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.

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 1

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.

Self-assembly of a model amphiphilic oligopeptide incorporating an arginine headgroup

The self-assembly in aqueous solution of the alanine-rich peptide A12R2 containing twelve alanine residues and two arginine residues has been investigated. This oligomeric peptide was synthesized via NCA-polymerization methods. The surfactant-like peptide is found via FTIR to form antiparallel dimers which aggregate into twisted fibrils, as revealed by cryogenic-transmission electron microscopy. The fibril substructure is probed via detailed X-ray scattering experiments, and are uniquely comprised of twisted tapes only 5 nm wide, set by the width of the antiparallel A12R2 dimers. The packing of the alanine residues leads to distinct “b-sheet” spacings compared to those for amyloid-forming peptides. For this peptide, b-sheet structure coexists with some a-helical content. These ultrafine amyloid fibrils present arginine at high density on their surfaces, and this may lead to applications in nanobiotechnology.

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.

Tuning self-assembled nanostructures through enzymatic degradation of a peptide amphiphile

The enzymatic cleavage of a peptide amphiphile (PA) is investigated. The self-assembly of the cleaved products is distinct from that of the PA substrate. The PA C16-KKFFVLK is cleaved by α-chymotrypsin at two sites leading to products C16-KKF with FVLK and C16-KKFF with VLK. The PA C16-KKFFVLK forms nanotubes and helical ribbons at room temperature. Both PAs C16-KKF and C16-KKFF corresponding to cleavage products instead self-assemble into 5-6 nm diameter spherical micelles, while peptides FVLK and VLK do not adopt well-defined aggregate structures. The secondary structures of the PAs and peptides are examined by FTIR and circular dichroism spectroscopy and X-ray diffraction. Only C16-KKFFVLK shows substantial β-sheet secondary structure, consistent with its self-assembly into extended aggregates, based on PA layers containing hydrogen-bonded peptide headgroups. This PA also exhibits a thermoreversible transition to twisted tapes on heating.

Toeplitz operators on Dirichlet-Besov spaces

We study Toeplitz operators on the Besov spaces in the case of the open unit disk. We prove that a symbol satisfying a weak Lipschitz type condition induces a bounded Toeplitz operator. Such symbols do not need to be bounded functions or have continuous extensions to the boundary of the open unit disk. We discuss the problem of the existence of nontrivial compact Toeplitz operators, and also consider Fredholm properties and prove an index formula.

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.