On ideals in the bidual of the Fourier algebra and related algebras

Let G be a compact nonmetrizable topological group whose local weight b(G) has uncountable cofinality. Let H be an amenable locally compact group, A(G×H) the Fourier algebra of G×H, and UC₂(G×H) the space of uniformly continuous functionals in VN(G×H)=A^∗(G×H). We use weak factorization of operators in the group von Neumann algebra VN(G×H) to prove that there exist at least ²b(G)² left ideals of dimensions at least ²b(G)² in A(G×H)^∗∗ and in UC₂^∗(G×H). We show that every nontrivial right ideal in A(G×H)^∗∗ and in UC₂^∗(G×H) has dimension at least ²b(G)².     

Optimal lower bound estimates for the blow-up rate for the Zakharov system in a nonhomogeneous medium

In this paper we obtain lower bound estimates for the blow-up rate of finite time blow-up solutions to the Cauchy problem for the Zakharov system in a nonhomogeneous medium in two space dimensions. By introducing suitable scale transformations of space and time, and the use of compactness arguments, we derive an optimal lower bound estimate in the energy space H²(R²)×L²(R²)×H¹(R²) for the blow-up rate for t near the finite blow-up time T. Also we give an application to the virial identity for the Zakharov system under study.     

$H^2$-Stabilization of the Isothermal Euler Equations: a Lyapunov Function Approach

The authors consider the problem of boundary feedback stabilization of the 1D Euler gas dynamics locally around stationary states and prove the exponential stability with respect to the H2-norm.To this...     

Admissible majorants for model subspaces, and arguments of inner functions

Let Θ be an inner function in the upper half-plane ?⁺ and let K _Θ denote the model subspace H ² ? Θ H ² of the Hardy space H ² = H ²(?⁺). A nonnegative function w on the real line is said to be an admissible majorant for K _Θ if there exists a nonzero function f ∈ K _Θ such that {f} ? w a.e. on ?. We prove a refined version of the parametrization formula for K _Θ-admissible majorants and simplify the admissibility criterion (in terms of arg Θ) obtained in [8]. We show that, for every inner function Θ, there exist minimal K _Θ-admissible majorants. The relationship between admissibility and some weighted approximation problems is considered.     

Hypoellipticity on the Heisenberg group

Let P be a left-invariant differential operator on the Heisenberg group Hⁿ, P homogeneous with respect to the dilations on Hⁿ. We show that a necessary and sufficient condition for the hypoellipticity of P is that π(P) be an injective operator for every irreducible unitary representation π of Hⁿ (except the trivial representation). Furthermore, hypoellipticity is preserved if the homogeneous operator P is perturbed by terms of lower order of homogeneity. (Homogeneity means homogeneity with respect to dilations of Hⁿ.) It is also shown that if P is homogeneous, left-invariant and hypoelliptic on Hⁿ, then its formal adjoint is hypoelliptic.     

Unconditional uniqueness in the charge class for the Dirac–Klein–Gordon equations in two space dimensions

Recently, Grünrock and Pecher proved global well-posedness of the 2d Dirac–Klein–Gordon equations given initial data for the spinor and scalar fields in H ^s and H ^s+1/2 × H ^s-1/2, respectively, where s ≥ 0, but uniqueness was only known in a contraction space of Bourgain type, strictly smaller than the natural solution space C([0,T]; H ^s × H ^s+1/2 × H ^s-1/2). Here we prove uniqueness in the latter space for s ≥ 0. This improves a recent result of Pecher, where the range s > 1/30 was covered.     

A sufficiency class for global (in time) solutions to the 3D Navier-Stokes equations

Let Ω be an open domain of class C² contained in R³, let L²(Ω)³ be the Hilbert space of square integrable functions on Ω and let H[Ω]?H be the completion of the set, , with respect to the inner product of L²(Ω)³. A well-known unsolved problem is that of the construction of a sufficient class of functions in H which will allow global, in time, strong solutions to the three-dimensional Navier-Stokes equations. These equations describe the time evolution of the fluid velocity and pressure of an incompressible viscous homogeneous Newtonian fluid in terms of a given initial velocity and given external body forces. In this paper, we use the analytic nature of the Stokes semigroup to construct an equivalent norm for H, which provides strong bounds on the nonlinear term. This allows us to prove that, under appropriate conditions, there exists a number u₊, depending only on the domain, the viscosity, the body forces and the eigenvalues of the Stokes operator, such that, for all functions in a dense set D contained in the closed ball B(Ω)?B of radius in H, the Navier-Stokes equations have unique, strong, solutions in C¹((0,∞),H).     

