Hankel measures on Hardy space

Toeplitz and asymptotic Toeplitz operators on [formula omitted

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.

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<∞.

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.

Abstract. AsequenceofcompositionoperatorsonHardyspaceiscon-sidered. Weprovethat,bynumericalrangeproperties,itisSOT-conver-gencebutnotconverge. 1. introductionLet ϕ be a holomorphic self-map of the unit disc U := {z ∈ C : |z| < 1}.The function ϕ induces the composition operator C ϕ , defined on the space ofholomorphic functions on Uby C ϕ f = f ◦ ϕ. The restriction of C ϕ to variousBanach spaces of holomorphic functions on U has been an active subject ofresearch for more than three decades and it will continue to be for decadesto come (see [11], [12] and [4]). Let H 2 denote the Hardy space of analyticfunctions on the open unit disc with square summable Taylor coefficients. Inrecent yearsthe study ofcomposition operatorson the Hardy space has receivedconsiderable attention.In this paper we consider the numerical range of elliptic composition opera-tors on H 2 . The numerical range of a bounded linear operator A on a Hilbertspace H is the set of complex numbersW(A) := {hAx,xi : x ∈ H,kxk = 1},where h·,· denotes the inner product in H.In [9] V. Matache determined the shape W(C

Boundedness and Compactness of Generalized Hankel Operators on Bounded Symmetric Domains

In this paper we investigate several interesting properties of the fractional calculus operators, the Carlson-Shaffer linear operator L(a,c) and a certain integral operator , associated with various subclasses of analytic and univalent functions. We also investigate the relationships of some of these classes with the Hardy space H ∞ (of bounded analytic functions in the open unit disk μ and with some families of multiplier transformations.

We show generic existence of functions f in the Hardy space \(H^p(0<p<1)\) on the open unit disc whose primitive F(f) satisfies the following. For every \(a>\dfrac{p}{1-p}\) and every \(A,B\in {\mathbb {R}}\), \(A<B\) it holds $$\begin{aligned} \sup _{0<r<1}\int ^B_A|F(f)(re^{i\theta })|^ad\theta =+\infty . \end{aligned}$$Results of similar nature are valid when the space \(H^p\) is replaced by localized versions of it, \(0<p<1\), or intersections of such spaces.

Isoperimetric type inequalities for harmonic functions

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

Complex symmetry of invertible composition operators

This paper is a survey of some works on paracommutators and Hankel operators in China, which are the subjects I have been studying since 1985. The classical Hankel operator is defined on the Hilbert space l 2 by the Hankel matrix Γ = {b n+k } n,k ≥0, and the Toeplitz operator is defined also on l 2 by the toeplitz matrix T = {b n−k } n≥k≥0. By the theory of the Fourier series both of them become the operators on the Hardy space \( {{H}^{2}}(T) = \{ f \in {{L}^{2}}(T):\hat{f}(n) = 0,n < 0\} . \). Let \( \overline H ^2 (T) \) denote the conjugate Hardy space and P,P denote the ortnogonal projections from L 2(T) into H 2(T),H 2 (T) respectively. Then the Toeplitz operator is defined as follows $$ {T_b}(f) = P(\bar bf),for f \in {H^2}(T) $$ (1.1) and the Hankel operator is defined as follows $$ {H_b}(f) = \bar P(\bar bf),for f \in {H^2}(T).$$ (1.2)

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.

In this paper one considers some general theorems of the theory of Hankel and Toeplitz operators in spaces of analytic functions. Under natural restrictions on the spaces X, it is shown that the symbols of the Toeplitz operators, acting in X, are bounded. One describes completely the symbols of the Hankel and Toeplitz operators, acting from Hp into ¯Hq (into Hq) for 0 < p, q < ∞.

Approximation numbers of weighted composition operators