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.

Condition number estimates for combined potential integral operators in acoustics and their boundary element discretisation

We consider the classical coupled, combined-field integral equation formulations for time-harmonic acoustic scattering by a sound soft bounded obstacle. In recent work, we have proved lower and upper bounds on the $L^2$ condition numbers for these formulations, and also on the norms of the classical acoustic single- and double-layer potential operators. These bounds to some extent make explicit the dependence of condition numbers on the wave number $k$, the geometry of the scatterer, and the coupling parameter. For example, with the usual choice of coupling parameter they show that, while the condition number grows like $k^{1/3}$ as $k\to\infty$, when the scatterer is a circle or sphere, it can grow as fast as $k^{7/5}$ for a class of `trapping' obstacles. In this paper we prove further bounds, sharpening and extending our previous results. In particular we show that there exist trapping obstacles for which the condition numbers grow as fast as $\exp(\gamma k)$, for some $\gamma>0$, as $k\to\infty$ through some sequence. This result depends on exponential localisation bounds on Laplace eigenfunctions in an ellipse that we prove in the appendix. We also clarify the correct choice of coupling parameter in 2D for low $k$. In the second part of the paper we focus on the boundary element discretisation of these operators. We discuss the extent to which the bounds on the continuous operators are also satisfied by their discrete counterparts and, via numerical experiments, we provide supporting evidence for some of the theoretical results, both quantitative and asymptotic, indicating further which of the upper and lower bounds may be sharper.

Commuting, CO2 and the location of offices

This paper investigates the extent to which office activity contributes to travel-related CO2 emission. Using ‘end-user’ figures[1], travel accounts for 32% of UK CO2 emission (Commission for Integrated Transport, 2007) and commuting and business travel accounts for a fifth of transport-related CO2 emissions, equating to 6.4% of total UK emissions (Building Research Establishment, 2000). Figures from the Department for Transport (2006) report that 70% of commuting trips were made by car, accounting for 73% of all commuting miles travelled. In assessing the environmental performance of an office building, the paper questions whether commuting and business travel-related CO2 emission is being properly assessed. For example, are office buildings in locations that are easily accessible by public transport being sufficiently rewarded? The de facto method for assessing the environmental performance of office buildings in the UK is the Building Research Establishment’s Environmental Assessment Method (BREEAM). Using data for Bristol, this paper examines firstly whether BREEAM places sufficient weight on travel-related CO2 emission in comparison with building operation-related CO2 emission, and secondly whether the methodology for assigning credits for travel-related CO2 emission efficiency is capable of discerning intra-urban differences in location such as city centre and out-of-town. The results show that, despite CO2 emission per worker from building operation and travel being comparable, there is a substantial difference in the credit-weighting allocated to each. Under the current version of BREEAM for offices, only a maximum of 4% of the available credits can be awarded for ensuring the office location is environmentally sustainable. The results also show that all locations within the established city centre of Bristol will receive maximum BREEAM credits. Given the parameters of the test there is little to distinguish one city centre location from another and out of town only one office location receives any credits. It would appear from these results that the assessment method is not able to discern subtle differences in the sustainability of office locations

Business parks and town centre workplaces in England: a comparative analysis of commuting-related energy consumption

To fully appreciate the environmental impact of a workplace the transport-related carbon dioxide (CO2) emissions resulting from its location should be considered in addition to the emissions that result from the occupation of the building itself. Since the first one was built in the early 1980s, business parks have become a significant workplace location for service-sector workers; a sector of the economy that grew rapidly at that time as the UK manufacturing output declined and the employment base shifted to retail services and de-regulated financial services. This paper examines the transport-related CO2 emissions associated with these workplace locations in comparison to town and city centre locations. Using 2001 Census Special Workplace Statistics which record people’s residence, usual workplace and mode of transport between them, distance travelled and mode of travel were calculated for a sample of city centre and out-of-town office locations. The results reveal the extent of the difference between transport-related CO2 emitted by commuters to out-of-town and city centre locations. The implications that these findings have for monitoring the environmental performance of workplaces are discussed.

The influence of office location on commuting behaviour

To fully appreciate the environmental impact of an office building, the transport-related carbon dioxide (CO2) emissions resulting from its location should be considered in addition to the emissions that result from the operation of the building itself. Travel-related CO2 emissions are a function of three criteria, two of which are influenced by physical location and one of which is a function of business practice. The two spatial criteria are, first, the location of the office relative to the location of the workforce, the market, complementary business activities (and the agglomeration benefits this offers) and, second, the availability and cost of transport modes. The business criterion is the need for, and therefore frequency of, visits and this, in turn, depends on the requirement for a physically present workforce and face-to-face contact with clients. This paper examines the commuting-related CO2 emissions that result from city centre and out-of-town office locations. Using 2001 Census Special Workplace Statistics which record people’s residence, usual workplace and mode of transport between them, distance travelled and mode of travel were calculated for a sample of city centre and out-of-town office locations. The results reveal the extent of the difference between transport-related CO2 emitted by commuters to out-of-town and city centre locations. The implications that these findings have for monitoring the environmental performance of offices are discussed.

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.

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.