Support projections on Banach spaces

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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