Hardy space interpolation in the unit ball

Interpolating sequences for the spaces H^p(Bⁿ), 1 <-p < ∞, n≥2, are studied. In contrast to the case n = 1, they are not the same for all p, but Varopoulos' necessary condition for H^∞ interpolation turns out to be necessary for H^p interpolation, for any p≥ 1, and sufficient for the sequence to be a finite union of H¹ interpolating sequences. Sufficient conditions are given for H^p interpolation, npn/(n-1), which are intermediary between Varopoulos' condition and Berndtsson's sufficient condition for H^∞(Bⁿ) interpolation.     

Holomorphic Hermitian vector bundles over the Riemann sphere

We give a complete classification of isomorphism classes of all SU(2)-equivariant holomorphic Hermitian vector bundles on CP¹. We construct a canonical bijective correspondence between the isomorphism classes of SU(2)-equivariant holomorphic Hermitian vector bundles on CP¹ and the isomorphism classes of pairs ({Hn}_n∈Z,T), where each Hn is a finite dimensional Hilbert space with Hn=0 for all but finitely many n, and T is a linear operator on the direct sum _⊕n∈ZHn such that T(Hn)⊂Hn+2 for all n.     

Equidistant hypersurfaces of the bidisk

The following are notes on the geometry of the bidisk, H ² × H ². In particular, we examine the properties of equidistant surfaces in the bidisk.     

Maps preserving operator pairs whose products are projections

Let B(H) be the algebra of all bounded linear operators on a complex Hilbert space H with dimH?2. It is proved that a surjective map φ on B(H) preserves operator pairs whose products are nonzero projections in both directions if and only if there is a unitary or an anti-unitary operator U on H such that φ(A)=λU^∗AU for all A in B(H) for some constants λ with λ²=1. Related results for surjective maps preserving operator pairs whose triple Jordan products are nonzero projections in both directions are also obtained. These show that the operator pairs whose products or triple Jordan products are nonzero projections are isometric invariants of B(H).     

The Smoothness of Scattering Operators for Sinh-Gordon and Nonlinear Schrödinger Equations

We show that the scattering operator carries a band in H ^s (? ⁿ ) ×H ^s?1 (? ⁿ ) into H ^s (? ⁿ ) ×H ^s?1(? ⁿ ) for the sinh-Gordon equation and an analogous result also holds true for the nonlinear Schrödinger equation with an exponential nonlinearity, where s≥n/2 is arbitrary and n≥ 2. Therefore, the scattering operators are infinitely smooth for the above two equations.     

On the essential commutant of analytic Toeplitz operators associated with spherical isometries

Let T∈Bn(H) be an essentially normal spherical isometry with empty point spectrum on a separable complex Hilbert space H, and let AT⊂B(H) be the unital dual operator algebra generated by T. In this note we show that every operator S∈B(H) in the essential commutant of AT has the form S=X+K with a T-Toeplitz operator X and a compact operator K. Our proof actually covers a larger class of subnormal operator tuples, called A-isometries, which includes for example the tuple T=(Mz₁,…,Mzn)∈B(H²n(σ)) consisting of the multiplication operators with the coordinate functions on the Hardy space H²(σ) associated with the normalized surface measure σ on the boundary ∂D of a strictly pseudoconvex domain D⊂Cn. As an application we determine the essential commutant of the set of all analytic Toeplitz operators on H²(σ) and thus extend results proved by Davidson (1977) [6] for the unit disc and Ding and Sun (1997) [11] for the unit ball.     

