A Gleason Solution Model for Row Contractions

Let ${\cal H}(b)$ denote the de Branges--Rovnyak space associated with a function $b$ in the unit ball of $H^\infty({\Bbb C}_+)$. We study the boundary behavior of the derivatives of functions in ${\cal H}(b)$ and obtain weighted norm estimates of the form $\|f^{(n)}\|_{L^2(\mu)} \le C\|f\|_{{\cal H}(b)}$, where $f \in {\cal H}(b)$ and $\mu$ is a Carleson-type measure on ${\Bbb C}_+\cup{\Bbb R}$. We provide several applications of these inequalities. We apply them to obtain embedding theorems for ${\cal H}(b)$ spaces. These results extend Cohn and Volberg--Treil embedding theorems for the model (star-invariant) subspaces which are special classes of de Branges--Rovnyak spaces. We also exploit the inequalities for the derivatives to study stability of Riesz bases of reproducing kernels $\{k^b_{\lambda_n}\}$ in ${\cal H}(b)$ under small perturbations of the points $\lambda_n$.

A quantitative majorization-subordination result of Goluzin and Tao Shah for univalent functions is generalized to H", the linear invariant family of locally univalent functions of finite order or.If f(z) is subordinate to F(z) in the open unit disc, /'(0) > 0, and F(z) is in U" 1.65 < a < oo, then f'(z) is majorized by F'(z) in \z\ < (a + 1) -(a2 + 2a)vi.The result is sharp.I. Introduction.Let <S denote the set of all normalized analytic univalent functions in the open unit disc D. Let/(z), F(z) and <p(z) be analytic in |z| < r.We say that/(z) is majorized byLet tla be the set of all locally univalent (f'(z) = 0) analytic functions in D with order < a which are of the form f(z) = z A-.The family U" is known as the universal linear invariant family of order a [4].A concise summary and introduction to properties of linear invariant families which relate to the following material is contained in [1].The present paper concludes the proof of results announced in [1].Majorization-subordination theory begins with Biernacki who showed in 1936 that if/(z) is subordinate in D to F(z) (F(z) G <S), then/(z) is majorized by F(z) in |z| < 1/4.In the succeeding years Goluzin, Tao Shah, Lewandowski and MacGregor examined various related problems but always under the stipulation that the dominant function F(z) is in <S (for greater detail see [1]).In 1951 Goluzin showed that if f(z) is majorized by a univalent function F(z), then f'(z) would be majorized by F'(z) in |z| < 0.12.He conjectured that majorization would always occur for jz| < 3 -y and this was proved by Tao Shah in 1958.In this paper we show that the result is actually true for functions in 11. and obtain the sharp radius of majorization as a + 1 -(a2 + 2a)^2 for 1.65 < o < oo.This yields 3 -y for the case a = 2.Our investigation shows that the important datum for majorization-subordination theory is not univalence, but the order of a linear invariant family.In particular, many classically derived estimates for univalent functions are true for functions of infinite valence.

We study the reproducing kernel Hilbert spaces \(\mathfrak{H}(\mathbb{D}^2 ,{\text{ }}S)\) with kernels of the form $$\frac{{I - S(z_1 ,z_2 > )S(w_1 ,w_2 )^* }}{{(1 - z_1 w_1^* )(1 - z_2 w_2^* )}}$$ where S(z1,z2) is a Schur function of two variables z1,z2l\(\mathbb{D}\). They are analogs of the spaces \(\mathfrak{H}\left( {\mathbb{D},S} \right)\) with reproducing kernel (1-S(z)S(w)*)/(1-zw*) introduced by de Branges and Rovnyak l. de Branges and J. Rovnyak, Square Summable Power Series Holt, Rinehart and Winston, New York, 1966. We discuss the characterization of \(\mathfrak{H}(\mathbb{D}^2 ,{\text{ }}S)\) as a subspace of the Hardy space on the bidisk. The spaces \(\mathfrak{H}(\mathbb{D}^2 ,{\text{ }}S)\) form a proper subset of the class of the so–called sub–Hardy Hilbert spaces of the bidisk.

This work presents a contemporary treatment of Krein's entire operators with deficiency indices $(1,1)$ and de Branges' Hilbert spaces of entire functions. Each of these theories played a central role in the research of both renown mathematicians. Remarkably, entire operators and de Branges spaces are intimately connected and the interplay between them has had an impact in both spectral theory and the theory of functions. This work exhibits the interrelation between Krein's and de Branges' theories by means of a functional model and discusses recent developments, giving illustrations of the main objects and applications to the spectral theory of difference and differential operators.

