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.

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.

Nonverbal predication in Erzya : Studies on morphosyntactic variation and part of speech distinctions

This dissertation consists of four articles and an introduction. The five parts address the same topic, nonverbal predication in Erzya, from different perspectives. The work is at the same time linguistic typology and Uralic studies. The findings based on a large corpus of empirical Erzya data, which was collected using several different methods and included recordings of the spoken language, made it possible for the present study to apply, then test and finally discuss the previous theories based on cross-linguistic data. Erzya makes use of multiple predication patterns which vary from totally analytic to the morphologically very complex. Nonverbal predicate clause types are classified on the basis of propositional acts in clauses denoting class-membership, identity, property and location. The predicates of these clauses are nouns, adjectives and locational expressions, respectively. The following three predication strategies in Erzya nonverbal predication can be identified: i. the zero-copula construction, ii. the predicative suffix construction and iii. the copula construction. It has been suggested that verbs and nouns cannot be clearly distinguished on morphological grounds when functioning as predicates in Erzya. This study shows that even though predicativity must not be considered a sufficient tool for defining parts of speech in any language, the Erzya lexical classes of adjective, noun and verb can be distinguished from each other also in predicate position. The relative frequency and degree of obligation for using the predicative suffix construction decreases when moving left to right on the scale verb adjective/locative noun ( identificational statement). The predicative suffix is the main pattern in the present tense over the whole domain of nonverbal predication in Standard Erzya, but if it is replaced it is most likely to be with a zero-copula construction in a nominal predication. This study exploits the theory of (a)symmetry for the first time in order to describe verbal vs. nonverbal predication. It is shown that the asymmetry of paradigms and constructions differentiates the lexical classes. Asymmetrical structures are motivated by functional level asymmetry. Variation in predication as such adds to the complexity of the grammar. When symmetric structures are employed, the functional complexity of grammar decreases, even though morphological complexity increases. The genre affects the employment of predication strategies in Erzya. There are differences in the relative frequency of the patterns, and some patterns are totally lacking from some of the data. The clearest difference is that the past tense predicative suffix construction occurs relatively frequently in Standard Erzya, while it occurs infrequently in the other data. Also, the predicative suffixes of the present tense are used more regularly in written Standard Erzya than in any other genre. The genre also affects the incidence of the translative in uľ(ń)ems copula constructions. In translations from Russian to Erzya the translative case is employed relatively frequently in comparison to other data. This study reveals differences between the two Mordvinic languages Erzya and Moksha. The predicative suffixes (bound person markers) of the present tense are used more regularly in Moksha in all kinds of nonverbal predicate clauses compared to Erzya. It should further be observed that identificational statements are encoded with a predicative suffix in Moksha, but seldom in Erzya. Erzya clauses are more frequently encoded using zero-constructions, displaying agreement in number only.

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.

Perspectives on prescribing in nursing homes in Helsinki

Prescribing for older patients is challenging. The prevalence of diseases increases with advancing age and causes extensive drug use. Impairments in cognitive, sensory, social and physical functioning, multimorbidity and comorbidities, as well as age-related changes in pharmacokinetics and pharmacodynamics all add to the complexity of prescribing. This study is a cross-sectional assessment of all long-term residents aged ≥ 65 years in all nursing homes in Helsinki, Finland. The residents’ health status was assessed and data on their demographic factors, health and medications were collected from their medical records in February 2003. This study assesses some essential issues in prescribing for older people: psychotropic drugs (Paper I), laxatives (Paper II), vitamin D and calcium supplements (Paper III), potentially inappropriate drugs for older adults (PIDs) and drug-drug interactions (DDIs)(Paper IV), as well as prescribing in public and private nursing homes. A resident was classified as a medication user if his or her medication record indicated a regular sequence for its dosage. Others were classified as non-users. Mini Nutritional Assessment (MNA) was used to assess residents’ nutritional status, Beers 2003 criteria to assess the use of PIDs, and the Swedish, Finnish, INteraction X-referencing database (SFINX) to evaluate their exposure to DDIs. Of all nursing home residents in Helsinki, 82% (n=1987) participated in studies I, II, and IV and 87% (n=2114) participated in the study III. The residents’ mean age was 84 years, 81% were female, and 70% were diagnosed with dementia. The mean number of drugs was 7.9 per resident; 40% of the residents used ≥ 9 drugs per day, and were thus exposed to polypharmacy. Eighty percent of the residents received psychotropics; 43% received antipsychotics, and 45% used antidepressants. Anxiolytics were prescribed to 26%, and hypnotics to 28% of the residents. Of those residents diagnosed with dementia, 11% received antidementia drugs. Fifty five percent of the residents used laxatives regularly. In multivariate analysis, those factors associated with regular laxative use were advanced age, immobility, poor nutritional status, chewing problems, Parkinson’s disease, and a high number of drugs. Eating snacks between meals was associated with lower risk for laxative use. Of all participants, 33% received vitamin D supplementation, 28% received calcium supplementation, and 20% received both vitamin D and calcium. The dosage of vitamin D was rather low: 21% received vitamin D 400 IU (10 µg) or more, and only 4% received 800 IU (20 µg) or more. In multivariate analysis, residents who received vitamin D supplementation enjoyed better nutritional status, ate snacks between meals, suffered no constipation, and received regular weight monitoring. Those residents receiving PIDs (34% of all residents) more often used psychotropic medication and were more often exposed to polypharmacy than residents receiving no PIDs. Residents receiving PIDs were less often diagnosed with dementia than were residents receiving no PIDs. The three most prevalent PIDs were short-acting benzodiazepine in greater dosages than recommended, hydroxyzine, and nitrofurantoin. These three drugs accounted for nearly 77% of all PID use. Of all residents, less than 5% were susceptible to a clinically significant DDI. The most common DDIs were related to the use of potassium-sparing diuretics, carbamazepine, and codeine. Residents exposed to potential DDIs were younger, had more often suffered a previous stroke, more often used psychotropics, and were more often exposed to PIDs and polypharmacy than were residents not exposed to DDIs. Residents in private nursing homes were less often exposed to polypharmacy than were residents in public nursing homes. Long-term residents in nursing homes in Helsinki use, on average, nearly eight drugs daily. The use of psychotropic drugs in our study was notably more common than in international studies. The prevalence of laxatives equaled other prior international studies. Regardless of the known benefit and recommendation of vitamin D supplementation for elderly residing mostly indoors, the proportion of nursing home residents receiving vitamin D and calcium was surprisingly low. The use of PIDs was common among nursing home residents. PIDs increased the likelihood of DDIs. However, DDIs did not seem a major concern among the nursing home population. Monitoring PIDs and potential drug interactions could improve the quality of prescribing.