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A generalization of the Aleksandrov operator and adjoints of weighted composition operators

Eva A. Gallardo-Gutiérrez, Jonathan R. Partington (2013)

Annales de l’institut Fourier

A generalization of the Aleksandrov operator is provided, in order to represent the adjoint of a weighted composition operator on 2 by means of an integral with respect to a measure. In particular, we show the existence of a family of measures which represents the adjoint of a weighted composition operator under fairly mild assumptions, and we discuss not only uniqueness but also the generalization of Aleksandrov–Clark measures which corresponds to the unweighted case, that is, to the adjoint of...

A Hankel matrix acting on Hardy and Bergman spaces

Petros Galanopoulos, José Ángel Peláez (2010)

Studia Mathematica

Let μ be a finite positive Borel measure on [0,1). Let μ = ( μ n , k ) n , k 0 be the Hankel matrix with entries μ n , k = [ 0 , 1 ) t n + k d μ ( t ) . The matrix μ induces formally an operator on the space of all analytic functions in the unit disc by the fomula μ ( f ) ( z ) = n = 0 i ( k = 0 μ n , k a k ) z , z ∈ , where f ( z ) = n = 0 a z is an analytic function in . We characterize those positive Borel measures on [0,1) such that μ ( f ) ( z ) = [ 0 , 1 ) f ( t ) / ( 1 - t z ) d μ ( t ) for all f in the Hardy space H¹, and among them we describe those for which μ is bounded and compact on H¹. We also study the analogous problem for the Bergman space A².

Admissible functions for the Dirichlet space

Javad Mashreghi, Mahmood Shabankhah (2010)

Studia Mathematica

Zero sets and uniqueness sets of the classical Dirichlet space are not completely characterized yet. We define the concept of admissible functions for the Dirichlet space and then apply them to obtain a new class of zero sets for . Then we discuss the relation between the zero sets of and those of .

Basic relations valid for the Bernstein spaces B ² σ and their extensions to larger function spaces via a unified distance concept

P. L. Butzer, R. L. Stens, G. Schmeisser (2014)

Banach Center Publications

Some basic theorems and formulae (equations and inequalities) of several areas of mathematics that hold in Bernstein spaces B σ p are no longer valid in larger spaces. However, when a function f is in some sense close to a Bernstein space, then the corresponding relation holds with a remainder or error term. This paper presents a new, unified approach to these errors in terms of the distance of f from B σ p . The difficult situation of derivative-free error estimates is also covered.

Boundary vs. interior conditions associated with weighted composition operators

Kei Izuchi, Yuko Izuchi, Shûichi Ohno (2014)

Open Mathematics

Associated with some properties of weighted composition operators on the spaces of bounded harmonic and analytic functions on the open unit disk 𝔻 , we obtain conditions in terms of behavior of weight functions and analytic self-maps on the interior 𝔻 and on the boundary 𝔻 respectively. We give direct proofs of the equivalence of these interior and boundary conditions. Furthermore we give another proof of the estimate for the essential norm of the difference of weighted composition operators.

Compactness of composition operators acting on weighted Bergman-Orlicz spaces

Ajay K. Sharma, S. Ueki (2012)

Annales Polonici Mathematici

We characterize compact composition operators acting on weighted Bergman-Orlicz spaces α ψ = f H ( ) : ψ ( | f ( z ) | ) d A α ( z ) < , where α > -1 and ψ is a strictly increasing, subadditive convex function defined on [0,∞) and satisfying ψ(0) = 0, the growth condition l i m t ψ ( t ) / t = and the Δ₂-condition. In fact, we prove that C φ is compact on α ψ if and only if it is compact on the weighted Bergman space ² α .

Convexities of Gaussian integral means and weighted integral means for analytic functions

Haiying Li, Taotao Liu (2019)

Czechoslovak Mathematical Journal

We first show that the Gaussian integral means of f : (with respect to the area measure e - α | z | 2 d A ( z ) ) is a convex function of r on ( 0 , ) when α 0 . We then prove that the weighted integral means A α , β ( f , r ) and L α , β ( f , r ) of the mixed area and the mixed length of f ( r 𝔻 ) and f ( r 𝔻 ) , respectively, also have the property of convexity in the case of α 0 . Finally, we show with examples that the range α 0 is the best possible.

