The hyperinvariant subspace problem on Hilbert space

Theorem 1.1 and Corollary 1.2: on $L^2(\mathbb Z_2;\ell^2(\mathbb Z))$ a fibered bilateral weighted shift $S$ with quadratically decaying weights over the 2-adic odometer is nonzero with $\|S^n\|\le\exp(-10\lfloor(n+1)^2/4\rfloor)$, hence norm-quasinilpotent, and its commutant (containing a backward intertwiner $V$ and all scalar multipliers) is transitive. So every infinite-dimensional separable Hilbert space carries a nonzero quasinilpotent operator with no nonzero proper closed hyperinvariant subspace. The operator has many ordinary invariant subspaces (the fiberwise tails), so the invariant subspace problem itself is untouched. The companion reaches the same existence statement by a different construction.

Theorem 1.1 and Corollary 1.2: on $L^2(\mathbb Z_2;\ell^2(\mathbb Z))$ a fibered bilateral weighted shift $S$ with quadratically decaying weights over the 2-adic odometer is nonzero with $\|S^n\|\le\exp(-10\lfloor(n+1)^2/4\rfloor)$, hence norm-quasinilpotent, and its commutant (containing a backward intertwiner $V$ and all scalar multipliers) is transitive. So every infinite-dimensional separable Hilbert space carries a nonzero quasinilpotent operator with no nonzero proper closed hyperinvariant subspace. The operator has many ordinary invariant subspaces (the fiberwise tails), so the invariant subspace problem itself is untouched. The companion reaches the same existence statement by a different construction.