Recent progress on truncated Toeplitz operators

This paper is a survey on the emerging theory of truncated Toeplitz operators. We begin with a brief introduction to the subject and then highlight the many recent developments in the field since Sarason's seminal paper in 2007.Comment: 46 page

Partial orders on partial isometries

This paper studies three natural pre-orders of increasing generality on the set of all completely non-unitary partial isometries with equal defect indices. We show that the problem of determining when one partial isometry is less than another with respect to these pre-orders is equivalent to the existence of a bounded (or isometric) multiplier between two natural reproducing kernel Hilbert spaces of analytic functions. For large classes of partial isometries these spaces can be realized as the well-known model subspaces and deBranges-Rovnyak spaces. This characterization is applied to investigate properties of these pre-orders and the equivalence classes they generate.Comment: 30 pages. To appear in Journal of Operator Theor

Real complex functions

We survey a few classes of analytic functions on the disk that have real boundary values almost everywhere on the unit circle. We explore some of their properties, various decompositions, and some connections these functions make to operator theory.Comment: 44 page

The Jordan curve theorem is non-trivial

The formal mathematical definition of a Jordan curve (a non-self-intersecting continuous loop in the plane) is so simple that one is often lead to the unimaginative view that a Jordan curve is nothing more than a circle or an ellipse. In this paper, we pursue the theme that a Jordan curve can be quite fantastical in the sense that there are some bizarre properties such a curve might have (jagged at every point, space filling, etc.) or that such a curve can have a difficult to discover inside and outside as promised by the celebrated Jordan Curve Theorem (JCT). We explore the JCT theorem through its history and some hand drawings which not only challenge the viewer\u27s preconceived notions of interior and exterior or that the JCT is a trivial result, but also challenge the reader\u27s notion that a curve is a cold boring object, incapable of telling an interesting story

Bergman Spaces on an Annulus and the Backward Bergman Shift

In this paper, we will give a complete characterization of the invariant subspaces M (under ƒ → zƒ) of the Bergman space Lpa(G), 1 \u3c p \u3c 2, G an annulus, which contain the constant function 1. As an application of this result, we will characterize the invariant subspaces of the adjoint of multiplication by z on the Dirichlet spaces Dq, q \u3e 2, as well as the invariant subspaces of the backward Bergman shift ƒ → (ƒ – ƒ(0))/z on Lpa(), 1 \u3c p \u3c 2

An invariant subspace problem for \u3cem\u3ep\u3c/em\u3e = 1 Bergman spaces on slit domains

In this paper, we characterize the z-invariant subspaces that lie between the Bergman spaces A1(G) and A1(G/K), where G is a bounded region in the complex plane and K is a compact subset of a simple arc of class C1