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

Model spaces: a survey

This is a brief and gentle introduction, aimed at graduate students, to the subject of model subspaces of the Hardy space.Comment: 55 page

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

C∗C^*-algebras generated by truncated Toeplitz operators

We obtain an analogue of Coburn's description of the Toeplitz algebra in the setting of truncated Toeplitz operators. As a byproduct, we provide several examples of complex symmetric operators which are not unitarily equivalent to truncated Toeplitz operators having continuous symbols.Comment: 12 page

Unitary equivalence to a truncated Toeplitz operator: analytic symbols

Unlike Toeplitz operators on $H^2$, truncated Toeplitz operators do not have a natural matricial characterization. Consequently, these operators are difficult to study numerically. In this note we provide criteria for a matrix with distinct eigenvalues to be unitarily equivalent to a truncated Toeplitz operator having an analytic symbol. This test is constructive and we illustrate it with several examples. As a byproduct, we also prove that every complex symmetric operator on a Hilbert space of dimension $\leq 3$ is unitarily equivalent to a direct sum of truncated Toeplitz operators.Comment: 15 page

Spatial isomorphisms of algebras of truncated Toeplitz operators

We examine when two maximal abelian algebras in the truncated Toeplitz operators are spatially isomorphic. This builds upon recent work of N. Sedlock, who obtained a complete description of the maximal algebras of truncated Toeplitz operators.Comment: 24 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

Price Impact and Survival of Irrational Traders

Milton Friedman argued that irrational traders will consistently lose money, won’t survive and, therefore, cannot influence long run equilibrium asset prices. Since his work, survival and price impact have been assumed to be the same. In this paper, we demonstrate that survival and price impact are two independent concepts. The price impact of irrational traders does not rely on their long-run survival and they can have a significant impact on asset prices even when their wealth becomes negligible. We also show that irrational traders ’portfolio policies can deviate from their limits long after the price process approaches its long-run limit. We show, in contrast to a partial equilibrium analysis, these general equilibrium considerations matter for the irrational traders long-run survival. In sum, we explicitly show that price impact can persist whether or not the irrational traders survive.