Limit operators, collective compactness, and the spectral theory of infinite matrices

In the first half of this memoir we explore the interrelationships between the abstract theory of limit operators (see e.g. the recent monographs of Rabinovich, Roch and Silbermann (2004) and Lindner (2006)) and the concepts and results of the generalised collectively compact operator theory introduced by Chandler-Wilde and Zhang (2002). We build up to results obtained by applying this generalised collectively compact operator theory to the set of limit operators of an operator (its operator spectrum). In the second half of this memoir we study bounded linear operators on the generalised sequence space , where and is some complex Banach space. We make what seems to be a more complete study than hitherto of the connections between Fredholmness, invertibility, invertibility at infinity, and invertibility or injectivity of the set of limit operators, with some emphasis on the case when the operator is a locally compact perturbation of the identity. Especially, we obtain stronger results than previously known for the subtle limiting cases of and . Our tools in this study are the results from the first half of the memoir and an exploitation of the partial duality between and and its implications for bounded linear operators which are also continuous with respect to the weaker topology (the strict topology) introduced in the first half of the memoir. Results in this second half of the memoir include a new proof that injectivity of all limit operators (the classic Favard condition) implies invertibility for a general class of almost periodic operators, and characterisations of invertibility at infinity and Fredholmness for operators in the so-called Wiener algebra. In two final chapters our results are illustrated by and applied to concrete examples. Firstly, we study the spectra and essential spectra of discrete Schrödinger operators (both self-adjoint and non-self-adjoint), including operators with almost periodic and random potentials. In the final chapter we apply our results to integral operators on .

On the numerical integration of a class of singular perturbation problems

A three-point difference scheme recently proposed in Ref. 1 for the numerical solution of a class of linear, singularly perturbed, two-point boundary-value problems is investigated. The scheme is derived from a first-order approximation to the original problem with a small deviating argument. It is shown here that, in the limit, as the deviating argument tends to zero, the difference scheme converges to a one-sided approximation to the original singularly perturbed equation in conservation form. The limiting scheme is shown to be stable on any uniform grid. Therefore, no advantage arises from using the deviating argument, and the most accurate and efficient results are obtained with the deviation at its zero limit.

Integration of a 3D Variational data assimilation scheme with a coastal area morphodynamic model of Morecambe Bay

This paper describes the implementation of a 3D variational (3D-Var) data assimilation scheme for a morphodynamic model applied to Morecambe Bay, UK. A simple decoupled hydrodynamic and sediment transport model is combined with a data assimilation scheme to investigate the ability of such methods to improve the accuracy of the predicted bathymetry. The inverse forecast error covariance matrix is modelled using a Laplacian approximation which is calibrated for the length scale parameter required. Calibration is also performed for the Soulsby-van Rijn sediment transport equations. The data used for assimilation purposes comprises waterlines derived from SAR imagery covering the entire period of the model run, and swath bathymetry data collected by a ship-borne survey for one date towards the end of the model run. A LiDAR survey of the entire bay carried out in November 2005 is used for validation purposes. The comparison of the predictive ability of the model alone with the model-forecast-assimilation system demonstrates that using data assimilation significantly improves the forecast skill. An investigation of the assimilation of the swath bathymetry as well as the waterlines demonstrates that the overall improvement is initially large, but decreases over time as the bathymetry evolves away from that observed by the survey. The result of combining the calibration runs into a pseudo-ensemble provides a higher skill score than for a single optimized model run. A brief comparison of the Optimal Interpolation assimilation method with the 3D-Var method shows that the two schemes give similar results.

