Jani Virtanen

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.

Hankel Operators on Fock Spaces

We study Hankel operators on the weighted Fock spaces Fp γ. 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.

Double-scaling limits of Toeplitz determinants and Fisher–Hartwig singularities

Double-scaling limits of Toeplitz determinants Dn(ft) generated by a set of functions ft ∈ L1 are discussed as both n → ∞ and t → 0 simultaneously, which is currently of great importance in mathematics and in physics. The main focus is on the cases where the number of Fisher–Hartwig singularities changes as t → 0. All the results on double-scaling limits are discussed in the context of applications in random matrix theory and in mathematical physics.

A note on structured pseudospectra of block matrices

In this note we consider the question of equivalence of pseudospectra and structured pseudospectra of block matrices. The structures we study are all so called double structures; that is, the blocks of the given matrix are of the same structure as the block matrix. The approach is based on that of non-block matrices, which are also briefly studied, and the use of distance to singularity. We also list some open problems and conjectures.

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

Abstract 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.