Spain, P. G.
(1977)
Support projections on Banach spaces.
*Glasgow Mathematical Journal*, 18(1),
pp. 13-15.
(doi:10.1017/S0017089500002974)

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## Abstract

Each bounded linear operator a on a Hilbert space K has a hermitian left-support projection p such that pK=aK = aa*K and (1— p)K = ker a* = keraa*. I demonstrate here that certain operators on Banach spaces also have left supports. Throughout this paper X will be a complex Banach space with norm-dual X', and L(X) will be the Banach algebra of bounded linear operators on X. Two linear subspaces Y and Z of X are orthogonal (in the sense of G. Birkhoff) if || j>|| g \\y+z \\(ye Y, zeZ); this orthogonality relation is not, in general, symmetric. It is easy to see that pX is orthogonal to (1 — p)X if and only if the norm of p is 0 or 1, when p is a projection on X. An element h of a complex unital Banach algebra A is hermitian if | exp(ith) || = l(ieR); equivalently, ft is hermitian if its numerical range, {f(h) :feA',f(l) = ||/|| = 1}, is real.

Item Type: | Articles |
---|---|

Status: | Published |

Refereed: | Yes |

Glasgow Author(s) Enlighten ID: | Spain, Dr Philip |

Authors: | Spain, P. G. |

College/School: | College of Science and Engineering > School of Mathematics and Statistics > Mathematics |

Journal Name: | Glasgow Mathematical Journal |

Publisher: | Cambridge University Press |

ISSN: | 0017-0895 |

ISSN (Online): | 1469-509X |

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