Distance formulae and invariant subspaces, with an application to localization of zeros of the Riemann ζ -function

Nikolai Nikolski (1995)

Annales de l'institut Fourier

It is proved that a subspace of a holomorphic Hilbert space is completely determined by their distances to the reproducing kernels. A simple rule is established to localize common zeros of a subspace of the Hardy space of the unit disc. As an illustration we show a series of discs of the complex plan free of zeros of the Riemann ζ -function.

Essential normality for certain finite linear combinations of linear-fractional composition operators on the Hardy space H 2

Mahsa Fatehi, Bahram Khani Robati (2012)

Czechoslovak Mathematical Journal

In 1999 Nina Zorboska and in 2003 P. S. Bourdon, D. Levi, S. K. Narayan and J. H. Shapiro investigated the essentially normal composition operator C ϕ , when ϕ is a linear-fractional self-map of 𝔻 . In this paper first, we investigate the essential normality problem for the operator T w C ϕ on the Hardy space H 2 , where w is a bounded measurable function on 𝔻 which is continuous at each point of F ( ϕ ) , ϕ 𝒮 ( 2 ) , and T w is the Toeplitz operator with symbol w . Then we use these results and characterize the essentially normal...

Estimates in the Hardy-Sobolev space of the annulus and stability result

Imed Feki (2013)

Czechoslovak Mathematical Journal

The main purpose of this work is to establish some logarithmic estimates of optimal type in the Hardy-Sobolev space H k , ; k * of an annular domain. These results are considered as a continuation of a previous study in the setting of the unit disk by L. Baratchart and M. Zerner, On the recovery of functions from pointwise boundary values in a Hardy-Sobolev class of the disk, J. Comput. Appl. Math. 46 (1993), 255–269 and by S. Chaabane and I. Feki, Optimal logarithmic estimates in Hardy-Sobolev spaces...

Hermitian composition operators on Hardy-Smirnov spaces

Gajath Gunatillake (2017)

Concrete Operators

Let Ω be an open simply connected proper subset of the complex plane and φ an analytic self map of Ω. If f is in the Hardy-Smirnov space defined on Ω, then the operator that takes f to f ⃘ φ is a composition operator. We show that for any Ω, analytic self maps that induce bounded Hermitian composition operators are of the form Φ(w) = aw + b where a is a real number. For ceratin Ω, we completely describe values of a and b that induce bounded Hermitian composition operators.

In a shadow of the RH: Cyclic vectors of Hardy spaces on the Hilbert multidisc

Nikolai Nikolski (2012)

Annales de l’institut Fourier

Completeness of a dilation system ( ϕ ( n x ) ) n 1 on the standard Lebesgue space L 2 ( 0 , 1 ) is considered for 2-periodic functions ϕ . We show that the problem is equivalent to an open question on cyclic vectors of the Hardy space H 2 ( 𝔻 2 ) on the Hilbert multidisc 𝔻 2 . Several simple sufficient conditions are exhibited, which include however practically all previously known results (Wintner; Kozlov; Neuwirth, Ginsberg, and Newman; Hedenmalm, Lindquist, and Seip). For instance, each of the following conditions implies cyclicity...

Invertible and normal composition operators on the Hilbert Hardy space of a half–plane

Valentin Matache (2016)

Concrete Operators

Operators on function spaces of form Cɸf = f ∘ ɸ, where ɸ is a fixed map are called composition operators with symbol ɸ. We study such operators acting on the Hilbert Hardy space over the right half-plane and characterize the situations when they are invertible, Fredholm, unitary, and Hermitian. We determine the normal composition operators with inner, respectively with Möbius symbol. In select cases, we calculate their spectra, essential spectra, and numerical ranges.

Monge-Ampère measures and Poletsky-Stessin Hardy spaces on bounded hyperconvex domains

Sibel Şahin (2015)

Banach Center Publications

Poletsky-Stessin Hardy (PS-Hardy) spaces are the natural generalizations of classical Hardy spaces of the unit disc to general bounded, hyperconvex domains. On a bounded hyperconvex domain Ω, the PS-Hardy space H u p ( Ω ) is generated by a continuous, negative, plurisubharmonic exhaustion function u of the domain. Poletsky and Stessin considered the general properties of these spaces and mainly concentrated on the spaces H u p ( Ω ) where the Monge-Ampère measure ( d d c u ) has compact support for the associated exhaustion...

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