## Numerical radius distance-preserving maps on $\mathcal {B}(H)$

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- by Zhaofang Bai and Jinchuan Hou
- Proc. Amer. Math. Soc.
**132**(2004), 1453-1461 - DOI: https://doi.org/10.1090/S0002-9939-03-07190-9
- Published electronically: September 30, 2003
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## Abstract:

Let $H$ be a complex Hilbert space, $\mathcal {B}(H)$ be the algebra of all bounded linear operators on $H$, $\mathcal {H}(H)$ be the subset of all selfadjoint operators in $\mathcal {B}(H)$ and $\mathcal {V}=\mathcal {B}(H)$ or ${\mathcal H}(H)$. Denote by $w(A)$ the numerical radius of $A\in \mathcal {B}(H)$. We characterize surjective maps $\Phi :{\mathcal V}\rightarrow {\mathcal V}$ that satisfy $w(\Phi (A)-\Phi (B))=w(A-B)$ for all $A,B\in {\mathcal V}$ without the linearity assumption.## References

- Jor-Ting Chan,
*Numerical radius preserving operators on $B(H)$*, Proc. Amer. Math. Soc.**123**(1995), no. 5, 1437–1439. MR**1231293**, DOI 10.1090/S0002-9939-1995-1231293-7 - P. A. Fillmore,
*Sums of operators with square zero*, Acta Sci. Math. (Szeged)**28**(1967), 285–288. MR**221301** - Paul Richard Halmos,
*A Hilbert space problem book*, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 17, Springer-Verlag, New York-Berlin, 1982. MR**675952** - P. Erdös and T. Grünwald,
*On polynomials with only real roots*, Ann. of Math. (2)**40**(1939), 537–548. MR**7**, DOI 10.2307/1968938 - P. Erdös and T. Grünwald,
*On polynomials with only real roots*, Ann. of Math. (2)**40**(1939), 537–548. MR**7**, DOI 10.2307/1968938 - A. R. Collar,
*On the reciprocation of certain matrices*, Proc. Roy. Soc. Edinburgh**59**(1939), 195–206. MR**8** - A. R. Collar,
*On the reciprocation of certain matrices*, Proc. Roy. Soc. Edinburgh**59**(1939), 195–206. MR**8** - Garrett Birkhoff and Morgan Ward,
*A characterization of Boolean algebras*, Ann. of Math. (2)**40**(1939), 609–610. MR**9**, DOI 10.2307/1968945 - Garrett Birkhoff and Morgan Ward,
*A characterization of Boolean algebras*, Ann. of Math. (2)**40**(1939), 609–610. MR**9**, DOI 10.2307/1968945 - Morgan Ward,
*Ring homomorphisms which are also lattice homomorphisms*, Amer. J. Math.**61**(1939), 783–787. MR**10**, DOI 10.2307/2371336 - Saunders MacLane,
*Steinitz field towers for modular fields*, Trans. Amer. Math. Soc.**46**(1939), 23–45. MR**17**, DOI 10.1090/S0002-9947-1939-0000017-3 - Richard V. Kadison and John R. Ringrose,
*Fundamentals of the theory of operator algebras. Vol. I*, Graduate Studies in Mathematics, vol. 15, American Mathematical Society, Providence, RI, 1997. Elementary theory; Reprint of the 1983 original. MR**1468229**, DOI 10.1090/gsm/015 - C. K. Li, L. Rodman, and P. Šemrl, Linear maps on selfadjoint operators preserving invertibility, positive definiteness, numerical range, Canad. Math. Bull. 46 (2003), 216-228.
- Chi-Kwong Li and Peter emrl,
*Numerical radius isometries*, Linear Multilinear Algebra**50**(2002), no. 4, 307–314. MR**1936954**, DOI 10.1080/03081080290025480 - S. Mazur and S. Ulam, Sur les transformations isométriques d$^{\prime }$espaces vectoriels normés, C. R. Acad. Sci. Paris 194 (1932), 946-948.
- Matjaž Omladič,
*On operators preserving the numerical range*, Linear Algebra Appl.**134**(1990), 31–51. MR**1060008**, DOI 10.1016/0024-3795(90)90004-V - Zhe-Xian Wan,
*Geometry of matrices*, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. In memory of Professor L. K. Hua (1910–1985). MR**1409610**, DOI 10.1142/9789812830234

## Bibliographic Information

**Zhaofang Bai**- Affiliation: School of Science, Xi$^{\prime }$an Jiaotong University, Xi$^{\prime }$an, 710049, P. R. China; Department of Mathematics, Shanxi Teachers University, Linfen, 041004, P. R. China
**Jinchuan Hou**- Affiliation: Department of Mathematics, Shanxi Teachers University, Linfen, 041004, P. R. China; Department of Mathematics, Shanxi University, Taiyuan, 030000, P. R. China
- Email: jhou@dns.sxtu.edu.cn
- Received by editor(s): September 27, 2002
- Received by editor(s) in revised form: January 11, 2003
- Published electronically: September 30, 2003
- Additional Notes: This work was supported partially by NNSFC and PNSFS
- Communicated by: Joseph A. Ball
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**132**(2004), 1453-1461 - MSC (2000): Primary 47H20, 47B49; Secondary 47A12
- DOI: https://doi.org/10.1090/S0002-9939-03-07190-9
- MathSciNet review: 2053353