Search

Let $\varphi,\psi$ be the analytic self-maps of the unit ball ${\mathbb B}$, we characterize the Hilbert-Schmidt differences of two composition operator $C_\varphi$ and $C_\psi$ on weighted Bergman space $A_\alpha^2$, and give some conclusions about the topological structure of $\mathcal{C}(A_\alpha^2)$, the space of all bounded composition operators on $A_\alpha^2$ endowed with operator norm.

Let D be a bounded symmetric domain and Σ be the Shilov boundary of D. For λ∈W, the Wallach set, and a nonnegative integer l, we study the weighted Bergman space Aλ2(D) and the weighted Bergman–Sobolev space A2,λ,l(D). For 0

Thèse en cotutelle avec l'université Mohammed V Rabat-Agdal; In this thesis, we study the cyclicity problem in some spaces of analytic functions on the open unit disc. We focus our attention on Korenblum type spaces and on weighted Bergman type spaces. First, we use the technique of premeasures, introduced and developed by Korenblum in the 1970-s and the 1980-s, to give a characterization of cyclic functions in the Korenblum type spaces. In particular, we give a positive answer to a conjecture by Deninger. Second, we use the so called resolvent transform method to study the cyclicity of the one point mass singular inner function in weighted Bergman type spaces, especially with weights depending on the distance to a subset of the unit circle.; Cette thèse est consacrée à l'étude du problème de la cyclicité dans certains espaces de fonctions analytiques sur le disque unité. Nous nous intéressons aux espaces de type Bergman et aux espaces de type Korenblum. Dans la première partie, nous étudions les fonctions cycliques dans les espaces de type Korenblum en utilisant la notion des prémesures. Cette notion a été introduite et développée par B. Korenblum au début des années 1970s. En particulier, nous donnons une réponse positive à une conjecture énoncée par C. Deninger. Dans la deuxième partie, nous utilisons la méthode de la résolvante pour étudier la cyclicité des fonctions intérieures singulières associées aux mesures de Dirac dans les espaces de type Bergman à poids.

We study mapping properties of Toeplitz operators associated to a finite positive Borel measure on a bounded strongly pseudoconvex domain D in n complex variables. In particular, we give sharp conditions on the measure ensuring that the associated Toeplitz operator maps the Bergman space A^p(D) into A^r(D) with r>p, generalizing and making more precise results by Cuckovic and McNeal. To do so, we give a geometric characterization of Carleson measures and of vanishing Carleson measures of weighted Bergman spaces in terms of the intrinsic Kobayashi geometry of the domain, generalizing to this setting results obtained by Kaptanoglu for the unit ball., 36 pages

Let $\varphi_{1}$ and $\varphi_{2}$ be holomorphic self-maps of the open unit ball $\mathbb B$ in $\mathbb C^N$, $u_{1}$ and $u_{2}$ be holomorphic functions on $\mathbb B$ and let weighted composition operators $W_{\varphi_{1},u_{1}}$; $W_{\varphi_{2},u_{2}}: A^{p}_{\alpha} \to H^{\infty}_{v}$ be bounded. This paper characterizes the compactness of the difference of these operators from the weighted Bergman space $A^p_\alpha$, $0-1$, to the weighted-type space $H^\infty_v$ of holomorphic functions on $\mathbb B$ in terms of inducing symbols $\varphi_{1}$, $\varphi_{2}$, $u_{1}$ and $u_{2}$. For the case $p \gt 1$ we find an asymptotically equivalent expression to the essential norm of the operator.

Let $\mathbb D = \{ z \in \mathbb C: |z| \lt 1\}$ be the open unit disk in the complex plane $\mathbb C$, $H(\mathbb D)$ be the space of all analytic functions on $\mathbb D$, $\varphi$ be an analytic self-map of $\mathbb D$ and $u \in H(\mathbb D)$. Define operators by $DW_{\varphi,u}f = (u \cdot f \circ \varphi)'$ and $W_{\varphi,u}Df = (u \cdot f' \circ \varphi)$ for $f \in H(\mathbb D)$. In this paper we characterize bounded operators $DW_{\varphi,u}$ and $W_{\varphi,u}D$ from weighted Bergman space to Zygmund-type space, Bloch-type space and Bers-type space on the open unit disk. We also give some sufficient and necessary conditions for these operators to be compact operators in terms of inducing maps $\varphi$ and $u$.

In this paper, we study the invariant subspace and reducing subspace of the weighted Bergman space over bidisk. The minimal reducing subspace of Toeplitz operator T z N = T z 1 N z 2 N is completely described, and Beurling-type theorem of some invariant subspace of the weighted Bergman space over bidisk is also obtained.

Carleson's corona theorem is used to obtain two results on cyclicity of singular inner functions in weighted Bergman-type spaces on the unit disk. Our method of proof requires no regularity conditions on the weights.

