Invertible weighted composition operators

A weighted composition operator Cψ,φ takes an analytic map f on the open unit disc of the complex plane to the analytic map ψ⋅f°φ where φ is an analytic map of the open unit disc into itself and ψ is an analytic map on the open unit disc. This paper studies the invertibility of such operators. The two maps ψ and φ are characterized when Cψ,φ acts on the Hardy-Hilbert space of the unit disc H²(D). Depending upon the nature of the fixed points of φ spectra are then investigated.     

球上Zygmund 型空间上的加权Cesàro 算子

该文在Cⁿ中单位球上讨论了Zygmund 型空间(小Zygmund 型空间)之间的加权Cesàro 算子T_g 的有界性和紧性特征, 得到了以下的结果: (1) T_g 是Z^p 到Z^q的有界算子或紧算子的充要条件; (2) T_g 是 Z^p₀ 到Z^q₀ 的有界算子或紧算子的充要条件.     

New criteria for boundedness and compactness of weighted composition operators mapping into the Bloch space

Let ψ and φ be analytic functions on the open unit disk $\mathbb{D}$ with φ( $\mathbb{D}$ ) ? $\mathbb{D}$ . We give new characterizations of the bounded and compact weighted composition operators W _ψ,? from the Hardy spaces H ^p , 1 ≤ p ≤ ∞, the Bloch space B, the weighted Bergman spaces A _α ^p , α > ? 1,1 ≤ p < ∞, and the Dirichlet space $\mathcal{D}$ to the Bloch space in terms of boundedness (respectively, convergence to 0) of the Bloch norms of W _ψ,? f for suitable collections of functions f in the respective spaces. We also obtain characterizations of boundedness for H ¹ as well as of compactness for H ^p , 1 ≤ p < ∞, and $\mathcal{D}$ purely in terms of the symbols ψ and φ.     

Some fixed point theorems of the Schauder and the Krasnosel'skii type and application to nonlinear transport equations

In [J. Math. Phys. 37 (1996) 1336-1348] the existence of solutions to the boundary value problem (1.1)-(1.2) was analyzed for isotropic scattering kernels on L_p spaces for p∈(1,∞). Due to the lack of compactness in L₁ spaces, the problem remains open for p=1. The purpose of this work is to extend this analysis to the case p=1 for anisotropic scattering kernels. Our strategy consists in establishing new variants of the Schauder and the Krasnosel'skii fixed point theorems in general Banach spaces involving weakly compact operators. In L₁ context these theorems provide an adequate tool to attack the problem. Our analysis uses the specific properties of weakly compacts sets on L₁ spaces and the weak compactness results for one-dimensional transport equations established in [J. Math. Anal. Appl. 252 (2000) 767-789].     

Contributions to the theory of the classical Banach spaces

The paper contains several results on the linear topological structure of the spaces C(K), K compact metric, and L_p(0, 1), 1 ? p < ∞. The topics which are studied include: complemented subspaces, special Schauder bases, and equivalent norms in these spaces.     

COMPOSITION OPERATORS FROM p-BLOCH SPACES TO LITTLE q-BLOCH SPACES ON THE UNIT BALL OF Cn

For all 0 p, q ∞, let C denote the composition operator from q-Bloch spaces βp to little p-Bloch spaces β0q on the unit ball of Cn. In this article, necessary and sufficient conditions for C to be a bounded or compact operator are given.     

Optimal spectral theory of the linear Boltzmann equation

We give optimal compactness results in L^p spaces ( 1<p<∞) related to spectral theory of general neutron transport equations on spatial domains with finite Lebesgue measure.     

