2-Complex Symmetric Composition Operators on H2

Complex symmetry of invertible composition operators

We present a unified approach to the problem of characterizing the Fredholm multiplication and composition operators on the Hardy spaceH 2.

Approximation numbers of weighted composition operators

The composition operator induced by a holomorphic self-map of the unit disc is compact on L 1 {L^1} of the unit circle if and only if it is compact on the Hardy space H 2 {H^2} of the disc. This answers a question posed by Donald Sarason: it proves that Sarason’s integral condition characterizing compactness on L 1 {L^1} is equivalent to the asymptotic condition on the Nevanlinna counting function which characterizes compactness on H 2 {H^2} .

In this paper, we investigate the conditions under which the Toeplitz composition operator on the Hardy space H2 becomes complex symmetric with respect to a certain conjugation. We also study various normality conditions for the Toeplitz composition operator on H2.

We investigate the composition operators on the weighted Hardy spaces H2(β). For any bounded weight sequence β, we give necessary conditions for those operators to be isometric. The sufficiency of those conditions is well‐known for the classical space H2. In the case where β is non‐decreasing or non‐increasing, their sufficiency holds only for very few weighted spaces. We find out such spaces by characterizing the isometric monomial composition operators, first for a general β, then for any β as before. With no restriction on β, we provide a complete description of all isometric composition operators. We also prove that the unitary monomial ones are the same as those acting on H2. Such a fact extends to general symbols in the case where β is bounded (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Given a unilateral forward shift S acting on a complex, separable, innite dimensional Hilbert space H, an asymptotically S-Toeplitz operator is a bounded linear operator T on H satisfying that {S* n TS n } is convergent with respect to one of the topologies commonly used in the algebra of bounded linear operators on H. In this paper, we study the asymptotic T u -Toeplitzness of weighted composition operators on the Hardy space H2, where u is a nonconstant inner function.

There is axiomatically described the class of spaces Υ (resp. Χ) of functions, analytic in the unit disk, for which the invariant subspaces of the shift operator f (z) → z f (z) (resp. the inverse shift f(z)→z−1(f(z)−f (0))) are constructed just like the Hardy space H2. It is proved that as Χ one can take, for example, the space H1, the disk-algebra CA, the space UA of all uniformly convergent power series; and as Υ the space of integrals of Cauchy type L1/H 1− , the space VMOA. There is also obtained an analog for the space UA of W. Rudin's theorem on z-invariant subspaces of the space CA.

The author has proved that an outer function in the Hardy space H1 can be factored into a product in which one factor is strongly outer and the other is the sum of two inner functions. In an endeavor to understand better the latter factor, we introduce a class of functions containing sums of inner functions as a special case. Using it, we describe the solutions of extremal problems in the Hardy spaces Hp for 1≦p<∞.

We characterise the complex measures μ on the open unit disk D such that for all f in the Hardy space H2. The characterisation involves Carleson measures, the duality between H1 and BMOA, and Hankel operators.

Toeplitz and asymptotic Toeplitz operators on [formula omitted

The paper contains a new proof for the sufficiency in Joel H. Shapiro’s recent characterization of inner functions saying that an analytic self-map φ of the open unit disk is an inner function if and only if the essential norm of the composition operator of symbol φ is equal to √ (1 + |φ(0)|)/(1 − |φ(0)|). The main ingredient in the proof is a formula for the essential norm of a composition operator in terms of Aleksandrov measures obtained by Cima and Matheson. The necessity was originally proved by Joel Shapiro in 1987. A short proof of the necessity, by Aleksandrov measure techniques, was obtained by Jonathan E. Shapiro in 1998. For each holomorphic self-map φ : U → U of the open unit disk U, the composition operator Cφ of symbol φ is defined as follows Cφf = f ◦ φ, f ∈ H. In this definition H is the Hilbert Hardy space on U, i.e., the set of all analytic functions on U with square summable Taylor coefficients. It is well known that each such φ induces a bounded composition operator Cφ on H. Recently Joel Shapiro obtained the following characterization of inner functions [7], (that is of self-maps φ whose radial limit function is unimodular almost everywhere on the unit circle T). Theorem 1. The function φ : U → U is inner if and only if (1) ‖Cφ‖e = √ 1 + |φ(0)| 1− |φ(0)| , where ‖Cφ‖e denotes the essential norm of Cφ. The necessity in the previous equivalence was originally proved in [6]. Jonathan Shapiro [5] observed that Aleksandrov measures can be used Received by the editors on May 7, 2003, and in revised form on May 30, 2003. Copyright c ©2005 Rocky Mountain Mathematics Consortium 1723

The function , where is Lerch's transcendent, satisfies the following two-dimensional formally self- adjoint second-order hyperbolic differential equation: where . The corresponding differential expression determines a densely defined symmetric operator (the minimal operator) on the Hilbert space , where . We obtain a description of the domains of definition of some symmetric extensions of the minimal operator. We show that formal solutions of the eigenvalue problem for these symmetric extensions are represented by functional series whose structure resembles that of the Fourier series of . We discuss sufficient conditions for these formal solutions to be eigenfunctions of the resulting symmetric differential operators. We also demonstrate a close relationship between the spectral properties of these symmetric differential operators and the distribution of the zeros of some special analytic functions analogous to the Riemann zeta function. Bibliography: 15 titles.

The topological structure of the set of (weighted) composition operators has been studied on various function spaces on the unit disc such as Hardy spaces, the space of bounded holomorphic functions, weighted Banach spaces of holomorphic functions with sup-norm, Hilbert Bergman spaces. In this paper we consider this problem for all Bergman spaces $$A_{\alpha }^p$$ with $$p \in (0, \infty )$$ and $$ \alpha \in (-1, \infty )$$ . In this setting we establish a criterion for two composition operators to be linearly connected in the space of composition operators; furthermore, for the space of weighted composition operators, we prove that the set of compact weighted composition operators is path connected, but it is not a component.

A necessary and sufficient condition is obtained for M‐quasi‐hyponormal composition operators. It has also been proved that the class of M‐quasi‐hyponormal composition operators coincides with the class of M‐paranormal composition operators. Existence of M‐hyponormal composition operators which are not hyponormal; and M‐quasihyponormal composition operators which are not M‐hyponormal and quasi‐hyponormal are also shown.