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,

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.

'Context' in Durham, E., 'Symbols of power: The Silchester Bronze Eagle and eagles in Roman Britain'

Those who study Roman art and religion in Britain will know that there are a relatively small number of pieces in stone and bronze which are regularly used to illustrate arguments on Romanization, provincialism and identity. However, while these objects become familiar through such use, they are, in fact, often little studied as pieces in their own right and the only description of their appearance and context are some fifty or more years old. Re-excavation of the context from which the Silchester eagle was recovered has raised questions about the date of its deposition, as well as its origin and use, and indeed the nature of its deposition at Silchester. This paper examines the figurine in detail, the role of the eagle at Silchester and explores the significance of the eagle more widely in Roman Britain.

Symbols of power: the Silchester bronze eagle and eagles in Roman Britain

Those who study Roman art and religion in Britain will know that there are a relatively small number of pieces in stone and bronze which are regularly used to illustrate arguments on Romanization, provincialism and identity. However, while these objects become familiar through such use, they are, in fact, often little studied as pieces in their own right and the only description of their appearance and context are some fifty or more years old. Re-excavation of the context from which the Silchester eagle was recovered has raised questions about the date of its deposition, as well as its origin and use, and indeed the nature of its deposition at Silchester. This paper examines the figurine in detail, the role of the eagle at Silchester and explores the significance of the eagle more widely in Roman Britain.

Science learning and graphic symbols: an exploration of early years teachers’ views and use of graphic symbols when teaching science

The study investigated early years teachers’ understanding and use of graphic symbols, defined as the visual representation(s) used to communicate one or more “linguistic” concepts, which can be used to facilitate science learning. The study was conducted in Cyprus where six early years teachers were observed and interviewed. The results indicate that the teachers had a good understanding of the role of symbols, but demonstrated a lack of understanding in regards to graphic symbols specifically. None of the teachers employed them in their observed science lesson, although some of them claimed that they did so. Findings suggest a gap in participants’ acquaintance with the terminology regarding different types of symbols and a lack of awareness about the use and availability of graphic symbols for the support of learning. There is a need to inform and train early years teachers about graphic symbols and their potential applications in supporting children’s learning.

Recognition of online handwritten music symbols

Paper submitted to MML 2013, 6th International Workshop on Machine Learning and Music, Prague, September 23, 2013.