For $\alpha>-1$, let $A^2_{\alpha}$ be the corresponding weighted Bergman space of the unit ball in $\mathbb{C}^n$. For a bounded measurable function $f$, let $T_f$ be the Toeplitz operator with symbol $f$ on $A^2_{\alpha}$. This paper describes all the functions $f$ for which $T_f$ commutes with a given $T_g$, where $g(z)=z_{1}^{L_1}... z_{n}^{L_n}$ for strictly positive integers $L_1,..., L_n$, or $g(z)=|z_1|^{s_1}... |z_n|^{s_n}h(|z|)$ for non-negative real numbers $s_1,..., s_n$ and a bounded measurable function $h$ on $[0,1)$., Comment: 18 pages

Let $\alpha \gt -1$, $U$ be the open unit disk in $\mathbb C$ and denote by $H(U)$ the set of all holomorphic functions on $U$. Let $C_\varphi$ be a composition operator induced by an analytic self-map $\varphi$ of $U$. Composition operators $C_\varphi$ on the weighted Hilbert Bergman space ${\mathcal A}^2_\alpha(U) = \big\{f \in H(U) \;|\; \int_U |f(z)|^2(1-|z|^2)^\alpha dm(z) \lt \infty \big\}$ are considered. We investigate when convergence of sequences $(\varphi_n)$ of symbols to a given symbol $\varphi$, implies the convergence of the induced composition operators. We give a necessary and sufficient condition for a sequence of Hilbert-Schmidt composition operators $(C_{\varphi_n})$ to converge in Hilbert-Schmidt norm to $C_\varphi$, and we obtain a sufficient condition for convergence in operator norm.

Let H(Dn) be the space of holomorphic functions on the unit polydisk Dn, and let , where p, q> 0, α = (α1,…,αn) with αj > -1, j =1,..., n, be the class of all measurable functions f defined on Dn such thatwhere Mp(f,r) denote the p-integral means of the function f. Denote the weighted Bergman space on . We provide a characterization for a function f being in . Using the characterization we prove the following result: Let p> 1, then the Cesàro operator is bounded on the space .

Balıkesir Üniversitesi, Fen Bilimleri Enstitüsü, Matematik Ana Bilim Dalı, Genel olarak L2(G) uzayında polinomlar ile yaklaşımı inceleyen bu çalışma beş bölümden oluşmaktadır. Birinci bölümde temel tanımlar, teoremler ve kullanılan gösterimler verilip, yaklaşımın incelendiği L2(G) uzayı tanımlanmıştır. Ayrıca bazı fonksiyon sınıflarının tanımları verilmiştir. İkinci bölümde, bir bölge üzerinden alınan Lebesgue integralin bilinen Riemann integrali yardımıyla ifade edilebileceği gösterilip L2(G) uzayının bir Hubert uzayı olduğu ispatlanmıştır. Üçüncü bölümde ortanormal ve tam ortanormal sistem tanımları verilip polinomlarla yaklaşımın mümkün olduğu bölgeler ( PA ) tanımlanıp PA özelliğine sahip bölgelere örnekler verilmiştir. Her Caratheodory bölgesinin PA özelliğine sahip olduğu ispatlandıktan sonra PA özelliği olmayan basit bağlantılı bölgelerden kesik bölge ve ayvari bölge incelenmiştir. Ayvari bölgenin PA özelliğine sahip olma ve olmama koşulları araştırılmıştır. Dördüncü bölümde Hubert uzaylarında ON sistemlere göre seri açılımı ile ilgili koşullar belirtildikten sonra özel olarak L2(G) uzayında ON seri açılımı incelenmiştir. ON sistem tam ise L2(G) uzayından bir fonksiyonun düzgün yakınsak ON seri açılımının var olduğu ispatlanmıştır. Daha sonra G de analitik bir fonksiyonun G da düzgün yakınsak ON seri açılımının varlığı, ∂G sınırının Jordan eğrisi ve ON sistemin G nin ON Pj, J=0, I, 2, 3,... polinomlar sistemi olması koşullan altında gösterilmiştir. Beşinci bölümde kvazi-konform sınıra sahip bir G bölgesinde tanımlı A2(G,ω) ağırlıklı Bergman uzayından bir fonksiyon için genelleştirilmiş Faber polinomları tanımlanıp, bunun yaklaşım özellikleri araştınlmıştır. A2(G,ω) uzayında bir fonksiyonun genelleştirilmiş Faber serisine açılımının kompakt altkümelerde düzgün yakınsak olduğu ispatlanıp genelleştirilmiş Faber serisinin teklik problemi incelenmiştir. Bu bölümde son olarak bir / fonksiyonunun genelleştirilmiş Faber serisinin kısmi toplamının/ ye A2(G) normunda yaklaşım hızı değerlendirilmiştir., This work in which generally concerns approximation by polinomials in L2(G) spaces, consist of five chapters. In the first chapter, basic definitions, theorems and notations which are used through in this work are given, the L2(G) spaces in which the approximation is studied are defined. Also definitions of some function classes are given. In the second chapter, the Lebesgue integral which is taken over any domain will be able to be expressed by the help of familiar Riemann integral is demonstrated after that it is proved that L2(G) is a Hubert space. In the thirth chapter definitions of orthanormal and complete orthanormal systems are given, after that, domains in which have got possibility to approximation by polinomials ( polinomial approximation ) are defined and various examples are given about domains which have the PA property. It is proved that every Caratheodory domains to have the PA property. After that simply connected Slit and Moon-shaped domains which have not PA property, are investigated. Conditions on the PA property of the moon-shaped domains that whether they have or not, are investigated, respectively. In the fourth chapter, first, conditions on expansion with respect to ON systems for functions in L2(G) spaces are investigated. If ON system is complete then for any function in L2(G) spaces, existence of uniformly convergent series expansion with respect to this system is proved. After that, for any function which is analitic in G, under the circumstences that ∂G boundary is Jordan curve and Pj, j=0, /, 2, 3,... is ON polinomials of domain G, existence of uniformly convergent series expansion with respect to this ON system are demonstrated. In the fifth chapter, the generalized Faber polinomials for the functions in A2(G,ω) weighted Bergman spaces are defined and their approximation properties are investigated. For any function in A2(G,ω) the expansion of generalized Faber series have been uniformly convergent on compact subsets of the domain G is proved and uniqueness problem of the generalized Faber series expansion is investigated. Finally in this chapter, for partial sum of the generalized Faber series of any function / rate of convergence to/with the A2(G) norm is evaluated.

