Convex-transitive Banach spaces and their hyperplanes

The topic of this dissertation is the geometric and isometric theory of Banach spaces. This work is motivated by the known Banach-Mazur rotation problem, which asks whether each transitive separable Banach space is isometrically a Hilbert space. A Banach space X is said to be transitive if the isometry group of X acts transitively on the unit sphere of X. In fact, some weaker symmetry conditions than transitivity are studied in the dissertation. One such condition is an almost isometric version of transitivity. Another investigated condition is convex-transitivity, which requires that the closed convex hull of the orbit of any point of the unit sphere under the rotation group is the whole unit ball. Following the tradition developed around the rotation problem, some contemporary problems are studied. Namely, we attempt to characterize Hilbert spaces by using convex-transitivity together with the existence of a 1-dimensional bicontractive projection on the space, and some mild geometric assumptions. The convex-transitivity of some vector-valued function spaces is studied as well. The thesis also touches convex-transitivity of Banach lattices and resembling geometric cases.

Aspects of atomic decompositions and Bergman projections

The concept of an atomic decomposition was introduced by Coifman and Rochberg (1980) for weighted Bergman spaces on the unit disk. By the Riemann mapping theorem, functions in every simply connected domain in the complex plane have an atomic decomposition. However, a decomposition resulting from a conformal mapping of the unit disk tends to be very implicit and often lacks a clear connection to the geometry of the domain that it has been mapped into. The lattice of points, where the atoms of the decomposition are evaluated, usually follows the geometry of the original domain, but after mapping the domain into another this connection is easily lost and the layout of points becomes seemingly random. In the first article we construct an atomic decomposition directly on a weighted Bergman space on a class of regulated, simply connected domains. The construction uses the geometric properties of the regulated domain, but does not explicitly involve any conformal Riemann map from the unit disk. It is known that the Bergman projection is not bounded on the space L-infinity of bounded measurable functions. Taskinen (2004) introduced the locally convex spaces LV-infinity consisting of measurable and HV-infinity of analytic functions on the unit disk with the latter being a closed subspace of the former. They have the property that the Bergman projection is continuous from LV-infinity onto HV-infinity and, in some sense, the space HV-infinity is the smallest possible substitute to the space H-infinity of analytic functions. In the second article we extend the above result to a smoothly bounded strictly pseudoconvex domain. Here the related reproducing kernels are usually not known explicitly, and thus the proof of continuity of the Bergman projection is based on generalised Forelli-Rudin estimates instead of integral representations. The minimality of the space LV-infinity is shown by using peaking functions first constructed by Bell (1981). Taskinen (2003) showed that on the unit disk the space HV-infinity admits an atomic decomposition. This result is generalised in the third article by constructing an atomic decomposition for the space HV-infinity on a smoothly bounded strictly pseudoconvex domain. In this case every function can be presented as a linear combination of atoms such that the coefficient sequence belongs to a suitable Köthe co-echelon space.

Turun linna kerrottuna ja kertovana tilana

The Turku castle, founded c. 1300, has changed over the centuries from a medieval defensive structure into a Renaissance palace and from a derelict jailhouse in the late 19th century into a prime example of the Medieval built heritage in Finland. Today, it is first and foremost a monument to the Medieval and Renaissance heyday of the castle. This is apparent in the architectural forms that have been carefully restored and reconstructed. It also becomes clear in all kinds of narratives, both visual (like the set of miniatures about the different stages of the construction of the castle) and textual (as during the guided tours). For the first time in the architectural history of the Turku castle, the Medieval, the Renaissance, the Modern, and the Present as architecturally constructed or reconstructed spaces can all be visited within the same hour. As a result, the monumental Turku castle may even be deemed anachronistic or inauthentic. In this study I look at the ways in which the Turku castle is, indeed, anachronistic and inauthentic. My main objective, however, is to find ways in which the anachronisms and inauthenticities are overcome in a positive way. I base my analysis of the Turku castle on three theoretical standpoints. First, I am studying the castle as space, described by Michel de Certeau as a practiced place (de Certeau 2002). Second, I am approaching the numerous narrative aspects of the castle following Paul Ricoeur s analysis of narrative as a threefold mimetic process (Ricoeur 1990). From these two theoretical settings I have summoned the concept of narrative space. The life and work at the castle are based on expectations and understandings of the historical surroundings. My third theoretical choice is to study this applied knowledge of the place as the management of blocks of knowledge in communication (Robert de Beaugrande 1980). Combining the theoretical starting points of space and narrative , I am approaching the castle as if it were an evolving set of narratives, narrated in space but also through space. Seeing e.g. the restoration teams of the mid-20th century and the present day tour guides as creative narrators, I am looking beyond the dilemma of the anachronistic spaces. What transpires is an inter-connected web of texts and spaces, tangible and intangible narratives. My analytical key to these narrative relationships is the threefold mimetic process of pre-figuration, con-figuration, and re-figuration, inspired by the writings of Paul Ricoeur (1990). This way, the past can be seen as a pool of endless possibilities to emplot place, time, and action into a narrative space. The narratives convey images of the past that may be contested by other images, and the power to narrate in the first place can be challenged and re-distributed.

