We consider Dirichlet spaces with superharmonic weights. This class contains both the harmonic weights and the power weights. Our main result is a characterization of the Dirichlet spaces with superharmonic weights that can be identified as de Branges-Rovnyak spaces. As an application, we obtain the dilation inequalityCommunicated by Isabelle Chalendar. B T. Ransford O. El-Fallah et al.where D ω denotes the Dirichlet integral with superharmonic weight ω, and f r (z) := f (r z) is the r -dilation of the holomorphic function f .

Let T be an operator on a Hilbert space H with numerical radius w(T ) ≤ 1. According to a theorem of Berger and Stampfli, if f is a function in the disk algebra such that f (0) = 0, then w(f (T )) ≤ f ∞ . We give a new and elementary proof of this result using finite Blaschke products.A well-known result relating numerical radius and norm says T ≤ 2w(T ). We obtain a local improvement of this estimate, namely, if w(T ) ≤ 1 thenUsing this refinement, we give a simplified proof of Drury's teardrop theorem, which extends the Berger-Stampfli theorem to the case f (0) = 0.

In this paper, a sufficient condition for the existence of hyperinvariant subspace of compact perturbations of multiplication operators on some Banach spaces is presented. An interpretation of this result for compact perturbations of normal and diagonal operators on Hilbert space is also discussed. An improvement of a result of [FX12] for compact perturbations of diagonal operators is also obtained. Keywords: invariant subspace problem; hyperinvariant subspace problem; compact perturbations of normal operators; compact perturbations of diagonal operators. MSC 2010 : 47A15, 47A10, 47B15, 47B38.

The aim of this paper is to describe the closure of the numerical range of the product of two orthogonal projections in Hilbert space as a closed convex hull of some explicit ellipses parametrized by points in the spectrum. Several improvements (removing the closure of the numerical range of the operator, using a parametrization after its eigenvalues) are possible under additional assumptions. An estimate of the least angular opening of a sector with vertex 1 containing the numerical range of a product of two orthogonal projections onto two subspaces is given in terms of the cosine of the Friedrichs angle. Applications to the rate of convergence in the method of alternating projections and to the uncertainty principle in harmonic analysis are also discussed.Previous results. Orthogonal projections in Hilbert space are basic objects of study in Operator theory. Products or sums of orthogonal projections, in finite or infinite dimensional Hilbert spaces, appear in various problems and in many different areas, pure or applied. We refer the reader to a book [Gal04] and two recent surveys [Gal08, BS10] for more information. The fact that the numerical range of a finite product of orthogonal projections is included in some sector of the complex plane with vertex at 1 was an essential ingredient in the proof by Delyon and Delyon [DD99] of a conjecture of Burkholder, saying that the iterates of a product of conditional expectations are almost surely convergent to some conditional expectation in an L 2 space (see also [Cro08,Coh07]). For a product of two orthogonal projections we know that the numerical range is included in a sector with vertex one and angle π/6 ([Cro08]).The spectrum of a product of two orthogonal projections appears naturally in the study of the rate of convergence in the strong operator topology of (P M2 P M1 ) n to P M1∩M2 (cf. [Deu01, BDH09, DH10a, DH10b, BGM, BGM10, BL10]). This is a particular instance of von Neumann-Halperin type theorems, sometimes called in the literature the method of alternating projections. The following dichotomy holds (see [BDH09]): either the sequence (P M2 P M1 ) n converge uniformly with an exponential speed to P M1∩M2 (if 1 / ∈ σ(P M2 P M1 )), or the sequence of alternating projections (P M2 P M1 ) n converges arbitrarily slowly in the strong operator topology (if 1 ∈ σ(P M2 P M1 )). We refer to [BGM, BGM10] for several possible meanings of "slow convergence".An occurrence of the numerical range of operators related to sums of orthogonal projections appears also in some Harmonic analysis problems. The uncertainty principle in Fourier analysis is the informal assertion that a function f ∈ L 2 (R) and its Fourier transform F (f ) cannot be too small simultaneously. Annihilating pairs and strong annihilating pairs are a way to formulate this idea (precise definitions will be given in Section 5). Characterizations of annihilating pairs and strong annihilating pairs (S, Σ) in terms of the numerical range of the operator P S +iP Σ , constructed using some associated orthogona...

In this paper, we prove that every diagonal operator on a Hilbert space of which is of multiplicity one and has perfect spectrum admits a rank one perturbation without eigenvalues. This answers a question of Ionascu.Mathematics Subject Classification. 47A10, 47B06, 47A75, 47B15.