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In this paper, we show that for \(T\in B({\mathcal {H}})\), 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.

A note about maximal almost-invariant subspaces and maximal hyperinvariant subspaces