Let D be the open unit disk in the complex plane C, H(D) the class of all analytic functions on D and φ an analytic self-map of D. In order to unify the products of composition, multiplication, and differentiation operators, Stević and Sharma introduced the following so-called Stević-Sharma operator on H(D): Tψ1,ψ2,φf(z) = ψ1(z)f(φ(z)) + ψ2(z)f ′(φ(z)), where ψ1, ψ2 ∈ H(D). By constructing some...

We give a short direct proof of the Agler and Nevanlinna factorization theorems that uses the Blecher-Ruan-Sinclair characterization of operator algebras. The key ingredient of this proof is an operator algebra factorization theorem. Our proof provides some additional information about these factorizations in the case of polynomials.

and Applied Analysis 3 Now we recall definitions and some properties of the Smirnov class, the Privalov class, the Bergman-Privalov class, and the Zygmund F-algebra on Bn or D. The space of all holomorphic functions on X Bn or D is denoted by H X . For each 0 < p ≤ ∞, the Hardy space is denoted by H X with the norm ‖ · ‖p. 2.1. Smirnov Class N∗ X Let X ∈ {Bn, Dn}. The Nevanlinna class N X on X ...

This is an expanded version of one of the Lectures in memory of Lars Ahlfors in Haifa in 1996. Some mistakes are corrected and references added. This article is an exposition for non-specialists of Ahlfors’ work in the theory of meromorphic functions. When the domain is not specified we mean meromorphic functions in the complex plane C. The theory of meromorphic functions probably begins with t...

is the Nevanlinna characteristic of f [13]. Meromorphic functions of finite order have been extensively studied and they have numerous applications in pure and applied mathematics, e.g. in linear differential equations. In many applications a major role is played by the logarithmic derivative of meromorphic functions and we need to obtain sharp estimates for the logarithmic derivative as we app...

The main objective is to derive a lower bound from an upper one for harmonic functions in the half space, which extends a result of B. Y. Levin from dimension 2 to dimension n ≥ 2. To this end, we first generalize the Carleman’s formula for harmonic functions in the half plane to higher dimensional half space, and then establish a Nevanlinna’s representation for harmonic functions in the half s...

Given a set of H∞ design specifications, the issue is to check whether there exists a controller, whose order is free, which satisfies these specifications. The classical solution, which is based on Youla parametrisation and convex closed loop design, is not really satisfactory since it should use an infinite dimensional basis of filters, which cannot be done in practice. Let J∗ the minimal val...

Let k be the reporducing kernel for a Hilbert space H(k) of nanlytic functions on Bd, the open unit ball in C, d ≥ 1. k is called a complete NP kernel, if k0 ≡ 1 and if 1 − 1/kλ(z) is positive definite on Bd × Bd. Let D be a separable Hilbert space, and consider H(k) ⊗ D ∼= H(k,D), and think of it as a space of D-valued H(k)-functions. A theorem of McCullough and Trent, [10], partially extends ...

The Nevanlinna–Pick interpolation problem is studied in the class of functions defined on the unit disk without a discrete set, with the property that all their Pick matrices have not more than a prescribed number of negative eigenvalues. It is shown, in particular, that the degenerate problem always has a unique solution, not necessarily meromorphic. A related extension problem to a maximal fu...