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.

Farmer experience of pluralistic agricultural extension, Malawi

Purpose: Malawi’s current extension policy supports pluralism and advocates responsiveness to farmer demand. We investigate whether smallholder farmers’ experience supports the assumption that access to multiple service providers leads to extension and advisory services that respond to the needs of farmers. Design/methodology/approach: Within a case study approach, two villages were purposively selected for in-depth qualitative analysis of available services and farmers’ experiences. Focus group discussions were held separately with male and female farmers in each village, followed by semi-structured interviews with 12 key informants selected through snowball sampling. Transcripts were analysed by themes and summaries of themes were made from cross case analysis. Findings: Farmers appreciate having access to a variety of sources of technical advice and enterprise specific technology. However, most service providers continue to dominate and dictate what they will offer. Market access remains a challenge, as providers still emphasize pushing a particular technology to increase farm productivity rather than addressing farmers’ expressed needs. Although farmers work in groups, providers do not seek to strengthen these to enable active interaction and to link them to input and produce markets. This limits farmers’ capacity to continue with innovations after service providers pull out. Poor coordination between providers limits exploitation of potential synergies amongst actors. Practical implications: Services providers can adapt their approach to engage farmers in discussion of their needs and work collaboratively to address them. At a system level, institutions that have a coordination function can play a more dynamic role in brokering interaction between providers and farmers to ensure coverage and responsiveness. Originality/value: The study provides a new farmer perspective on the implementation of extension reforms.

Demand, supply and willingness-to-pay for extension services in an emerging-market setting

Although it may be wholly inappropriate to generalize, the most important resource available to a subsistence household is the total amount of time that its members have available to spend in productive enterprises. In this context, services that minimize the time that it takes to perform productive activities are valuable to the household. Consequently the household is willing to relinquish quantities of other resources in exchange for quantities of the time-saving service. These simple observations motivate a search for the values that subsistence households place on time-saving services. This search is especially important when it is realized that extension services promote productivity, enhance the surplus-generating potential of the household and can, as a consequence, promote immersion into markets that are currently constrained by thinness and instability. In this capacity, extension visitation has the potential to overcome one of the principal impediments to economic development, namely lack of density of market participation. In this article, we consider this issue in the context of a rich data set on milk-market participation by small-holder dairy producers in the Ethiopian highlands.

Femtosecond pulse shaping using differential evolutionary algorithm and wavelet operators

Evolutionary meta-algorithms for pulse shaping of broadband femtosecond duration laser pulses are proposed. The genetic algorithm searching the evolutionary landscape for desired pulse shapes consists of a population of waveforms (genes), each made from two concatenated vectors, specifying phases and magnitudes, respectively, over a range of frequencies. Frequency domain operators such as mutation, two-point crossover average crossover, polynomial phase mutation, creep and three-point smoothing as well as a time-domain crossover are combined to produce fitter offsprings at each iteration step. The algorithm applies roulette wheel selection; elitists and linear fitness scaling to the gene population. A differential evolution (DE) operator that provides a source of directed mutation and new wavelet operators are proposed. Using properly tuned parameters for DE, the meta-algorithm is used to solve a waveform matching problem. Tuning allows either a greedy directed search near the best known solution or a robust search across the entire parameter space.

Can valuation factors be prescribed? The case of collective enfranchisement and leasehold extension of flats in England and Wales

Recent concerns over the valuation process in collective leasehold enfranchisement and lease extension cases have culminated in new legislation. To underpin this, the Government (Department of Environment Transport and the Regions (DETR)) commissioned new research, which examined whether the valuation of the freehold in such cases could be simplified through the prescription of either yield or marriage value/relativity. This paper, which is based on that research, examines whether it is possible or desirable to prescribe such factors in the valuation process. Market, settlement and Local Valuation Tribunal (LVT) decisions are analysed, and the basis of 'relativity charts' used in practice is critically examined. Ultimately the imperfect nature of the market in freehold investment sales and leasehold vacant possession sales means that recommendations must rest on an analysis of LVT data. New relativity curves are developed from this data and used in conjunction with an alternative approach to valuation yields (based on other investment assets). However, the paper concludes that although the prescription of yields and relativity is possible, it is not fully defensible because of problems in determining risk premia; that the evidential basis for relativity consists of LVT decisions; and that a formula approach would tend to 'lead' the market as a whole.

Spectral theory of some non-selfadjoint linear differential operators

We give a characterisation of the spectral properties of linear differential operators with constant coefficients, acting on functions defined on a bounded interval, and determined by general linear boundary conditions. The boundary conditions may be such that the resulting operator is not selfadjoint. We associate the spectral properties of such an operator $S$ with the properties of the solution of a corresponding boundary value problem for the partial differential equation $\partial_t q \pm iSq=0$. Namely, we are able to establish an explicit correspondence between the properties of the family of eigenfunctions of the operator, and in particular whether this family is a basis, and the existence and properties of the unique solution of the associated boundary value problem. When such a unique solution exists, we consider its representation as a complex contour integral that is obtained using a transform method recently proposed by Fokas and one of the authors. The analyticity properties of the integrand in this representation are crucial for studying the spectral theory of the associated operator.

Casimirs and Lax operators from the structure of Lie algebras

This paper uses the structure of the Lie algebras to identify the Casimir invariant functions and Lax operators for matrix Lie groups. A novel mapping is found from the cotangent space to the dual Lie algebra which enables Lax operators to be found. The coordinate equations of motion are given in terms of the structure constants and the Hamiltonian.

A Riemann-Hilbert problem with a vanishing coefficient and applications to Toeplitz operators

We study the homogeneous Riemann-Hilbert problem with a vanishing scalar-valued continuous coefficient. We characterize non-existence of nontrivial solutions in the case where the coefficient has its values along several rays starting from the origin. As a consequence, some results on injectivity and existence of eigenvalues of Toeplitz operators in Hardy spaces are obtained.

Glimpsing regular lattice arrangements of primary rat hippocampal astrocytes on ultra-thin nodes of Parylene-C

In our seminal work, we reported how the biomaterial Parylene-C has the unique ability to coerce neurons and glial cells to migrate to and then grow in straight lines along serum coated rectangular parylene-C structures mounted on an oxidised silicon substrate. In this brief communication, we report how astrocyte cell bodies, from the dissociated postnatal rat hippocampus, can now to be successfully localised on an ultra-thin 13nm layer of parylene-C mounted on oxidised silicon (Figure 1). What is extremely interesting about this finding is that the astrocyte processes extended mainly in horizontal and vertical directions from the cell body thus creating a regular lattice network of individual cells. In addition, they comfortably extended a 50μm gap (equivalent to ~ 10 cell body diameters) to connect to adjacent astrocytes on neighbouring Parylene-C structures. This was found to occur repeatedly on circular geometries of 20μm diameter. In comparison to our previous work [1], we have decreased the thickness of the parylene-C structures by a factor of 10, to allow such technology to be able to be utilised for passive electrode design that requires extremely thin structures such as these. Thus, being able to culture astrocytes in regular lattice networks will pave the way for precise monitoring and stimulation of such ensembles via multi-electrode arrays, allowing a closer insight into their dynamic behaviour and their network properties.