This study focuses on deriving Fekete-Szegö inequalities for a specific subclass of univalent functions with complex order, related to quasi-subordination. It investigates the geometric properties of these functions and the conditions for subordination between analytic functions within the open unit disk. The research also discusses generalized starlikeness and convexity as examples of subordination and explores the role of analytic Schwarz functions in this context. New results are provided for superior functions of various orders, contributing to the field of geometric function theory. The findings offer insights into quasi-subordination and its practical applications in pure mathematics, advancing the understanding of function behavior under subordination constraints.

In this article, we introduce a general family of analytic and bi-univalent functions in the open unit disk, which is defined by applying the principle of differential subordination between analytic functions and the Tremblay fractional derivative operator. The upper bounds for the general coefficients | a n | of functions in this subclass are found by using the Faber polynomial expansion. We have thereby generalized and improved some of the previously published results.

The goal of this paper is to investigate some of the features of differential subordination of analytic univalent functions in an open unit disc. In addition, it has shed light on geometric features such as coefficient inequality, Hadamard product qualities, and the Komatu integral operator. Some intriguing results for third-order differential subordination and superordination of analytic univalent functions have been installed. Then, using the convolution of two linear operators, certain results of third order differential subordination involving linear operators were reported. As a result, we use features of the Komatu integral operator to analyze and study third-order subordinations and superordinations in relation to the convolution. Finally, several results for third order differential subordination in the open unit disk using generalized hypergeometric function have been addressed using the convolution operator.

Let Ω denote the class of functions f ( z ) = z + a 2 z 2 + a 3 z 3 + ⋯ belonging to the normalized analytic function class A in the open unit disk U = z : z &lt; 1 , which are bi-univalent in U , that is, both the function f and its inverse f − 1 are univalent in U . In this paper, we introduce and investigate two new subclasses of the function class Ω of bi-univalent functions defined in the open unit disc U , which are associated with a new differential operator of analytic functions involving binomial series. Furthermore, we find estimates on the Taylor–Maclaurin coefficients | a 2 | and | a 3 | for functions in these new subclasses. Several (known or new) consequences of the results are also pointed out.

Let the differential system where D is the diagonal matrix and the elements αij (Z) of the matrix A(z) are analytic functions in some neighborhood of the closed unit disc. In this paper under several assumptions with respect to the constant matrices and the diagonal D, it is proved that the conjugate system of (1) has exactly k(p − l) linearly independent solutions in the product space: where is the usual Hilbert space of analytic functions in the open unit disc.

Let 𝔻 be the open unit disk in the complex plane ℂ and \(dA(z) = \frac {1}{\pi }dx dy\), the normalized area measure on 𝔻. Let \({L_{a}^{2}}(\mathbb {D}, dA)\) be the Bergman space consisting of analytic functions on 𝔻 that are also in \(L^{2}(\mathbb {D}, dA)\). Let \(\mathcal {L}({L_{a}^{2}}(\mathbb {D}))\) be the set of all bounded linear operators from the Hilbert space \({L_{a}^{2}}(\mathbb {D})\) into itself. For \(T\in \mathcal {L}({L_{a}^{2}}(\mathbb {D}))\), let \(\widetilde {T}\) denote the Berezin transform of T. In this paper, we find conditions on positive operators \(S, T\in \mathcal {L}({L_{a}^{2}}(\mathbb {D}))\) such that \(\widetilde {S}(z)\geq \widetilde {T}(z)\) for all z ∈ 𝔻.

Trace ideal criteria for Toeplitz operators

A normalized analytic function f defined on the open unit disc D is called Ma-Minda convex if 1 + zf′′(z)/f′(z) is subordinate to the function φ. For 0 ⩽ α ⩽ β, the Kaplan class K(α, β) of type α and β consists of normalized analytic functions of the form p^{α}g defined on D where p with p(0) = 1 is an analytic function taking values in the right half-plane and g is an analytic function with g(0) = 1 satisfying Re(zg′(z)/g(z)) &gt; (α − β)/2. For functions f with f′ ∈ K(α, β), we obtain the radius of Ma-Minda convexity for various choices of φ. The radius of lemniscate convexity, lune convexity, nephroid convexity, exponential convexity and several other radius estimates are examined. The results obtained are sharp.

Let f be an analytic function in the open unit disk that assumes the values 0 and 1 at least once and an unequal number of rimes in a disk of radius 1 10-5 entered at the origin. We prove that fmust assume the value 0 or 1 elsewhere in the unit disk. This result is motivated by a control engineering design problem which is also presented.

We give a controllable energy-preserving and an observable co-energy-preserving de Branges-Rovnyak functional model realization of an arbitrary given operator Schur function defined on the complex right-half plane. We work the theory out fully in the right-half plane, without using results for the disk case, in order to expose the technical details of continuous-time systems theory. At the end of the article, we make explicit the connection to the corresponding classical de Branges-Rovnyak realizations for Schur functions on the complex unit disk.

Neighborhoods of certain analytic functions