The study of quasianalytic contractions, motivated by the hyperinvariant subspace problem, is continued. Special emphasis is put on the case when the contraction is asymptotically cyclic. New properties of the functional commutant are explored. Analytic contractions and bilateral weighted shifts are discussed as illuminating examples.

We pose, and answer partially, questions in connection with the spectral behaviour of quasianalytic contractions. These problems are related to the hyperinvariant subspace problem in the class of asymptotically nonvanishing contractions. T is not a scalar multiple of the identity operator I. These problems are arguably the most challenging open questions in operator theory. They can be reduced to the case when T is an absolutely continuous (a.c.) contraction, that is T ≤ 1 and T splits into the orthogonal sum T = U ⊕ T c , where U is a unitary operator whose spectral measure is a.c. with respect to the normalized Lebesgue measure m on the unit circle T, and T c is completely nonunitary, i.e. the restriction of T c to any non-zero invariant subspace is not unitary.We focus on the particular case when the a.c. contraction T is asymptotically non-vanishing (a.n.v.), that is when lim n→∞ T n h > 0 holds for some vector h ∈ H. Then a non-trivial unitary asymptote can be associated with T , which is a pair (X, V ), where V is an a.c. unitary operator acting on a non-zero Hilbert space K, X is a bounded linear transformation from H into K, Xh = lim n→∞ T n h holds for all h ∈ H, XT = XV and ∨ ∞ n=1 V −n XH = K. The pair (X, V ) is unique up to isomorphism; for details see Chapter IX in [NFBK], and [Ker13].Proposition 5. For every T 0 ∈ L 0 (H), there exists T 1 ∈ L 1 (H) such that {T 0 } = {T 1 } and so Hlat T 0 = Hlat T 1 .This proposition makes it especially important to answer the following question.Question 4. What are the possible spectra of the contractions belonging to L 1 (H)?We know that for every 0 ≤ δ < 1 there is a contraction T δ ∈ L 1 (H) such that σ(T δ ) = {z ∈ C : δ ≤ |z| ≤ 1}; see Example 5.8 in [Ker11] and Example 24 in [KSz]. Now we show that the spectrum can be the unit circle T too. The following theorem also gives a positive answer for Question 2 in the special case, when the arc J is the whole circle T.

We examine the convexity of the hitting distribution of the real axis for symmetric random walks on Z 2 . We prove that for a random walk starting at (0, h), the hitting distribution is convex on [h − 2, ∞) ∩ Z if h ≥ 2. We also show an analogous fact for higher-dimensional discrete random walks. This paper extends the results of a recent paper [NT].

No abstract

No abstract