# Different Types of Weighted Composition Operators on Banach Spaces of Analytic Functions

##### Norrbo, David (2023-05-26)

Norrbo, David

Åbo Akademi - Åbo Akademi University

26.05.2023

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https://urn.fi/URN:ISBN:978-952-12-4290-8

##### Abstract

Many problems in real life are, at least approximatively, of linear nature and can be mathematically examined with the aid of linear spaces. It is natural to measure the size of objects occurring in the problems and such an operation is called a norm. If the space and the norm fit well together, they constitute a Banach space. A norm associates every vector with a nonnegative number or infinity, and the Banach space consists of those vectors whose associated number is finite. Different norms give rise to different Banach spaces.

In this dissertation, which contains three articles, different types of weighted composition operators on Banach spaces, consisting of analytic functions defined on the unit disc of the complex plane, are examined. Since the vectors are functions, there are two basic linear operations to consider. One way to modify the vector is by multiplying it with another function. Such an operator is said to be a multiplication operator if it is well defined. Another way to modify the vector is to first transform the input via a function and make the original vector act on the modified input. Such a transformation is done by a composition operator, since the resultant vector is a composition of the original vector and the function transforming the input. A combination of a multiplication operator and a composition operator is said to be a weighted composition operator.

In one of the articles, a certain class of integral operators on weighted Bergman spaces are examined. The exact value of the essential norm of such operators, which can be represented as a mean of weighted composition operators, is calculated. Another article deals with the connection between some operator-theoretic properties of a weighted composition operator on the Banach space BMOA and the behaviour of corresponding functions. Compactness, weak compactness and complete continuity are examined. In the so far not mentioned article, the spectrum and essential spectrum are determined for multiplication operators on some Banach spaces.

In this dissertation, which contains three articles, different types of weighted composition operators on Banach spaces, consisting of analytic functions defined on the unit disc of the complex plane, are examined. Since the vectors are functions, there are two basic linear operations to consider. One way to modify the vector is by multiplying it with another function. Such an operator is said to be a multiplication operator if it is well defined. Another way to modify the vector is to first transform the input via a function and make the original vector act on the modified input. Such a transformation is done by a composition operator, since the resultant vector is a composition of the original vector and the function transforming the input. A combination of a multiplication operator and a composition operator is said to be a weighted composition operator.

In one of the articles, a certain class of integral operators on weighted Bergman spaces are examined. The exact value of the essential norm of such operators, which can be represented as a mean of weighted composition operators, is calculated. Another article deals with the connection between some operator-theoretic properties of a weighted composition operator on the Banach space BMOA and the behaviour of corresponding functions. Compactness, weak compactness and complete continuity are examined. In the so far not mentioned article, the spectrum and essential spectrum are determined for multiplication operators on some Banach spaces.

##### Collections

- 111 Mathematics [12]