Transcending “the Impossible” : From Dead End to Expansive Learning

This dissertation investigates changes in bank work and the experience of impossibility attached to these by workers at the local level from the viewpoint of work-related well-being and collective learning. A special challenge in my work is to conceptualize the experience of impossibility as related to change, and as a starting point and tool for development work. The subject of the dissertation, solving the impossible as a collective learning process, came up as a central theme in an earlier project: Work Units between the Old and the New (1997 – 1999). Its aim was to investigate how change is constructed as a long-term process, starting from the planning of the change until its final realization in everyday banking work. I studied changes taking place in the former Postipankki (Postal Bank), later called Leonia. The three-year study involved the Branch Office of Martinlaakso, and was conducted from the perspective of well-being in a change process. The sense of impossibility involved in changes turned out to be one of the most crucial factors impairing the sense of well-being. The work community that was the target of my study did not have the available tools to construct the change locally, or to deal with the change-related impossibility by solving it through a mutual process among themselves. During the last year of the project, I carried out an intervention for development in the Branch Office, as collaboration between the researchers and the workers. The purpose of the intervention was to resolve such perceived change-related impossibility as experienced repeatedly and considered by the work community as relevant to work-related well-being. The documentation of the intervention – audio records from development sessions, written assignments by workers and assessment or evaluation interviews – constitute the essential data for my dissertation. The earlier data, collected and analysed during the first two years, provides a historical perspective on the process, all the way from construction of the impossibility towards resolving and transcending it. The aim of my dissertation is to understand the progress of developmental intervention as a shared, possibly expansive learning process within a work community and thus to provide tools for perceiving and constructing local change. I chose the change-related impossibility as a starting point for development work in the work community and as a target of conceptualization. This, I feel, is the most important contribution of my dissertation. While the intervention was in progress, the concept of impossibility started emerging as a stimulating tool for development work. An understanding of such a process can be applied to development work outside banking work as well. According to my results, it is pivotal that a concept stimulating development is strongly connected with everyday experiences of and speech about changes in work activity, as well as with the theoretical framework of work development. During this process, development work on a local level became of utmost interest as a case study for managing change. Theoretically, this was conceptualized as so-called second-order work and this concept accompanies us all the way through the research process. Learning second-order work and constructing tools based on this work have proved crucial for promoting well-being in the change circumstances in a local work unit. The lack of second-order work has led to non-well-being and inability to transcend the change-related sense of impossibility in the work community. Solving the impossible, either individually or situationally, did not orient the workers towards solving problems of impossibility together as a work community. Because the experience of the impossibility and coming to terms with transcending it are the starting point and the target of conceptualization in this dissertation, the research provides a fresh viewpoint on the theoretical framework of change and developmental work. My dissertation can facilitate construction of local changes necessitated by the recent financial crisis, and thus promote fluency and well-being in work units. It can also support change-related well-being in other areas of working life.

