Composition operators on vector-valued BMOA and related function spaces

A composition operator is a linear operator between spaces of analytic or harmonic functions on the unit disk, which precomposes a function with a fixed self-map of the disk. A fundamental problem is to relate properties of a composition operator to the function-theoretic properties of the self-map. During the recent decades these operators have been very actively studied in connection with various function spaces. The study of composition operators lies in the intersection of two central fields of mathematical analysis; function theory and operator theory. This thesis consists of four research articles and an overview. In the first three articles the weak compactness of composition operators is studied on certain vector-valued function spaces. A vector-valued function takes its values in some complex Banach space. In the first and third article sufficient conditions are given for a composition operator to be weakly compact on different versions of vector-valued BMOA spaces. In the second article characterizations are given for the weak compactness of a composition operator on harmonic Hardy spaces and spaces of Cauchy transforms, provided the functions take values in a reflexive Banach space. Composition operators are also considered on certain weak versions of the above function spaces. In addition, the relationship of different vector-valued function spaces is analyzed. In the fourth article weighted composition operators are studied on the scalar-valued BMOA space and its subspace VMOA. A weighted composition operator is obtained by first applying a composition operator and then a pointwise multiplier. A complete characterization is given for the boundedness and compactness of a weighted composition operator on BMOA and VMOA. Moreover, the essential norm of a weighted composition operator on VMOA is estimated. These results generalize many previously known results about composition operators and pointwise multipliers on these spaces.

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.

Properties of Toeplitz operators on analytic function spaces : from function symbols to distributions

Toeplitz operators are among the most important classes of concrete operators with applications to several branches of pure and applied mathematics. This doctoral thesis deals with Toeplitz operators on analytic Bergman, Bloch and Fock spaces. Usually, a Toeplitz operator is a composition of multiplication by a function and a suitable projection. The present work deals with generalizing the notion to the case where the function is replaced by a distributional symbol. Fredholm theory for Toeplitz operators with matrix-valued symbols is also considered. The subject of this thesis belongs to the areas of complex analysis, functional analysis and operator theory. This work contains five research articles. The articles one, three and four deal with finding suitable distributional classes in Bergman, Fock and Bloch spaces, respectively. In each case the symbol class to be considered turns out to be a certain weighted Sobolev-type space of distributions. The Bergman space setting is the most straightforward. When dealing with Fock spaces, some difficulties arise due to unboundedness of the complex plane and the properties of the Gaussian measure in the definition. In the Bloch-type spaces an additional logarithmic weight must be introduced. Sufficient conditions for boundedness and compactness are derived. The article two contains a portion showing that under additional assumptions, the condition for Bergman spaces is also necessary. The fifth article deals with Fredholm theory for Toeplitz operators having matrix-valued symbols. The essential spectra and index theorems are obtained with the help of Hardy space factorization and the Berezin transform, for instance. The article two also has a part dealing with matrix-valued symbols in a non-reflexive Bergman space, in which case a condition on the oscillation of the symbol (a logarithmic VMO-condition) must be added.

Boundedness of weakly singular integral operators on domains

The monograph dissertation deals with kernel integral operators and their mapping properties on Euclidean domains. The associated kernels are weakly singular and examples of such are given by Green functions of certain elliptic partial differential equations. It is well known that mapping properties of the corresponding Green operators can be used to deduce a priori estimates for the solutions of these equations. In the dissertation, natural size- and cancellation conditions are quantified for kernels defined in domains. These kernels induce integral operators which are then composed with any partial differential operator of prescribed order, depending on the size of the kernel. The main object of study in this dissertation being the boundedness properties of such compositions, the main result is the characterization of their Lp-boundedness on suitably regular domains. In case the aforementioned kernels are defined in the whole Euclidean space, their partial derivatives of prescribed order turn out to be so called standard kernels that arise in connection with singular integral operators. The Lp-boundedness of singular integrals is characterized by the T1 theorem, which is originally due to David and Journé and was published in 1984 (Ann. of Math. 120). The main result in the dissertation can be interpreted as a T1 theorem for weakly singular integral operators. The dissertation deals also with special convolution type weakly singular integral operators that are defined on Euclidean spaces.