Doubly nonlinear evolution equations governed by time-dependent subdifferentials in reflexive Banach spaces

We prove the existence of solutions of the Cauchy problem for the doubly nonlinear evolution equation: dv(t)/dt+V_∂φt(u(t))∋f(t), v(t)∈H_∂ψ(u(t)), 0<t<T, where H_∂ψ (respectively, V_∂φt) denotes the subdifferential operator of a proper lower semicontinuous functional ψ (respectively, φt explicitly depending on t) from a Hilbert space H (respectively, reflexive Banach space V) into (−∞,+∞] and f is given. To do so, we suppose that V?H≡H^∗?V^∗ compactly and densely, and we also assume smoothness in t, boundedness and coercivity of φt in an appropriate sense, but use neither strong monotonicity nor boundedness of H_∂ψ. The method of our proof relies on approximation problems in H and a couple of energy inequalities. We also treat the initial-boundary value problem of a non-autonomous degenerate elliptic-parabolic problem.     

Convergent sequences of composition operators

Composition operators Cφ on the Hilbert Hardy space H² over the unit disk are considered. We investigate when convergence of sequences {φn} of symbols, (i.e., of analytic selfmaps of the unit disk) towards a given symbol φ, implies the convergence of the induced composition operators, Cφn→Cφ. If the composition operators Cφn are Hilbert-Schmidt operators, we prove that convergence in the Hilbert-Schmidt norm, ‖Cφn−CφHS_‖→0 takes place if and only if the following conditions are satisfied: ‖φn−φ_‖2→0, ∫1/(1−²|φ|)<∞, and ∫1/(1−²|φn|)→∫1/(1−²|φ|). The convergence of the sequence of powers of a composition operator is studied.     

A Class of Operators on Lh 2

Let D be the open unit disc in ? and let L_h ² be the space of quadratic integrable harmonic functions defined on D. Let \(\varphi: {\bar D}\rightarrow {\rm C}\) be a function in L^∞(D) with the property that φ(b) = lim_x→b,x?Dφ(x) for all b ? ?D. Define the operator C_φ in L_h ² as follows: C_φf = Q(φ·f),f ? L_h ², where Q is the orthogonal projection from L² (D) on L_h ². The following results are proved. If φ¦_?D ≡ 0, then C_φ is a compact linear operator and if φ¦_?D vanishes nowhere, then C_φ is a Fredholm operator.     

Compact and Weakly Compact Composition Operators on BMOA

Any analytic map φ of the unit disc ${\mathbb{D}}$ into itself induces a composition operator C _φ on BMOA, mapping ${f \mapsto f \circ \varphi}$ , where BMOA is the Banach space of analytic functions ${f\colon \mathbb{D} \to \mathbb{C}}$ whose boundary values have bounded mean oscillation on the unit circle. We show that C _φ is weakly compact on BMOA precisely when it is compact on BMOA, thus solving a question initially posed by Tjani and by Bourdon, Cima and Matheson in the special case of VMOA. As a crucial step of our argument we simplify the compactness criterion due to Smith for C _φ on BMOA and show that his condition on the Nevanlinna counting function alone characterizes compactness. Additional equivalent compactness criteria are established. Furthermore, we prove the unexpected result that compactness of C _φ on VMOA implies compactness even from the Bloch space into VMOA.     

Singular values and Schmidt pairs of composition operators on the Hardy space

We find the singular values and corresponding Schmidt pairs of a compact composition operator Cφ induced by φ(z)=az+b, where |a|+|b|<1, on the classical Hardy space. We do so by solving a functional equation that is a generalization of Schröder's equation: find a function f, holomorphic on the open unit disc, and a complex number λ such that G(z)f(ψ(z))=λf(ψ(z)), where ψ is a holomorphic self-map of the open unit disc with an interior fixed point and G is a bounded holomorphic function on the open unit disc. In addition, we find the spectrum of the weighted composition operator MGCψ.     

Endpoint estimates for the commutator of pseudo-differential operators

It is well known that the commutator Tb of the Calderón-Zygmund singular integral operator is bounded on L^p(Rⁿ) for 1 < p < +∞ if and only if b ∈ BMO [1]. On the other hand, the commutator T_b is bounded from H¹(Rⁿ) into L¹(Rⁿ) only if the function b is a constant [2]. In this article, we will discuss the boundedness of commutator of certain pseudo-differential operators on Hardy spaces H¹. Let T^σ be the operators that its symbol is S⁰_1,δ with 0 ≤ δ < 1, if b ∈ LMO_∞, then, the commutator [b, T^σ] is bounded from H¹(Rⁿ) into L¹(Rⁿ) and from L¹(Rⁿ) into BMO(Rⁿ); If [b, T^σ] is bounded from H¹(Rⁿ) into L¹(Rⁿ) or L¹(Rⁿ) into BMO(Rⁿ), then, b ∈ LMO_loc.     

