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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A characterization of the range of a bounded linear transformation in Hilbert space


Author: George O. Golightly
Journal: Proc. Amer. Math. Soc. 79 (1980), 591-592
MSC: Primary 47A05
MathSciNet review: 572309
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Abstract: It is a theorem of Smul'jan and Mac Nerney that for B a bounded linear transformation from a complete complex inner product space $ \{S,(\cdot , \cdot )\} $ to S, with adjoint transformation $ {B^ \ast },B(S)$ is the set of all z in S for which there is a nonnegative number b such that for all x in $ S,\vert(z,x){\vert^2} \leqslant b\left\Vert{B^ \ast }x\right\Vert{^2}$, in which case if w is that point of $ {(\ker B)^ \bot }$ such that $ Bw = z$ then the least such b is $ \left\Vert w\right\Vert{^2}$. This paper provides another description of $ B(S)$ and formula for $ \left\Vert w\right\Vert{^2}$.


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DOI: https://doi.org/10.1090/S0002-9939-1980-0572309-X
Keywords: Complete inner product space, bounded linear transformation, nonnegative operator, spectral resolution
Article copyright: © Copyright 1980 American Mathematical Society