Trace ideal criteria for Toeplitz operators

Products of multiplication composition and differentiation operators on weighted Bergman spaces

Introduction 1 Some analysis in Hilbert spaces and scales 1.1 Smooth and analytic maps 1.2 Scales of Hilbert spaces and interpolation 1.3 Differential forms 2 Symplectic structures and Hamiltonian equations 2.1 Basic definitions 2.2 Symplectic transformations 2.3 Darboux lemmas Appendix. Time-quasiperiodic solutions 3 Lax-integrable Hamiltonian equations and their integrable subsystems 3.1 Examples of Hamiltonian PDEs 3.2 Lax-integrable equations 3.3 Integrable subsystems 4 Finite-gap manifolds and theta-formulas 4.1 Finite-gap manifolds for the KdV equation 4.2 The Its-Matveev theta-formulas 4.3 Higher equations from the KdV hierarchy 4.4 Sine-Gordon equation under Dirichlet boundary conditions 5 Linearised equations and their Floquet solutions 5.1 The linearised equation 5.2 Floquet solutions 5.3 Complete systems of Floquet solutions 5.4 Lower-dimensional invariant tori of finite-dimensional systems and Floquet’s theorem 6 Linearised Lax-integrable equations 6.1 Abstract situation 6.2 Linearised KdV equation 6.3 Higher KdV-equations 6.4 Linearised SG equation 7 Normal form 7.1 A normal form theorem 7.2 Examples 8 The KAM theorem 8.1 The main theorem and related results 8.2 Reduction to a parameter-depending case 8.3 A KAM-theorem for parameter-depending equations 8.4 Completion of the Main Theorem’s proof (Step 4) 9 Examples 9.1 Perturbed KdV equation 9.2 Higher KdV equations 9.3 Perturbed SG equation 9.4 KAM-persistence of lower-dimensional invariant tori of nonlinear finite-dimensional systems References

Let L2 = L2(D, r dr dθ/π) be the Lebesgue space on the open unit disc and let be the Bergman space. Let P be the orthogonal projection of L2 onto and let Q be the orthogonal projection onto . Then I − P ≥ Q. The big Hankel operator and the small Hankel operator on are defined as: for ϕ in L∞, and . In this paper, the finite‐rank intermediate Hankel operators between and are studied. We are working on the more general space, that is, the weighted Bergman space.

Symplectic structures and hamiltonian systems in scales of hilbert spaces

We consider semilinear evolution equations for which the linear part is normal and generates a strongly continuous semigroup, and the nonlinear part is sufficiently smooth on a scale of Hilbert spaces. We approximate their semi-flow by an implicit A-stable Runge–Kutta discretization in time and a spectral Galerkin truncation in space. We show regularity of the Galerkin-truncated semi-flow and its time discretization on open sets of initial values with bounds that are uniform in the spatial resolution and the initial value. We also prove convergence of the space-time discretization without any condition that couples the time step to the spatial resolution. We then estimate the Galerkin truncation error for the semi-flow of the evolution equation, its Runge–Kutta discretization and their respective derivatives, showing how the order of the Galerkin truncation error depends on the smoothness of the initial data. Our results apply, in particular, to the semilinear wave equation and to the nonlinear Schrodinger equation.

We require some preparation for the proof of the splitting theorem. We begin with the introduction of certain scales of Hilbert spaces.

G. Point interactions and scales of Hilbert spaces

Isometric weighted composition operators on weighted Bergman spaces

We characterize bounded and compact weighted composition operators acting between weighted Bergman spaces and between Hardy spaces. Our results use certain integral transforms that generalize the Berezin transform. We also estimate the essential norms of these operators. As applications, we characterize bounded and compact pointwise multiplication operators between weighted Bergman spaces and estimate their essential norms.

Joint Carleson measure and the difference of composition operators on [formula omitted

The commutative C*-algebras of Toeplitz operators on the classical (weightless) Bergman space were classified in Chapter 9 by pencils of geodesics on the unit disk, considered as the hyperbolic plane. Theorem 10.4.1 shows that the same classes of defining symbols generate commutative C*-algebras of Toeplitz operators on each weighted Bergman space. At the same time the principal question, whether the above cases are the only possible sets of defining symbols which might generate the commutative C * -algebras of Toeplitz operators on each weighted Bergman space, has remained open.

Norm Approximation by Polynomials in Some Weighted Bergman Spaces

In this paper we consider linear operators M ψ C ϕ D and M ψ DC ϕ acting between weighted Bergman spaces, where M ψ , C ϕ and D are multiplication, composition and differentation operators respectively.Our goal is to characterize those holomorphic self maps ϕ of D for which M ψ DC ϕ and M ψ C ϕ D acts boundedly and compactly between weighted Bergman spaces.

This paper gives a note on weighted composition operators on the weighted Bergman space, which shows that for a fixed composition symbol, the weighted composition operators are bounded on the weighted Bergman space only with bounded weighted symbols if and only if the composition symbol is a finite Blaschke product.

Let $\varphi,\psi$ be the analytic self-maps of the unit ball ${\mathbb B}$, we characterize the Hilbert-Schmidt differences of two composition operator $C_\varphi$ and $C_\psi$ on weighted Bergman space $A_\alpha^2$, and give some conclusions about the topological structure of $\mathcal{C}(A_\alpha^2)$, the space of all bounded composition operators on $A_\alpha^2$ endowed with operator norm.