Expert Identity in Development of Core-Task-Oriented Working Practices for Mastering Demanding Situations

The point of departure in this dissertation was the practical safety problem of unanticipated, unfamiliar events and unexpected changes in the environment, the demanding situations which the operators should take care of in the complex socio-technical systems. The aim of this thesis was to increase the understanding of demanding situations and of the resources for coping with these situations by presenting a new construct, a conceptual model called Expert Identity (ExId) as a way to open up new solutions to the problem of demanding situations and by testing the model in empirical studies on operator work. The premises of the Core-Task Analysis (CTA) framework were adopted as a starting point: core-task oriented working practices promote the system efficiency (incl. safety, productivity and well-being targets) and that should be supported. The negative effects of stress were summarised and the possible countermeasures related to the operators' personal resources such as experience, expertise, sense of control, conceptions of work and self etc. were considered. ExId was proposed as a way to bring emotional-energetic depth into the work analysis and to supplement CTA-based practical methods to discover development challenges and to contribute to the development of complex socio-technical systems. The potential of ExId to promote understanding of operator work was demonstrated in the context of the six empirical studies on operator work. Each of these studies had its own practical objectives within the corresponding quite broad focuses of the studies. The concluding research questions were: 1) Are the assumptions made in ExId on the basis of the different theories and previous studies supported by the empirical findings? 2) Does the ExId construct promote understanding of the operator work in empirical studies? 3) What are the strengths and weaknesses of the ExId construct? The layers and the assumptions of the development of expert identity appeared to gain evidence. The new conceptual model worked as a part of an analysis of different kinds of data, as a part of different methods used for different purposes, in different work contexts. The results showed that the operators had problems in taking care of the core task resulting from the discrepancy between the demands and resources (either personal or external). The changes of work, the difficulties in reaching the real content of work in the organisation and the limits of the practical means of support had complicated the problem and limited the possibilities of the development actions within the case organisations. Personal resources seemed to be sensitive to the changes, adaptation is taking place, but not deeply or quickly enough. Furthermore, the results showed several characteristics of the studied contexts that complicated the operators' possibilities to grow into or with the demands and to develop practices, expertise and expert identity matching the core task. They were: discontinuation of the work demands, discrepancy between conceptions of work held in the other parts of organisation, visions and the reality faced by the operators, emphasis on the individual efforts and situational solutions. The potential of ExId to open up new paths to solving the problem of the demanding situations and its ability to enable studies on practices in the field was considered in the discussion. The results were interpreted as promising enough to encourage the conduction of further studies on ExId. This dissertation proposes especially contribution to supporting the workers in recognising the changing demands and their possibilities for growing with them when aiming to support human performance in complex socio-technical systems, both in designing the systems and solving the existing problems.

