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Locating the Range of an Operator on a Hilbert Space
Author(s) -
Bridges Douglas,
Ishihara Hajime
Publication year - 1992
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/24.6.599
Subject(s) - the arts , range (aeronautics) , mathematical sciences , library science , mathematics , citation , hilbert space , operator (biology) , computer science , mathematics education , visual arts , engineering , art , pure mathematics , biochemistry , chemistry , repressor , transcription factor , gene , aerospace engineering
exists (is computable) for each x in H; if P is that projection, / is the identity operator on H, and the adjoint T*ofT exists, then / P is the projection of H on ker(3*), the kernel of T*. (For an example to show that the existence of the adjoint of a bounded operator on a Hilbert space is not automatic in constructive mathematics, see Brouwerian Example 3 in [7]; see also Example 2 below.) This much is neither surprising nor particularly interesting. Of more interest is the observation that, constructively, the existence of the projection on ker(T*) is no guarantee of the existence of the projection on the closure of ran(7), unless we are prepared to accept within our constructive mathematics Markov's principle: