The theory of almost invariant half-spaces for operators on Banach spaces was begun recently and is now under active development. Much less attention has been given to almost invariant half-spaces for operators on Hilbert space, where some techniques and results are available that are not present in the more general context of Banach spaces. In this note, we begin such a study. Our much simpler and shorter proofs of the main theorems have important consequences for the matricial structure of arbitrary operators on Hilbert space.
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