Biseparating linear maps between continuous vector-valued function spaces

Haw-Long Gau Jyh-Shyang Jeang Ngai-Ching Wong

Spectral Theory and Operator Algebra mathscidoc:1910.43633

Journal of the Australian Mathematical Society, 74, (1), 101-110, 2003.2
Let <i>X, Y</i> be compact Hausdorff spaces and <i>E, F</i> be Banach spaces. A linear map <i>T: C(X, E) C(Y, F)</i> is separating if <i>Tf, Tg</i> have disjoint cozeroes whenever <i>f, g</i> have disjoint cozeroes. We prove that a biseparating linear bijection <i>T</i> (that is, <i>T</i> and <i>T</i><sup>-1</sup> are separating) is a weighted composition operator <i>Tf = h f</i> o . Here, <i>h</i> is a function from <i>Y</i> into the set of invertible linear operators from <i>E</i> onto <i>F</i>, and , is a homeomorphism from <i>Y</i> onto <i>X</i>. We also show that <i>T</i> is bounded if and only if <i>h(y)</i> is a bounded operator from <i>E</i> onto <i>F</i> for all <i>y</i> in <i>Y</i>. In this case, <i>h</i> is continuous with respect to the strong operator topology.
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@inproceedings{haw-long2003biseparating,
  title={Biseparating linear maps between continuous vector-valued function spaces},
  author={Haw-Long Gau, Jyh-Shyang Jeang, and Ngai-Ching Wong},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020205008940429162},
  booktitle={Journal of the Australian Mathematical Society},
  volume={74},
  number={1},
  pages={101-110},
  year={2003},
}
Haw-Long Gau, Jyh-Shyang Jeang, and Ngai-Ching Wong. Biseparating linear maps between continuous vector-valued function spaces. 2003. Vol. 74. In Journal of the Australian Mathematical Society. pp.101-110. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020205008940429162.
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