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.

Below-ground processes in meadow soil under elevated ozone and carbon dioxide : - Greenhouse gas fluxes, N cycling and microbial communities

This thesis focuses on how elevated CO2 and/or O3 affect the below-ground processes in semi-natural vegetation, with an emphasis on greenhouse gases, N cycling and microbial communities. Meadow mesocosms mimicking lowland hay meadows in Jokioinen, SW Finland, were enclosed in open-top chambers and exposed to ambient and elevated levels of O3 (40-50 ppb) and/or CO2 (+100 ppm) for three consecutive growing season, while chamberless plots were used as chamber controls. Chemical and microbiological analyses as well as laboratory incubations of the mesocosm soils under different treatments were used to study the effects of O3 and/or CO2. Artificially constructed mesocosms were also compared with natural meadows with regards to GHG fluxes and soil characteristics. In addition to research conducted at the ecosystem level (i.e. the mesocosm study), soil microbial communities were also examined in a pot experiment with monocultures of individual species. By comparing mesocosms with similar natural plant assemblage, it was possible to demonstrate that artificial mesocosms simulated natural habitats, even though some differences were found in the CH4 oxidation rate, soil mineral N, and total C and N concentrations in the soil. After three growing seasons of fumigations, the fluxes of N2O, CH4, and CO2 were decreased in the NF+O3 treatment, and the soil NH4+-N and mineral N concentrations were lower in the NF+O3 treatment than in the NF control treatment. The mesocosm soil microbial communities were affected negatively by the NF+O3 treatment, as the total, bacterial, actinobacterial, and fungal PLFA biomasses as well as the fungal:bacterial biomass ratio decreased under elevated O3. In the pot survey, O3 decreased the total, bacterial, actinobacterial, and mycorrhizal PLFA biomasses in the bulk soil and affected the microbial community structure in the rhizosphere of L. pratensis, whereas the bulk soil and rhizosphere of the other monoculture, A. capillaris, remained unaffected by O3. Elevated CO2 caused only minor and insignificant changes in the GHG fluxes, N cycling, and the microbial community structure. In the present study, the below-ground processes were modified after three years of moderate O3 enhancement. A tentative conclusion is that a decrease in N availability may have feedback effects on plant growth and competition and affect the N cycling of the whole meadow ecosystem. Ecosystem level changes occur slowly, and multiplication of the responses might be expected in the long run.