Math. Anal. 1(2), 154–175 (2007)

    Article  MathSciNet  Google Scholar 

  3. Grover, P.: Orthogonality of matrices in the Ky Fan \(k\)-norms. Linear Multilinear Algebra 65(3), 496–509 (2017)

    Article  MathSciNet  Google Scholar 

  4. Gustafson, K.E., Rao, D.K.M.: Numerical range. The field of values of linear operators and matrices. Universitext. Springer, New York (1997)

    Google Scholar 

  5. Halmos, P.R.: A Hilbert space problem book, 2nd edn. Springer, New York (1982)

    Book  Google Scholar 

  6. He, K., Hou, J.C., Zhang, X.L.: Maps preserving numerical radius or cross norms of products of self-adjoint operators. Acta Math. Sin. 26(6), 1071–1086 (2010). (English Series)

    Article  MathSciNet  Google Scholar 

  7. Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, New York (1991)

    Book  Google Scholar 

  8. Kittaneh, F., Moslehian, M.S., Yamazaki, T.: Cartesian decomposition and numerical radius inequalities. Linear Algebra Appl. 471, 46–53 (2015)

    Article  MathSciNet  Google Scholar 

  9. Mal, A., Sain, D., Paul, K.: On some geometric properties of operator spaces. Banach J. Math. Anal. 13(1), 174–191 (2019)

  10. Seddik, A.: Rank one operators and norm of elementary operators. Linear Algebra Appl. 424, 177–183 (2007)

    Article  MathSciNet  Google Scholar 

  11. Wójcik, P.: Norm-parallelism in classical \(M\)-ideals. Indag. Math. (N.S.) 28(2), 287–293 (2017)

    Article  MathSciNet  Google Scholar 

  12. Yamazaki, T.: On upper and lower bounds of the numerical radius and an equality condition. Stud. Math. 178(1), 83–89 (2007)

    Article  MathSciNet  Google Scholar 

  13. Zamani, A.: The operator-valued parallelism. Linear Algebra Appl. 505, 282–295 (2016)

    Article  MathSciNet  Google Scholar 

  14. Zamani, A., Moslehian, M.S.: Exact and approximate operator parallelism. Can. Math. Bull. 58(1), 207–224 (2015)

    Article  MathSciNet  Google Scholar 

  15. Zamani, A., Moslehian, M.S.: Norm-parallelism in the geometry of Hilbert \(C^*\)-modules. Indag. Math. (N.S.) 27(1), 266–281 (2016)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to thank the referees for their comments that helped us improve this article.

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Correspondence to Maryam Amyari.

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Communicated by Hamid Reza Ebrahimi Vishki.

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Mehrazin, M., Amyari, M. & Zamani, A. Numerical Radius Parallelism of Hilbert Space Operators. Bull. Iran. Math. Soc. 46, 821–829 (2020). https://doi.org/10.1007/s41980-019-00295-3

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