On operators commuting with Toeplitz operators modulo the compact operators

We prove that an operator on H² of the disc commutes modulo the compacts with all analytic Toeplitz operators if and only if it is a compact perturbation of a Toeplitz operator with symbol in H^∞ + C. Consequently, the essential commutant of the whole Toeplitz algebra is the algebra of Toeplitz operators with symbol in QC. The image in the Calkin algebra of the Toeplitz operators with symbol in H^∞ + C is a maximal abelian algebra. These results lead to a characterization of automorphisms of the algebra of compact perturbations of the analytic Toeplitz operators.     

HFD groups in the Solovay model

Hajnal and Juhász proved that under CH there is a hereditarily separable, hereditarily normal topological group without non-trivial convergent sequences that is countably compact and not Lindelöf. The example constructed is a topological subgroup H⊆ω₁² that is an HFD with the following property

(P)

the projection of H onto every partial product I² for I∈ω[ω₁] is onto.

Any such group has the necessary properties. We prove that if κ is a cardinal of uncountable cofinality, then in the model obtained by forcing over a model of CH with the measure algebra on κ², there is an HFD topological group in ω₁² which has property (P).     

Extension of isometries on the unit sphere of L p spaces

In this paper we study the isometric extension problem and show that every surjective isometry between the unit spheres of Lp (μ) (1 p ∞, p≠2) and a Banach space E can be extended to a linear isometry from Lp(μ) onto E, which means that if the unit sphere of E is (metrically) isometric to the unit sphere of Lp(μ), then E is linearly isometric to Lp(μ). We also prove that every surjective 1-Lipschitz or anti-1-Lipschitz map between the unit spheres of Lp (μ1, H1) and Lp(μ2,H2) must be an isometry and can be extended to a linear isometry from Lp (μ1,H1) to Lp (μ2,H2), where H1 and H2 are Hilbert spaces.     

A constructive procedure to recover asymptotics of short-range or long-range potentials

We study an inverse scattering problem for a pair of Hamiltonians (H,H₀) on L²(Rⁿ), where H₀=-Δ and H=H₀+V, V being a short- or long-range potential. By an elementary constructive method, we show that the scattering operator S, which is localized near a fixed energy λ>0, determines the asymptotics of the potential V at infinity, in dimension n?3. This is done by studying the action of the scattering operator on suitable wave packets.     

Extension of derivations in the algebra of compact operators

Let C(H) denote the C¹-algebra of all compact linear operators on a complex Hilbert space H. If δ is a closable ¹-derivation in C(H) which anti-commutes with an involutive ¹-antiautomorphism α and has finite spatial deficiency-indices, then there exists an infinitesimal generator δ₀ of a continuous action of R on C(H) which extends δ and anti-commutes with α. This is an analogue of the von Neumann's theorem which states that a symmetric operator commuting with a conjugation J has a self-adjoint extension which also commutes with J.     

Norms of composition operators on weighted Hardy spaces

The norm of a bounded composition operator induced by a disc automorphism is estimated on weighted Hardy spaces H ²(β) in which the classical Hardy space is continuously embedded. The estimate obtained is accurate in the sense that it provides the exact norm for particular instances of the sequence β. As a by-product of our results, an estimate for the norm of any bounded composition operator on H ²(β) is obtained.     

Spectral gaps for periodic Schrödinger operators with hypersurface magnetic wells: Analysis near the bottom

We consider a periodic magnetic Schrödinger operator Hh, depending on the semiclassical parameter h>0, on a noncompact Riemannian manifold M such that H¹(M,R)=0 endowed with a properly discontinuous cocompact isometric action of a discrete group. We assume that there is no electric field and that the magnetic field has a periodic set of compact magnetic wells. We suppose that the magnetic field vanishes regularly on a hypersurface S. First, we prove upper and lower estimates for the bottom λ₀(Hh) of the spectrum of the operator Hh in L²(M). Then, assuming the existence of non-degenerate miniwells for the reduced spectral problem on S, we prove the existence of an arbitrarily large number of spectral gaps for the operator Hh in the region close to λ₀(Hh), as h→0. In this case, we also obtain upper estimates for the eigenvalues of the one-well problem.