Design and management of sustainable built environments

The United Nation Intergovernmental Panel on Climate Change (IPCC) makes it clear that climate change is due to human activities and it recognises buildings as a distinct sector among the seven analysed in its 2007 Fourth Assessment Report. Global concerns have escalated regarding carbon emissions and sustainability in the built environment. The built environment is a human-made setting to accommodate human activities, including building and transport, which covers an interdisciplinary field addressing design, construction, operation and management. Specifically, Sustainable Buildings are expected to achieve high performance throughout the life-cycle of siting, design, construction, operation, maintenance and demolition, in the following areas: • energy and resource efficiency; • cost effectiveness; • minimisation of emissions that negatively impact global warming, indoor air quality and acid rain; • minimisation of waste discharges; and • maximisation of fulfilling the requirements of occupants’ health and wellbeing. Professionals in the built environment sector, for example, urban planners, architects, building scientists, engineers, facilities managers, performance assessors and policy makers, will play a significant role in delivering a sustainable built environment. Delivering a sustainable built environment needs an integrated approach and so it is essential for built environment professionals to have interdisciplinary knowledge in building design and management . Building and urban designers need to have a good understanding of the planning, design and management of the buildings in terms of low carbon and energy efficiency. There are a limited number of traditional engineers who know how to design environmental systems (services engineer) in great detail. Yet there is a very large market for technologists with multi-disciplinary skills who are able to identify the need for, envision and manage the deployment of a wide range of sustainable technologies, both passive (architectural) and active (engineering system),, and select the appropriate approach. Employers seek applicants with skills in analysis, decision-making/assessment, computer simulation and project implementation. An integrated approach is expected in practice, which encourages built environment professionals to think ‘out of the box’ and learn to analyse real problems using the most relevant approach, irrespective of discipline. The Design and Management of Sustainable Built Environment book aims to produce readers able to apply fundamental scientific research to solve real-world problems in the general area of sustainability in the built environment. The book contains twenty chapters covering climate change and sustainability, urban design and assessment (planning, travel systems, urban environment), urban management (drainage and waste), buildings (indoor environment, architectural design and renewable energy), simulation techniques (energy and airflow), management (end-user behaviour, facilities and information), assessment (materials and tools), procurement, and cases studies ( BRE Science Park). Chapters one and two present general global issues of climate change and sustainability in the built environment. Chapter one illustrates that applying the concepts of sustainability to the urban environment (buildings, infrastructure, transport) raises some key issues for tackling climate change, resource depletion and energy supply. Buildings, and the way we operate them, play a vital role in tackling global greenhouse gas emissions. Holistic thinking and an integrated approach in delivering a sustainable built environment is highlighted. Chapter two demonstrates the important role that buildings (their services and appliances) and building energy policies play in this area. Substantial investment is required to implement such policies, much of which will earn a good return. Chapters three and four discuss urban planning and transport. Chapter three stresses the importance of using modelling techniques at the early stage for strategic master-planning of a new development and a retrofit programme. A general framework for sustainable urban-scale master planning is introduced. This chapter also addressed the needs for the development of a more holistic and pragmatic view of how the built environment performs, , in order to produce tools to help design for a higher level of sustainability and, in particular, how people plan, design and use it. Chapter four discusses microcirculation, which is an emerging and challenging area which relates to changing travel behaviour in the quest for urban sustainability. The chapter outlines the main drivers for travel behaviour and choices, the workings of the transport system and its interaction with urban land use. It also covers the new approach to managing urban traffic to maximise economic, social and environmental benefits. Chapters five and six present topics related to urban microclimates including thermal and acoustic issues. Chapter five discusses urban microclimates and urban heat island, as well as the interrelationship of urban design (urban forms and textures) with energy consumption and urban thermal comfort. It introduces models that can be used to analyse microclimates for a careful and considered approach for planning sustainable cities. Chapter six discusses urban acoustics, focusing on urban noise evaluation and mitigation. Various prediction and simulation methods for sound propagation in micro-scale urban areas, as well as techniques for large scale urban noise-mapping, are presented. Chapters seven and eight discuss urban drainage and waste management. The growing demand for housing and