Abstract
In this paper, we show that for Tin B({mathcal {H}}), if {mathcal {M}} is almost-invariant for T, then every maximal almost-invariant subspace of {mathcal {M}} is of codimension 1 in {mathcal {M}}, where {mathcal {H}} is a separable, infinite-dimensional Hilbert space. We also describe the maximal hyperinvariant subspaces for normal operators with all the dimensions of eigenspaces at most 1 acting on {mathcal {H}}. Our result is that for each hyperinvariant subspace, all its maximal hyperinvariant subspaces are also of codimension 1 in it.
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