Open Access Published by De Gruyter Open Access

Volume 5 Issue 1

April 2018

Issue of Concrete Operators

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Contents
Journal Overview

Special Issue on 13th Advanced Courses in Operator Theory and Complex Analysis

Open Access April 28, 2018

On a class of shift-invariant subspaces of the Drury-Arveson space

Nicola Arcozzi, Matteo Levi

Page range: 1-8

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Abstract

In the Drury-Arveson space, we consider the subspace of functions whose Taylor coefficients are supported in a set Y⊂ ℕ d with the property that ℕ\X + e j ⊂ ℕ\X for all j = 1, . . . , d. This is an easy example of shift-invariant subspace, which can be considered as a RKHS in is own right, with a kernel that can be explicitly calculated for specific choices of X. Every such a space can be seen as an intersection of kernels of Hankel operators with explicit symbols. Finally, this is the right space on which Drury’s inequality can be optimally adapted to a sub-family of the commuting and contractive operators originally considered by Drury.

Open Access April 28, 2018

Hardy spaces of generalized analytic functions and composition operators

Elodie Pozzi

Page range: 9-23

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Abstract

We present some recent results on Hardy spaces of generalized analytic functions on D specifying their link with the analytic Hardy spaces. Their definition can be extended to more general domains Ω . We discuss the way to extend such definitions to more general domains that depends on the regularity of the boundary of the domain ∂Ω. The generalization over general domains leads to the study of the invertibility of composition operators between Hardy spaces of generalized analytic functions; at the end of the paper, we discuss invertibility and Fredholm property of the composition operator C ∅ on Hardy spaces of generalized analytic functions on a simply connected Dini-smooth domain for an analytic symbol ∅.

Open Access May 24, 2018

Iteration of Composition Operators on small Bergman spaces of Dirichlet series

Jing Zhao

Page range: 24-34

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Abstract

The Hilbert spaces ℋw consisiting of Dirichlet series F(s)=∑n=1∞ann-s$F(s) = \sum\nolimits_{n = 1}^\infty {{a_n}{n^{ - s}}}$ that satisfty ∑n=1∞|an|2/wn<∞${\sum\nolimits_{n = 1}^\infty {\left| {{a_n}} \right|} ^2}/{w_n} < \infty $ with { wn } n of average order log j n (the j -fold logarithm of n ), can be embedded into certain small Bergman spaces. Using this embedding, we study the Gordon–Hedenmalm theorem on such ℋw from an iterative point of view. By that theorem, the composition operators are generated by functions of the form Φ ( s ) = c 0 s + ϕ ( s ), where c 0 is a nonnegative integer and ϕ is a Dirichlet series with certain convergence and mapping properties. The iterative phenomenon takes place when c 0 = 0. It is verified for every integer j ⩾ 1, real α > 0 and { wn } n having average order (logj+n)α${(\log _j^ + n)^\alpha }$, that the composition operators map ℋw into a scale of ℋw ’ with w ’ n having average order (logj+1+n)α${(\log _{j + 1}^ + n)^\alpha }$. The case j = 1 can be deduced from the proof of the main theorem of a recent paper of Bailleul and Brevig, and we adopt the same method to study the general iterative step.

Open Access November 21, 2018

The Distribution Function for a Polynomial

Joseph A. Cima, William Derrick

Page range: 35-41

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Abstract

This paper explores the continuity and differentiability properties for the distribution function for a polynomial

Open Access December 31, 2018

A note on bi-contractive projections on spaces of vector valued continuous functions

Fernanda Botelho, T.S.S.R.K. Rao

Page range: 42-49

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Abstract

This paper concerns the analysis of the structure of bi-contractive projections on spaces of vector valued continuous functions and presents results that extend the characterization of bi-contractive projections given by the first author. It also includes a partial generalization of these results to affine and vector valued continuous functions from a Choquet simplex into a Hilbert space.

About this journal

Concrete Operators is a fully peer-reviewed open access journal publishing articles devoted to all special classes of linear operators acting on function spaces. This includes, but is not limited to, Perron-Frobenius operators, multiplication operators, operators in reproducing Kernel Hilbert spaces, weighted composition operators, integral operators, kernel operators, Toeplitz operators, Hankel operators, Wiener-Hopf operators, Jacobi operators, weighted shift operators, operators on function spaces.