commercial developments in the 21st century, as well as the environmental pressure caused by climate change, has increased the focus on sustainable urban drainage systems (SUDS). Chapter seven discusses the SUDS concept which is an integrated approach to surface water management. It takes into consideration quality, quantity and amenity aspects to provide a more pleasant habitat for people as well as increasing the biodiversity value of the local environment. Chapter eight discusses the main issues in urban waste management. It points out that population increases, land use pressures, technical and socio-economic influences have become inextricably interwoven and how ensuring a safe means of dealing with humanity’s waste becomes more challenging. Sustainable building design needs to consider healthy indoor environments, minimising energy for heating, cooling and lighting, and maximising the utilisation of renewable energy. Chapter nine considers how people respond to the physical environment and how that is used in the design of indoor environments. It considers environmental components such as thermal, acoustic, visual, air quality and vibration and their interaction and integration. Chapter ten introduces the concept of passive building design and its relevant strategies, including passive solar heating, shading, natural ventilation, daylighting and thermal mass, in order to minimise heating and cooling load as well as energy consumption for artificial lighting. Chapter eleven discusses the growing importance of integrating Renewable Energy Technologies (RETs) into buildings, the range of technologies currently available and what to consider during technology selection processes in order to minimise carbon emissions from burning fossil fuels. The chapter draws to a close by highlighting the issues concerning system design and the need for careful integration and management of RETs once installed; and for home owners and operators to understand the characteristics of the technology in their building. Computer simulation tools play a significant role in sustainable building design because, as the modern built environment design (building and systems) becomes more complex, it requires tools to assist in the design process. Chapter twelve gives an overview of the primary benefits and users of simulation programs, the role of simulation in the construction process and examines the validity and interpretation of simulation results. Chapter thirteen particularly focuses on the Computational Fluid Dynamics (CFD) simulation method used for optimisation and performance assessment of technologies and solutions for sustainable building design and its application through a series of cases studies. People and building performance are intimately linked. A better understanding of occupants’ interaction with the indoor environment is essential to building energy and facilities management. Chapter fourteen focuses on the issue of occupant behaviour; principally, its impact, and the influence of building performance on them. Chapter fifteen explores the discipline of facilities management and the contribution that this emerging profession makes to securing sustainable building performance. The chapter highlights a much greater diversity of opportunities in sustainable building design that extends well into the operational life. Chapter sixteen reviews the concepts of modelling information flows and the use of Building Information Modelling (BIM), describing these techniques and how these aspects of information management can help drive sustainability. An explanation is offered concerning why information management is the key to ‘life-cycle’ thinking in sustainable building and construction. Measurement of building performance and sustainability is a key issue in delivering a sustainable built environment. Chapter seventeen identifies the means by which construction materials can be evaluated with respect to their sustainability. It identifies the key issues that impact the sustainability of construction materials and the methodologies commonly used to assess them. Chapter eighteen focuses on the topics of green building assessment, green building materials, sustainable construction and operation. Commonly-used assessment tools such as BRE Environmental Assessment Method (BREEAM), Leadership in Energy and Environmental Design ( LEED) and others are introduced. Chapter nineteen discusses sustainable procurement which is one of the areas to have naturally emerged from the overall sustainable development agenda. It aims to ensure that current use of resources does not compromise the ability of future generations to meet their own needs. Chapter twenty is a best-practice exemplar - the BRE Innovation Park which features a number of demonstration buildings that have been built to the UK Government’s Code for Sustainable Homes. It showcases the very latest innovative methods of construction, and cutting edge technology for sustainable buildings. In summary, Design and Management of Sustainable Built Environment book is the result of co-operation and dedication of individual chapter authors. We hope readers benefit from gaining a broad interdisciplinary knowledge of design and management in the built environment in the context of sustainability. We believe that the knowledge and insights of our academics and professional colleagues from different institutions and disciplines illuminate a way of delivering sustainable built environment through holistic integrated design and management approaches. Last, but not least, I would like to take this opportunity to thank all the chapter authors for their contribution. I would like to thank David Lim for his assistance in the editorial work and proofreading.

