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 .

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.

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.

Schatten class Toeplitz operators on generalized Fock spaces

In this paper we characterize the Schatten p class membership of Toeplitz operators with positive measure symbols acting on generalized Fock spaces for the full range p>0.

The counter-propagating Rossby-wave perspective on baroclinic instability. I: Mathematical basis

It is shown that Bretherton's view of baroclinic instability as the interaction of two counter-propagating Rossby waves (CRWs) can be extended to a general zonal flow and to a general dynamical system based on material conservation of potential vorticity (PV). The two CRWs have zero tilt with both altitude and latitude and are constructed from a pair of growing and decaying normal modes. One CRW has generally large amplitude in regions of positive meridional PV gradient and propagates westwards relative to the flow in such regions. Conversely, the other CRW has large amplitude in regions of negative PV gradient and propagates eastward relative to the zonal flow there. Two methods of construction are described. In the first, more heuristic, method a ‘home-base’ is chosen for each CRW and the other CRW is defined to have zero PV there. Consideration of the PV equation at the two home-bases gives ‘CRW equations’ quantifying the evolution of the amplitudes and phases of both CRWs. They involve only three coefficients describing the mutual interaction of the waves and their self-propagation speeds. These coefficients relate to PV anomalies formed by meridional fluid displacements and the wind induced by these anomalies at the home-bases. In the second method, the CRWs are defined by orthogonality constraints with respect to wave activity and energy growth, avoiding the subjective choice of home-bases. Using these constraints, the same form of CRW equations are obtained from global integrals of the PV equation, but the three coefficients are global integrals that are not so readily described by ‘PV-thinking’ arguments. Each CRW could not continue to exist alone, but together they can describe the time development of any flow whose initial conditions can be described by the pair of growing and decaying normal modes, including the possibility of a super-modal growth rate for a short period. A phase-locking configuration (and normal-mode growth) is possible only if the PV gradient takes opposite signs and the mean zonal wind and the PV gradient are positively correlated in the two distinct regions where the wave activity of each CRW is concentrated. These are easily interpreted local versions of the integral conditions for instability given by Charney and Stern and by Fjørtoft.

Sufficiency of Favard's condition for a class of band-dominated operators on the axis

The purpose of this paper is to show that, for a large class of band-dominated operators on $\ell^\infty(Z,U)$, with $U$ being a complex Banach space, the injectivity of all limit operators of $A$ already implies their invertibility and the uniform boundedness of their inverses. The latter property is known to be equivalent to the invertibility at infinity of $A$, which, on the other hand, is often equivalent to the Fredholmness of $A$. As a consequence, for operators $A$ in the Wiener algebra, we can characterize the essential spectrum of $A$ on $\ell^p(Z,U)$, regardless of $p\in[1,\infty]$, as the union of point spectra of its limit operators considered as acting on $\ell^p(Z,U)$.

Condition number estimates for combined potential boundary integral operators in acoustic scattering

We study the classical combined field integral equation formulations for time-harmonic acoustic scattering by a sound soft bounded obstacle, namely the indirect formulation due to Brakhage-Werner/Leis/Panic, and the direct formulation associated with the names of Burton and Miller. We obtain lower and upper bounds on the condition numbers for these formulations, emphasising dependence on the frequency, the geometry of the scatterer, and the coupling parameter. Of independent interest we also obtain upper and lower bounds on the norms of two oscillatory integral operators, namely the classical acoustic single- and double-layer potential operators.