Environmental change and anthropogenic impact on lake sediments during the Holocene in the Finnish - Karelian inland area

This thesis discusses the prehistoric human disturbance during the Holocene by means of case studies using detailed high-resolution pollen analysis from lake sediment. The four lakes studied are situated between 61o 40' and 61o 50' latitudes in the Finnish Karelian inland area and vary between 2.4 and 28.8 ha in size. The existence of Early Metal Age population was one important question. Another study question concerned the development of grazing, and the relationship between slash-and-burn cultivation and permanent field cultivation. The results were presented as pollen percentages and pollen concentrations (grains cm 3). Accumulation values (grains cm 2 yr 1) were calculated for Lake Nautajärvi and Lake Orijärvi sediment, where the sediment accumulation rate was precisely determined. Sediment properties were determined using loss-on-ignition (LOI) and magnetic susceptibility (k). Dating methods used include both conventional and AMS 14C determinations, paleomagnetic dating and varve choronology. The isolation of Lake Kirjavalampi on the northern shore of Lake Ladoga took place ca. 1460 1300 BC. The long sediment cores from Finland, Lake Kirkkolampi and Lake Orijärvi in southeastern Finland and Lake Nautajärvi in south central Finland all extended back to the Early Holocene and were isolated from the Baltic basin ca. 9600 BC, 8600 BC and 7675 BC, respectively. In the long sediment cores, the expansion of Alnus was visible between 7200 - 6840 BC. The spread of Tilia was dated in Lake Kirkkolampi to 6600 BC, in Lake Orijärvi to 5000 BC and at Lake Nautajärvi to 4600 BC. Picea is present locally in Lake Kirkkolampi from 4340 BC, in Lake Orijärvi from 6520 BC and in Lake Nautajärvi from 3500 BC onwards. The first modifications in the pollen data, apparently connected to anthropogenic impacts, were dated to the beginning of the Early Metal Period, 1880 1600 BC. Anthropogenic activity became clear in all the study sites by the end of the Early Metal Period, between 500 BC AD 300. According to Secale pollen, slash-and-burn cultivation was practised around the eastern study lakes from AD 300 600 onwards, and at the study site in central Finland from AD 880 onwards. The overall human impact, however, remained low in the studied sites until the Late Iron Age. Increasing human activity, including an increase in fire frequency was detected from AD 800 900 onwards in the study sites in eastern Finland. In Lake Kirkkolampi, this included cultivation on permanent fields, but in Lake Orijärvi, permanent field cultivation became visible as late as AD 1220, even when the macrofossil data demonstrated the onset of cultivation on permanent fields as early as the 7th century AD. On the northern shore of Lake Ladoga, local activity became visible from ca. AD 1260 onwards and at Lake Nautajärvi, sediment the local occupation was traceable from 1420 AD onwards. The highest values of Secale pollen were recorded both in Lake Orijärvi and Lake Kirjavalampi between ca. AD 1700 1900, and could be associated with the most intensive period of slash-and-burn from AD 1750 to 1850 in eastern Finland.

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.

Dirichlet problem at infinity for p-harmonic functions on negatively curved spaces

The object of this dissertation is to study globally defined bounded p-harmonic functions on Cartan-Hadamard manifolds and Gromov hyperbolic metric measure spaces. Such functions are constructed by solving the so called Dirichlet problem at infinity. This problem is to find a p-harmonic function on the space that extends continuously to the boundary at inifinity and obtains given boundary values there. The dissertation consists of an overview and three published research articles. In the first article the Dirichlet problem at infinity is considered for more general A-harmonic functions on Cartan-Hadamard manifolds. In the special case of two dimensions the Dirichlet problem at infinity is solved by only assuming that the sectional curvature has a certain upper bound. A sharpness result is proved for this upper bound. In the second article the Dirichlet problem at infinity is solved for p-harmonic functions on Cartan-Hadamard manifolds under the assumption that the sectional curvature is bounded outside a compact set from above and from below by functions that depend on the distance to a fixed point. The curvature bounds allow examples of quadratic decay and examples of exponential growth. In the final article a generalization of the Dirichlet problem at infinity for p-harmonic functions is considered on Gromov hyperbolic metric measure spaces. Existence and uniqueness results are proved and Cartan-Hadamard manifolds are considered as an application.

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.

