Results on weighted shifts, fields of operators and the transitive commutant, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• The commutant of one operator — Lemma 1.1. {S}' is a weakly and strongly closed unital algebra, different from B(H) unless S is a scalar; it is transitive exactly when S has no nontrivial hyperinvariant subspace.
• Hyperinvariant projections — Lemma 1.2. The projection onto a hyperinvariant subspace of T lies in every von Neumann algebra containing T.
• Weighted shifts and weighted sequence spaces — Lemma 2.1. A bilateral weighted shift with weights in (0,1] is unitarily equivalent to right translation on l^2(w) for a positive decreasing w, tails to tails; bounded functionals and annihilators of l^2(w).
• Greatest convex minorant — Lemma 3.1. For f_0 = 0 and f_n in [0, infinity], finite infinitely often and above a convex G with G_n/n -> infinity: the greatest convex minorant is finite, its differences tend to infinity, and it equals f at infinitely many contact indices.
• The root condition — Lemma 4.1. If log w is convex on j <= 0 and concave on j >= 0 and sum over j != 0 of w_j^(1/j) is finite, the ratios w_(j+1)/w_j are summable.
• Support separation (Domar) — Proposition 4.2. w positive and decreasing, log w convex on j <= 0 and concave on j >= 0, the root condition, and limsup log w_(-n)/n^2 > log 3 or liminf log w_n/n^2 < -log 3; nonzero b, c with (c_j w_j) and (b_j/w_j) bounded and sum_j b_j c_(j-r) = 0 for r >= 1: max supp b <= min supp c.
• Invariant subspaces of weighted translations (Domar, l^2 case) — Theorem 5.1. The same hypotheses on w: the closed invariant subspaces of right translation on l^2(w) are exactly the tails.
• Weighted shifts whose invariant subspaces are tails — Corollary 5.2. Logarithmic weights a_j nondecreasing for j >= 0 and for j <= -1 away from the centre, with a_j >= kappa(j+1) for j >= 0 and a_j >= kappa|j| for j <= -1, kappa > 2 log 3.
• The odometer — Lemma 1.1. Binary digit sequences with the fair-coin product measure and x -> x + 1: residue classes generate the sigma-algebra; the odometer is bimeasurable, measure preserving and ergodic; L^2 is separable and infinite-dimensional.
• Decomposable operators — Lemma 2.1. An essentially bounded measurable field of operators on l^2(Z) defines an operator on L^2(X x Z) commuting with the scalar multipliers; products and adjoints act fibrewise.
• Fields from scalar multipliers — Theorem 2.2. Every operator on L^2(X x Z) that commutes with all scalar multipliers is given by an almost everywhere unique measurable field of operators, bounded by its norm; a projection has projection fibres.
• The fibres — Proposition 3.1. With c = 10 and D = 100, every fibre shift S_x has norm e^(-10) and exactly the tails as closed invariant subspaces.
• The power estimate — Proposition 4.1. ||S^n|| <= exp(-10 floor((n+1)^2/4)) for n >= 1; S is nonzero, quasinilpotent and not a scalar.
• Two-ended drift — Lemma 1.2. For x not eventually constant, under x -> x + 1 the logarithmic weights on [0, n) gain at least Dn - C(x) and those on [-q, 0) lose at least Dq - C(x).
• Decay of the intertwining coefficients — Lemma 2.1. The solution of the commutation recursion decays at both ends of the integer line, because D > c.
• The backward intertwiner — Proposition 2.2. A contraction V commuting with S that carries the fibre over x to the fibre over x + 1 and each tail K_(>=h) onto a dense subspace of K_(>=h-1).
• Strict descent — Lemma 3.1. A measurable label with values in Z or +-infinity that strictly decreases along an ergodic invertible measure-preserving map, -infinity being absorbing, is almost everywhere -infinity or almost everywhere +infinity.
• Common invariant subspaces — Proposition 4.1. S, V and the scalar multipliers have no common closed invariant subspace except 0 and H_0.
• The direct model (OpenAI) — Theorem 5.1. Fibre classification, ||S^n|| <= exp(-10 floor((n+1)^2/4)), the backward intertwiner, and transitivity of the unital algebra generated by S, V and the scalar multipliers.
• The hyperinvariant subspace problem (OpenAI) — Corollary 5.2. Every infinite-dimensional separable complex Hilbert space carries a nonzero quasinilpotent operator with no nontrivial hyperinvariant subspace; its commutant is a proper weakly closed unital transitive algebra.
• Invariant subspaces of S and the multipliers — Proposition 6.1. They are the spaces of fields with values in K_(>=sigma(x)) for measurable sigma; only the trivial ones are V-invariant.
• Polar conjugates — Proposition 6.2. Finitely many polar conjugates of |S| generate an atomic abelian von Neumann algebra; all of them generate L^infinity(X x Z), which has no minimal projection.

Reading order

Read the lessons in order. The first sets up commutants and hyperinvariant subspaces and proves Domar's theorem on weighted shifts whose invariant subspaces are tails. The second constructs the base space, the 2-adic integers with the odometer, the field of weighted shifts and the power estimate, and proves that operators commuting with the scalar multipliers are fields of operators. The third constructs the backward intertwiner and proves that the commutant is transitive.