We study increasing sequences of positive integers (n k ) k 1 with the following property: every bounded linear operator T acting on a separable Banach (or Hilbert) space with sup k 1 T n k < ∞ has a countable set of unimodular eigenvalues. Whether this property holds or not depends on the distribution (modulo one) of sequences (n k α) k 1 , α ∈ R, or on the growth of n k+1 /n k . Counterexamples to some conjectures in linear dynamics are given. For instance, a Hilbert space operator which is frequently hypercyclic, chaotic, but not topologically mixing is constructed. The situation of C 0 -semigroups is also discussed.

A generalization of the cosine of the Friedrichs angle between two subspaces to a parameter associated to several closed subspaces of a Hilbert space is given. This parameter is used to analyze the rate of convergence in the von Neumann-Halperin method of cyclic alternating projections. General dichotomy theorems are proved, in the Hilbert or Banach space situation, providing conditions under which the alternative QUC/ASC (quick uniform convergence versus arbitrarily slow convergence) holds. Several meanings for ASC are proposed.

We characterize Jamison sequences, that is sequences (n k ) of positive integers with the following property: every bounded linear operator T acting on a separable Banach space with sup k T n k < +∞ has a countable set of peripheral eigenvalues. We also discuss partially power-bounded operators acting on Banach or Hilbert spaces having peripheral point spectra with large Hausdorff dimension. For a Lavrentiev domain Ω in the complex plane, we show the uniform minimality of some families of eigenvectors associated with peripheral eigenvalues of operators satisfying the Kreiss resolvent condition with respect to Ω. We introduce and study the notion of Ω-Jamison sequence, which is defined by replacing the partial power-boundedness condition sup k T n k < +∞ by sup k F Ω n k (T ) < +∞, where F Ω n is the nth Faber polynomial of Ω. A characterization of Ω-Jamison sequences is obtained for domains with sufficiently smooth boundary.

the introduction was changed and some remarks have been added. 26 pages ; to appear in Math. ZInternational audienceLet $\Omega$ be an open convex domain of the complex plane. We study constants K such that $\Omega$ is K-spectral or complete K-spectral for each continuous linear Hilbert space operator with numerical range included in $\Omega$. Several approaches are discussed

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W e prove a Korovkin-type theorem on approximation of bivariate functions in the space of B-continuous functions (introduced by K. Bogel in 1934). As consequences, some sequences of uniformly approximating pseudopolynomials are obtained.

Exploiting a construction of rigidity sequences for weakly mixing dynamical systems by Fayad and Thouvenot, we show that for every integers p1, . . . , pr there exists a continuous probability measure µ on the unit circle T such that inf k 1 ≥0,...,kr ≥0

Abstract. Given N ≥ 2 closed subspaces M1, . . . , MN of a Hilbert space X, let P k denote the orthogonal projection onto M k , 1 ≤ k ≤ N . It is known that the sequence (xn), defined recursively by x0 = x and xn+1 = PN · · · P1xn for n ≥ 0, converges in norm to PM x as n → ∞ for all x ∈ X, where PM denotes the orthogonal projection onto M = M1 ∩ . . . ∩ MN . Moreover, the rate of convergence is either exponentially fast for all x ∈ X or as slow as one likes for appropriately chosen initial vectors x ∈ X. We give a new estimate in terms of natural geometric quantities on the rate of convergence in the case when it is known to be exponentially fast. More importantly, we then show that even when the rate of convergence is arbitrarily slow there exists, for each real number α > 0, a dense subset Xα of X such that xn − PM x = o(n −α ) as n → ∞ for all x ∈ Xα. Furthermore, there exists another dense subset X∞ of X such that, if x ∈ X∞, then xn −PM x = o(n −α ) as n → ∞ for all α > 0. These latter results are obtained as consequences of general properties of Ritt operators. As a by-product, we also strengthen the unquantified convergence result by showing that PM x is in fact the limit of a series which converges unconditionally.