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 .

A moment problem for random discrete measures

Let X be a locally compact Polish space. A random measure on X is a probability measure on the space of all (nonnegative) Radon measures on X. Denote by K(X) the cone of all Radon measures η on X which are of the form η =

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.

CSR: the co-evolution of grocery multiples in the UK (2005-2010)

Purpose – The purpose of this paper is to investigate if CSR is balanced between firm and wider society interests. Design/methodology/approach – A qualitative interpretive hermeneutic approach is used to analyse a variety of publically published secondary sources on the CSR of Tesco, Sainsburys, Morrisons and Co-operative in the UK grocery multiple sector (2005-2010). Findings – CSR strategic outcomes currently favour the firm more than society interests. A multilayered framework in the form of Social Responsibility of the Corporation (SRC) is designed and offered in support of balancing the business-society relationship more evenly. Research limitations/implications – This study is limited to firms originating from within the UK grocery multiple sector. Asda could not be included in the study as it does not publish CSR reports annually in the UK after becoming part of Walmart group. Practical implications – A framework for multi-level standardised definition of CSR in the form of SRC is offered. The inclusion of employees and members of the public on CSR/SRC boards is recommended to foster wider collaboration. The SRC framework promotes standardisation at global level while respecting diversity and firm heterogeneity at firm level. The findings may further contribute to GRI; UN Global Compact; WEF dialogues. Social implications – Recommendations are made to extend CSR board diversity for improved dialogue with communities. The SRC framework may be applied at global; national; industry and firm level. The framework can be applied internationally or locally. Future studies may offer quantitative attributes for balancing CSR/SRC. Originality/value – A globally unique and universally applicable framework for evaluating CSR activities is proposed. Future studies may extend the authors' framework to other industries, national environments or globally in the pursuit of balance between firm and society. Furthermore, firms may also adopt the framework to support CSR activities.