Subspace maps of operators on Hilbert space

Longstaff, W. E.

doi:10.1090/S0002-9939-1982-0637168-7

Subspace maps of operators on Hilbert space
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by W. E. Longstaff

Proc. Amer. Math. Soc. 84 (1982), 195-201

DOI: https://doi.org/10.1090/S0002-9939-1982-0637168-7

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Abstract:

An operator $A$ acting on a Hilbert space $H$ gives rise to a map ${\varphi _A}$ on the set of subspaces of $H$ given by ${\varphi _A}(M) = \overline {AM}$, where ${\text {’}} - {\text {’}}$ denotes norm closure. This map is called the subspace map of $A$. By identifying subspaces with projections in the usual way it is shown that for $A \ne 0$, ${\varphi _A}$ is uniformly (respectively, strongly) continuous if and only if the approximate point spectrum of $A$ does not contain 0. In this case it is proved that ${\varphi _A}$ preserves the property of being uniformly (respectively, strongly, weakly) closed and its effect on reflexivity is described.

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