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.

Multiplication of hybrid aspen (Populus tremula L. x P. tremuloides Michx.) from cuttings

The primary aim of the present study was to find an efficient and simple method of vegetative propagation for producing large numbers of hybrid aspen (Populus tremuloides L. x P. tremula Michx.) plants for forest plantations. The key objectives were to investigate the main physiological factors that affect the ability of cuttings to regenerate and to determine whether these factors could be manipulated by different growth conditions. In addition, clonal variation in traits related to propagation success was examined. According to our results, with the stem cutting method, depending on the clone, it is possible to obtain only 1−8 plants from one stock plant per year. With the root cutting method the corresponding values for two-year-old stock plants are 81−207 plants. The difference in number of cuttings between one- and two-year-old stock plants is so pronounced that it is economically feasible to grow stock plants for two years. There is no reason to use much older stock plants as a source of cuttings, as it has been observed that rooting ability diminishes as root diameter increases. Clonal variation is the most important individual factor in propagation of hybrid aspen. The fact that the efficiently sprouted clones also rooted best facilitates the selection of clones for large-scale propagation. In practice, root cuttings taken from all parts of the root system of hybrid aspen were capable of producing new shoots and roots. However, for efficient rooting it is important to use roots smaller than one centimeter in diameter. Both rooting and sprouting, as well as sprouting rate, were increased by high soil temperature; in our studies the highest temperature tested (30ºC) was the best. Light accelerated the sprouting of root cuttings, but they rooted best in dark conditions. Rooting is essential because without roots the sprouted cutting cannot survive long. For aspen the criteria for clone selection are primarily fiber qualities and growth rate, but ability to regenerate efficiently is also essential. For large-scale propagation it is very important to find clones from which many cuttings per stock plant can be obtained. In light of production costs, however, it is even more important that the regeneration ability of the produced cuttings be high.

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.

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.

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.

Groteski suomalaisessa kirjallisuudessa : Neljä tapaustutkimusta

The grotesque in Finnish literature. Four case studies The topic of the dissertation is the grotesque in Finnish literature. The dissertation is twofold. Firstly, it focuses on the genre tradition of the grotesque, especially its other main branch, which has been named, following in Bakhtin s footsteps, subjective ( chamber ) grotesque, to be distinguished from carnivalistic ( public square ) grotesque. Secondly, the dissertation analyses and interprets four fictional literary works within the context of the grotesque genre, constructed on the basis of previous research and literature. These works are the novel Rakastunut rampa (1922) by Joel Lehtonen, the novel Prins Efflam (1953, transl. into Finnish as Kalastajakylän prinssi) by Sally Salminen, the short story Orjien kasvattaja (1965) by Juhani Peltonen, and the novel Veljeni Sebastian (1985) by Annika Idström. What connects these stirring novels, representing early or full modernism, is the supposition that they belong to the tradition of the subjective grotesque, not only due to occasional details, but also in a more comprehensive manner. The premises are that genre is a significant part of the work and that reading a novel in the context of the genre tradition adds something essential to the interpretation of individual texts and reveals meanings that might otherwise go unnoticed. The main characteristic of the grotesque is breaking the norm. This is accomplished through different means: degradation, distortion, inversion, combination, exaggeration and multiplication. The most significant strategy for breaking the norm is incongruence: the grotesque combines conflicting or mutually exclusive categories and elements on different levels. Simultaneously, the grotesque unravels categorisations and questions our way of perceiving the world. The grotesque not only poses a threat to one s identity, but can also pose a threat to the cognitive process. An analysis of the fictional works is presented as case studies of each chosen work as a whole. The analysis is based on the method of close reading, which draws on both classical and postclassical narratology, and the analysis and interpretation are expanded within the genre tradition of the grotesque. The grotesque is also analysed in terms of its relationship to the neighbouring categories and genre traditions, such as the tragic, the sublime, the horror story and the coming-of-age story. This dissertation shows how the grotesque is constructed repeatedly on deviations from the norm as well as on incongruence, also in the works analysed, and how it stratifies in these novels on and between different levels, such as the story, text, narration, composition and the world of the novels. In all the works analysed, the grotesque reduces and subverts. Again and again it reveals different sides of humanity stripped of idealisation and glorification. The dissertation reveals that Finnish literature is not a solitary island, even regarding the grotesque, for it continues and offers variations of the common tradition of grotesque literature, and likewise draws on grotesque visual arts. This dissertation is the first monograph in Finnish literature research focusing on the subjective grotesque.

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.