2,721 research outputs found
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
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