Essential norm of products of multiplication composition and differentiation operators on weighted Bergman spaces

Let ψ be a holomorphic function on the open unit disk D and φ a holomorphic self-map of D. Let C_φ,M_ψ and D denote the composition, multiplication and differentiation operator, respectively. We find an asymptotic expression for the essential norm of products of these operators on weighted Bergman spaces on the unit disk. This paper is a continuation of our recent paper concerning the boundedness of these operators on weighted Bergman spaces.     

Some Remarks on Operator-Norm Convergence of the Trotter Product Formula on Banach Spaces

We extend the Trotter-Kato-Chernoff theory of strong approximation of C₀ semigroups on Banach spaces to operator-norm approximation of analytic semigroups with error estimate. As application we obtain a criterion for the operator-norm convergence of the Trotter product formula on Banach spaces with error estimate n⁻¹ log n, provided one of the generators has a bounded H^∞ functional calculus. For both results, we present versions for approximation in operator-ideal-norms such as the trace norm or the Hilbert-Schmidt norm. Finally, we give some remarks on the operator-norm convergence of the Trotter product formula for semigroups acting on a scale of L_p-spaces.     

Essential spectra of quasi-parabolic composition operators on Hardy spaces of analytic functions

In this work we study the essential spectra of composition operators on Hardy spaces of analytic functions which might be termed as “quasi-parabolic.” This is the class of composition operators on H² with symbols whose conjugate with the Cayley transform on the upper half-plane are of the form φ(z)=z+ψ(z), where ψ∈H^∞(H) and ℑ(ψ(z))>?>0. We especially examine the case where ψ is discontinuous at infinity. A new method is devised to show that this type of composition operator fall in a C*-algebra of Toeplitz operators and Fourier multipliers. This method enables us to provide new examples of essentially normal composition operators and to calculate their essential spectra.     

Extended Cesàro operators on Bergman spaces

We define an extended Cesàro operator T_g with holomorphic symbol g in the unit ball B of Cⁿ. For a large class of weights w we characterize those g for which T_g is bounded (or compact) from Bergman space L^p_a,w(B) to L^q_a,w(B), 0<p,q<∞. In addition, we obtain some results about equivalent norms, the norm of point evaluation functionals, and the interpolation sequences on L^p_a,w(B).     

Exposed points and extremal problems in H1

If φ ∈ L_∞, we denote by T_φ the functional defined on the Hardy space H¹ by

$\text{T}{}_{\mathrm{\phi }}\text{(?) =}\text{∫}\text{?π}\text{π}\text{?(e}{}^{\mathrm{i\theta }}\text{) φ(e}{}^{\mathrm{i\theta }}\text{)}\text{dθ}\text{2π}$

. Let S_φ be the set of functions in H¹ which satisfy $\text{T}{}_{\mathrm{\phi }}\text{(?) = ∥T}{}_{\mathrm{\phi }}\text{∥}\text{and}\text{∥? ∥}{}_1\text{? 1}$. It is known that if φ is continuous, then S_φ is weak-¹ compact and not empty. For many noncontinuous φ each S_φ is weak-¹ compact and not empty. A complete descr ption of S_φ if S_φ is weak-¹ compact and not empty is obtained. S_φ is not empty if and only if $\text{S}{}_{\mathrm{\phi }}\text{= S}{}_{\mathrm{\psi }}\text{and}\text{ψ = ¦ ?¦}\text{?}$ for some nonzero ? in H¹. It is shown that if $\text{φ = ¦? ¦}\text{?}$ and $\text{? = pg}$, where p is an analytic polynomial and g is a strong outer function, then S_φ is weak-¹ compact. As the consequence, if $\text{? = p}$, then S_φ is weak-¹ compact.