Agronomic challenges for organic crop husbandry

With respect to resource management and environmental impact, organic farming offers rationales for agricultural sustainability. However, agronomic productivity is usually higher with conventional farming. This work aimed at investigating two factors of major importance for the agronomic productivity of organic crop husbandry, nitrogen (N) supply through symbiotic N fixation (SNF) and weed occurrence. Perennial red clover-grass leys and spring cereal crops subjected to regular agricultural practices were studied on 34 organic farms located in the southern and the north-western coastal regions of Finland. Herbage growth, clover content as a proportion of the ley and extent of SNF in perennial leys, and the occurrence of weed species and weed-crop competition in spring cereal stands were related to climate conditions, soil properties, and management measures. The herbage accumulated from the first and the second cut of one- and two-year-old leys averaged 7.5 t DM ha-1 (SD ± 1.7 t DM ha-1); the clover content averaged 43.9% (SD ± 18.8%). Along with the clover content, herbage production decreased with ley age. Radiation use efficiency (RUE) correlated positively with clover proportion but despite low clover contents, three-year-old leys were still productive with regard to RUE. SNF in the accumulated annual growth of one- and two-year-old leys averaged 247.5 kg N ha-1 yr-1 (SD ± 114.4 kg N ha-1 yr-1). It was supposed that if red clover-grass leys constituted 40% of the rotation, then the mean N supply by SNF would be able to sustain two or three succeeding cereal crops (green manure and forage ley, respectively), yielding 3.0 to 4.0 t grain ha-1. Being a function of clover biomass, the SNF increased from the first to the second cut and thereafter declined with ley age. Coefficients of variation of clover contents (and SNF) between and within fields were around 50%, which was about twice as high as those of herbage production. The lower were the clover contents, the higher were the within-field variations of clover as a proportion of the ley. Low clover contents in one-year-old leys and increasing variability with ley age suggested that red clover growth was limited by poor establishment and poor overwintering. The proportions of clover in leys were lower and their variability was higher in the northwest than in the south. Soil properties, primarily texture and structure, had a major impact on clover proportion and herbage production, which largely explained regional differences in ley growth. Within-field variability of soil properties can be amended through site-specific measures, including drainage, liming, and applications of organic manures and mineral fertilizers. Overwintering and the persistence of leys can be improved by the choice of winter-hardy varieties, careful establishment and the appropriate harvest regime. Mean grain yields of spring cereal crops amounted to 3.2 t ha-1 in the south and 3.6 t ha-1 in the northwest. At 570 and 565 m-2 for the south and northwest respectively, mean weed densities did not differ between the regions, whereas the respective mean weed biomass of 697 and 1594 kg dry weight ha-1, respectively did differ. Weed abundance varied remarkably between single fields. The number of weed species was higher in the south than in the northwest. For example, Fumaria officinalis and Lamium spp. were found only in the south. Frequencies and abundances of Lapsana communis, Myosotis arvensis, Polygonum aviculare, Tripleurospermum inodorum, and Vicia spp. were higher in the south, whereas those of Elymus repens, Persicaria spp. and Spergula arvensis were higher in the northwest. The number of years since conversion to organic farming, i.e. long-term management, was one of the variables that explained the abundance of single weed species. E. repens was the weed species whose biomass increased most with the duration of organic farming. Another significant variable was crop biomass, which was affected by short-term management. The presence of different weed species was related to the duration of organic farming and to low crop yield. This finding demonstrated that it was not the organic farming regime per se, which resulted in high weed infestation and low yielding crops, but failures in the understanding and the management of organic farming systems. Successful weed control relies on farm- and field-specific long- and short-term management approaches. The agronomic productivity of ley and spring cereal crops managed by full-time farmers with an interest in organic farming was on the same level as of the mean for conventional farming. Given the many options for further improvements of the agronomic performance of organic arable systems, organic farming offers foundations for the development of sustainable agriculture. The main threat to the sustainability of farming in Finland, both conventional and organic, is the spatial separation of crop production and animal husbandry by region, along with the simplification of associated crop rotations.

Boundedness and convergence of singular integrals on fractal type sets

The topic of this dissertation lies in the intersection of harmonic analysis and fractal geometry. We particulary consider singular integrals in Euclidean spaces with respect to general measures, and we study how the geometric structure of the measures affects certain analytic properties of the operators. The thesis consists of three research articles and an overview. In the first article we construct singular integral operators on lower dimensional Sierpinski gaskets associated with homogeneous Calderón-Zygmund kernels. While these operators are bounded their principal values fail to exist almost everywhere. Conformal iterated function systems generate a broad range of fractal sets. In the second article we prove that many of these limit sets are porous in a very strong sense, by showing that they contain holes spread in every direction. In the following we connect these results with singular integrals. We exploit the fractal structure of these limit sets, in order to establish that singular integrals associated with very general kernels converge weakly. Boundedness questions consist a central topic of investigation in the theory of singular integrals. In the third article we study singular integrals of different measures. We prove a very general boundedness result in the case where the two underlying measures are separated by a Lipshitz graph. As a consequence we show that a certain weak convergence holds for a large class of singular integrals.