Over-parameterised, uncertain 'mathematical marionettes' - How can we best use catchment water quality models? An example of an 80-year catchment-scale nutrient balance

The Integrated Catchment Model of Nitrogen (INCA-N) was applied to the River Lambourn, a Chalk river-system in southern England. The model's abilities to simulate the long-term trend and seasonal patterns in observed stream water nitrate concentrations from 1920 to 2003 were tested. This is the first time a semi-distributed, daily time-step model has been applied to simulate such a long time period and then used to calculate detailed catchment nutrient budgets which span the conversion of pasture to arable during the late 1930s and 1940s. Thus, this work goes beyond source apportionment and looks to demonstrate how such simulations can be used to assess the state of the catchment and develop an understanding of system behaviour. The mass-balance results from 1921, 1922, 1991, 2001 and 2002 are presented and those for 1991 are compared to other modelled and literature values of loads associated with nitrogen soil processes and export. The variations highlighted the problem of comparing modelled fluxes with point measurements but proved useful for identifying the most poorly understood inputs and processes thereby providing an assessment of input data and model structural uncertainty. The modelled terrestrial and instream mass-balances also highlight the importance of the hydrological conditions in pollutant transport. Between 1922 and 2002, increased inputs of nitrogen from fertiliser, livestock and deposition have altered the nitrogen balance with a shift from possible reduction in soil fertility but little environmental impact in 1922, to a situation of nitrogen accumulation in the soil, groundwater and instream biota in 2002. In 1922 and 2002 it was estimated that approximately 2 and 18 kg N ha(-1) yr(-1) respectively were exported from the land to the stream. The utility of the approach and further considerations for the best use of models are discussed. (C) 2008 Elsevier B.V. All rights reserved.

Over-parameterised, uncertain 'mathematical marionettes' - How can we best use catchment water quality models? An example of an 80-year catchment-scale nutrient balance

The Integrated Catchment Model of Nitrogen (INCA-N) was applied to the River Lambourn, a Chalk river-system in southern England. The model's abilities to simulate the long-term trend and seasonal patterns in observed stream water nitrate concentrations from 1920 to 2003 were tested. This is the first time a semi-distributed, daily time-step model has been applied to simulate such a long time period and then used to calculate detailed catchment nutrient budgets which span the conversion of pasture to arable during the late 1930s and 1940s. Thus, this work goes beyond source apportionment and looks to demonstrate how such simulations can be used to assess the state of the catchment and develop an understanding of system behaviour. The mass-balance results from 1921, 1922, 1991, 2001 and 2002 are presented and those for 1991 are compared to other modelled and literature values of loads associated with nitrogen soil processes and export. The variations highlighted the problem of comparing modelled fluxes with point measurements but proved useful for identifying the most poorly understood inputs and processes thereby providing an assessment of input data and model structural uncertainty. The modelled terrestrial and instream mass-balances also highlight the importance of the hydrological conditions in pollutant transport. Between 1922 and 2002, increased inputs of nitrogen from fertiliser, livestock and deposition have altered the nitrogen balance with a shift from possible reduction in soil fertility but little environmental impact in 1922, to a situation of nitrogen accumulation in the soil, groundwater and instream biota in 2002. In 1922 and 2002 it was estimated that approximately 2 and 18 kg N ha(-1) yr(-1) respectively were exported from the land to the stream. The utility of the approach and further considerations for the best use of models are discussed. (C) 2008 Elsevier B.V. All rights reserved.

Mathematical modelling of decision making processes in construction projects

Many different individuals, who have their own expertise and criteria for decision making, are involved in making decisions on construction projects. Decision-making processes are thus significantly affected by communication, in which a dynamic performance of human intentions leads to unpredictable outcomes. In order to theorise the decision making processes including communication, it is argued here that the decision making processes resemble evolutionary dynamics in terms of both selection and mutation, which can be expressed by the replicator-mutator equation. To support this argument, a mathematical model of decision making has been made from an analogy with evolutionary dynamics, in which there are three variables: initial support rate, business hierarchy, and power of persuasion. On the other hand, a survey of patterns in decision making in construction projects has also been performed through self-administered mail questionnaire to construction practitioners. Consequently, comparison between the numerical analysis of mathematical model and the statistical analysis of empirical data has shown a significant potential of the replicator-mutator equation as a tool to study dynamic properties of intentions in communication.