Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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}$.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47A05

Retrieve articles in all journals with MSC: 47A05

Additional Information

Keywords: Complete inner product space, bounded linear transformation, nonnegative operator, spectral resolution
Article copyright: © Copyright 1980 American Mathematical Society

American Mathematical Society