Tiedostumaton nykytaiteessa : Katse, ääni ja aika vuosituhannen taitteen suomalaisessa nykytaiteessa

Leevi Haapala explores moving image works, sculptures and installations from a psychoanalytic perspective in his study The Unconscious in Contemporary Art. The Gaze, Voice and Time in Finnish Contemporary Art at the Turn of the Millennium . The artists included in the study are Eija-Liisa Ahtila, Hans-Christian Berg, Markus Copper, Liisa Lounila and Salla Tykkä. The theoretical framework includes different psychoanalytic readings of the concepts of the gaze, voice and temporality. The installations are based on spatiality and temporality, and their detailed reading emphasizes the medium-specific features of the works as well as their fragmentary nature, heterogeneity and affectivity. The study is cross-disciplinary in that it connects perspectives from the visual culture, new art history and theory to the interpretation of contemporary art. The most important concepts from psychoanalysis, affect theory and trauma discourse used in the study include affect, object a (objet petit a) as articulated by Jacques Lacan, Sigmund Freud s uncanny (das Unheimliche) and trauma. Das Unheimliche has been translated as uncanny in art history under the influence of Rosalind Krauss. The object of the study, the unconscious in contemporary art, is approached through these concepts. The study focuses on Lacan s additions to the list of partial drives: the gaze and voice as scopic and invocative drives and their interpretations in the studies of the moving image. The texts by the American film theorist and art historian Kaja Silverman are in crucial role. The study locates contemporary art as part of trauma culture, which has a tendency to define individual and historical experiences through trauma. Some of the art works point towards trauma, which may appear as a theoretic or fictitious construction. The study presents a comprehensive collection of different kinds of trauma discourse in the field of art research through the texts of Hal Foster, Cathy Caruth, Ruth Leys and Shoshana Felman. The study connects trauma theory with the theoretical analysis of the interference and discontinuity of the moving image in the readings by Susan Buck-Morss, Mary Ann Doane and Peter Osborn among others. The analysis emphasizes different ways of seeing and multisensoriality in the reception of contemporary art. With their reflections and inverse projections, the surprising mechanisms of Hans-Christian Berg s sculptures are connected with Lacan s views on the early mirroring and imitation attempts of the individual s body image. Salla Tykkä s film trilogy Cave invites one to contemplate the Lacanian theory of the gaze in relation to the experiences of being seen. The three oceanic sculpture installations by Markus Copper are studied through the vocality they create, often through an aggressive way of acting, as well as from the point of view of the functioning of an invocative drive. The study compares the work of fiction and Freud s texts on paranoia and psychosis to Eija-Liisa Ahtila s manuscripts and moving image installations about the same topic. The cinematic time in Liisa Lounila s time-slice video installations is approached through the theoretical study of the unconscious temporal structure. The viewer of the moving image is inside the work in an in-between state: in a space produced by the contents of the work and its technology. The installations of the moving image enable us to inhabit different kinds of virtual bodies or spaces, which do not correspond with our everyday experiences. Nevertheless, the works of art often try to deconstruct the identification to what has been shown on screen. This way, the viewer s attention can be fixed on his own unconscious experiences in parallel with the work s deconstructed nature as representation. The study shows that contemporary art is a central cultural practice, which allows us to discuss the unconscious in a meaningful way. The study suggests that the agency that is discursively diffuse and consists of several different praxes should be called the unconscious. The emergence of the unconscious can happen in two areas: in contemporary art through different senses and discursive elements, and in the study of contemporary art, which, being a linguistic activity is sensitive to the movements of the unconscious. One of the missions of art research is to build different kinds of articulated constructs and to open an interpretative space for the nature of art as an event.

Locally checkable proofs

This work studies decision problems from the perspective of nondeterministic distributed algorithms. For a yes-instance there must exist a proof that can be verified with a distributed algorithm: all nodes must accept a valid proof, and at least one node must reject an invalid proof. We focus on locally checkable proofs that can be verified with a constant-time distributed algorithm. For example, it is easy to prove that a graph is bipartite: the locally checkable proof gives a 2-colouring of the graph, which only takes 1 bit per node. However, it is more difficult to prove that a graph is not bipartite—it turns out that any locally checkable proof requires Ω(log n) bits per node. In this work we classify graph problems according to their local proof complexity, i.e., how many bits per node are needed in a locally checkable proof. We establish tight or near-tight results for classical graph properties such as the chromatic number. We show that the proof complexities form a natural hierarchy of complexity classes: for many classical graph problems, the proof complexity is either 0, Θ(1), Θ(log n), or poly(n) bits per node. Among the most difficult graph properties are symmetric graphs, which require Ω(n2) bits per node, and non-3-colourable graphs, which require Ω(n2/log n) bits per node—any pure graph property admits a trivial proof of size O(n2).