Plan Ekaterinoslavskoĭ gubernii Krivorozhskogo okruga Mikhaĭlovskogo raĭona na uchastok zemli otgranichennoĭ v pori︠a︡dke ustroĭstva kolonizatsionnogo fonda

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Karta raspolozhenii︠a︡ zemelʻ Evreǐskiikh koloniiǐ kollektivov i uchastkov kolfondov v raǐonakh Mikhaǐlovskom, Sofievskom i Shirokovskom Krivorogskogo okruga

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Karta raspolozhenii︠a︡ zemelʻ Evropeĭskikh koloniĭ kollektivov i uchastkov kolfondov v raĭonokh Mikhaĭlovskom, Sofievskom i Shirokovskom, Krivorogskogo okruga

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Plan Kryvorizʹkoĭ okrugi, Shyrokivsʹkogo raĭonu, Sverdlovsʻkoĭ silʻrady na uchastok zemli zemelʹnoï gromady vys. U 1 (Voroshilosʻkyĭ)z dorizannym do nʻogo uchaskom zemli Kolfonda Y. 52oskolfonda Nº 35 otgranichennogo v pori︠a︡dke ustroĭstva kolonizat︠s︡ionnogo fonda

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Composition operators, Aleksandrov measures and value distribution of analytic maps in the unit disc

A composition operator is a linear operator that precomposes any given function with another function, which is held fixed and called the symbol of the composition operator. This dissertation studies such operators and questions related to their theory in the case when the functions to be composed are analytic in the unit disc of the complex plane. Thus the subject of the dissertation lies at the intersection of analytic function theory and operator theory. The work contains three research articles. The first article is concerned with the value distribution of analytic functions. In the literature there are two different conditions which characterize when a composition operator is compact on the Hardy spaces of the unit disc. One condition is in terms of the classical Nevanlinna counting function, defined inside the disc, and the other condition involves a family of certain measures called the Aleksandrov (or Clark) measures and supported on the boundary of the disc. The article explains the connection between these two approaches from a function-theoretic point of view. It is shown that the Aleksandrov measures can be interpreted as kinds of boundary limits of the Nevanlinna counting function as one approaches the boundary from within the disc. The other two articles investigate the compactness properties of the difference of two composition operators, which is beneficial for understanding the structure of the set of all composition operators. The second article considers this question on the Hardy and related spaces of the disc, and employs Aleksandrov measures as its main tool. The results obtained generalize those existing for the case of a single composition operator. However, there are some peculiarities which do not occur in the theory of a single operator. The third article studies the compactness of the difference operator on the Bloch and Lipschitz spaces, improving and extending results given in the previous literature. Moreover, in this connection one obtains a general result which characterizes the compactness and weak compactness of the difference of two weighted composition operators on certain weighted Hardy-type spaces.

Karta Krivorozhskogo okryga

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Map of the Kryvyĭ Rih area

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Negative photo map of Jewish settlements in the Krivoy Rog district

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Schematicheskai︠a︡ karta chasti territorii Zaporozhskogo i Mariupolʻskogo okrugov s ukazaniem evreǐskikh zemelʻpolʻzovaniǐ i gosfondov v blizi nikh

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Schematicheskai︠a︡ karta chasti territorii Zaporozhskogo i Mariupolʻskogo okrugov s ukazaniem evreǐskikh zemelʻpolʻzovaniǐ i gosfondov v blizi nikh

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Planting schema of the Novai︠a︡ Zari︠a︡ settlement

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Schematicheskiǐ plan otvoda zemli kollektiva "Novyǐ Putʻ"

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Plan zemelʻnogo otvoda kollektiva "Novyǐ Putʻ" Krivorogskogo okruga Mikhaǐlovskogo raǐona

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Plan zemelʻnogo otvoda kolfonda "Novai︠a︡ Zhiznʻ" Krivorogskogo okruga Mikhaĭlovskogo raĭona ploshchadi︠u︡ 303 desi︠a︡tin

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