We prove a new sharp inequality for norms in weighted Bergman space. This inequality is then used to derive isoperimetric inequalities for geometric functionals which are closely related to the torsional rigidity of a simply connected domain.

Balıkesir Üniversitesi, Fen Bilimleri Enstitüsü, Matematik Ana Bilim Dalı, Bu çalışma üç bölümden oluşmaktadır. Birinci bölümde, önce ilerideki bölümlerde kullanılacak olan temel tanımlar, gösterimler ve teoremler verilmiş, daha sonra yaklaşımın incelendiği ağırlıklı Bergman uzayları tanımlanmış, yaklaşımın derecesi ve en iyi yaklaşım polinomu ile ilgili gereken bilgiler verilmiştir. Bölümün son kısmında, yaklaşım teorisinde kvazikonform dönüşümler teorisinin kullanılması ile yeni uygulama boyutu kazanan kvazikonform eğriler ve kvazikonform yansımalar tanımlanmıştır. İkinci bölümde yaklaşım teorisinde yaklaşan polinomlann inşası için vazgeçilmez bir obje haline gelmiş olan Faber polinomları ve genelleştirilmiş Faber polinomlan incelenmiştir. İlk önce, Faber ve genelleştirilmiş Faber polinomlanmn asimptotik özellikleri araştırılmıştır. Bu bölümün son kısmında Faber serileri ve genelleştirilmiş Faber serilerinin yaklaşım özellikleri, karmaşık düzlemin basit bağlantılı bölgelerinde incelenmiştir. Son bölümde, kvazikonform sınırlı bölgelerde geçerli olan, analitik fonksiyonların bölge üzerinden bir integral gösteriminin doğurduğu genelleştirilmiş Faber serilerinin yaklaşım özellikleri incelenmiştir. Bu bölümde önce bazı gerekli yardımcı sonuçlar elde edilmiştir. Daha sonra ağırlıklı Bergman uzaylarında genelleştirilmiş Faber serilerinin yaklaşım özellikleri ve teklik problemleri incelenmiş ve yaklaşım hatası değerlendirilmiştir., This work consists of three chapters. In the first chapter, basic definitions, notations and theorems which are used in the following two chapters are given. After that, the weighted Bergman spaces, in which the approximation is investigated, are defined, and the necessary information about the degree of approximation and the best approximant polynomial are given. At the final part of the chapter, quasiconformal curves and quasiconformal reflections used in the approximation theory by means of quasiconformal mapping theory are given. In the second chapter, Faber polynomials and generalized Faber polynomials which has been used without alternative in the construction of approximant polynomials in approximation theory are investigated. Firstly, asymptotic properties of Faber and generalized Faber polynomials are investigated. Secondly, approximation properties of Faber and the generalized Faber series are considered in the simply connected regions in the complex plane. In the final chapter, the approximation properties of generalized Faber series which are generated by an integral representation over a region, which is valid for bounded regions with quasiconformal boundary are investigated. Firstly, some necessary lemmas are obtained and then the approximation properties and uniqueness problems of the generalized Faber series and the error of approximation in the weighted Bergman spaces are given.

We study the action and properties of a differential operator in the polydisk, extending some classical results from the unit disk. Using so called dyadic decomposition of the polydisk we find precise connections between quazinorms of holomorphic function in the polydisk with quazinorms on the subframe and the unit disk. All our results were previously well-known in the unit disk.