Essential selfadjointness of singular magnetic Schrödinger operators on Riemannian manifolds

In this paper we extend the well-known Leinfelder–Simader theorem on the essential selfadjointness of singular Schrödinger operators to arbitrary complete Riemannian manifolds. This improves some earlier results of Shubin, Milatovic and others.

On the strong uniqueness of some finite dimensional Dirichlet operators

We prove essential self-adjointness of a class of Dirichlet operators in ℝn using the hyperbolic equation approach. This method allows one to prove essential self-adjointness under minimal conditions on the logarithmic derivative of the density and a condition of Muckenhoupt type on the density itself.

Toeplitz operators with distributional symbols on Bergman spaces

We study the boundedness and compactness of Toeplitz operators Ta on Bergman spaces , 1 < p < ∞. The novelty is that we allow distributional symbols. It turns out that the belonging of the symbol to a weighted Sobolev space of negative order is sufficient for the boundedness of Ta. We show the natural relation of the hyperbolic geometry of the disc and the order of the distribution. A corresponding sufficient condition for the compactness is also derived.

A note on the Fredholm properties of Toeplitz operators on weighted Bergman spaces with matrix-valued symbols

We characterize the essential spectra of Toeplitz operators Ta on weighted Bergman spaces with matrix-valued symbols; in particular we deal with two classes of symbols, the Douglas algebra C+H∞ and the Zhu class Q := L∞ ∩VMO∂ . In addition, for symbols in C+H∞ , we derive a formula for the index of Ta in terms of its symbol a in the scalar-valued case, while in the matrix-valued case we indicate that the standard reduction to the scalar-valued case fails to work analogously to the Hardy space case. Mathematics subject classification (2010): 47B35,

New results and open problems on Toeplitz operators in Bergman spaces

We discuss some of the recent progress in the field of Toeplitz operators acting on Bergman spaces of the unit disk, formulate some new results, and describe a list of open problems -- concerning boundedness, compactness and Fredholm properties -- which was presented at the conference "Recent Advances in Function Related Operator Theory'' in Puerto Rico in March 2010.

Toeplitz operators on Bergman spaces with locally integrable symbols

We study the boundedness of Toeplitz operators $T_a$ with locally integrable symbols on Bergman spaces $A^p(\mathbb{D})$, $1 < p < \infty$. Our main result gives a sufficient condition for the boundedness of $T_a$ in terms of some ``averages'' (related to hyperbolic rectangles) of its symbol. If the averages satisfy an ${o}$-type condition on the boundary of $\mathbb{D}$, we show that the corresponding Toeplitz operator is compact on $A^p$. Both conditions coincide with the known necessary conditions in the case of nonnegative symbols and $p=2$. We also show that Toeplitz operators with symbols of vanishing mean oscillation are Fredholm on $A^p$ provided that the averages are bounded away from zero, and derive an index formula for these operators.

Spectral theory of Toeplitz and Hankel operators on the Bergman space A1

The Fredholm properties of Toeplitz operators on the Bergman space A2 have been well-known for continuous symbols since the 1970s. We investigate the case p=1 with continuous symbols under a mild additional condition, namely that of the logarithmic vanishing mean oscillation in the Bergman metric. Most differences are related to boundedness properties of Toeplitz operators acting on Ap that arise when we no longer have 1operators on A1 were characterized completely very recently but only for bounded symbols. We also consider compactness of Hankel operators on A1.

Hankel operators on Fock spaces

We study Hankel operators on the weighted Fock spaces Fp. The boundedness and compactness of these operators are characterized in terms of BMO and VMO, respectively. Along the way, we also study Berezin transform and harmonic conjugates on the plane. Our results are analogous to Zhu's characterization of bounded and compact Hankel operators on Bergman spaces of the unit disk.