January 172:30 PM(Note special day and time) |
Christopher Hammond, Connecticut CollegeNorm inequalities for composition operators on Hardy and weighted Bergman spacesIt is well known that any analytic self-map of the open unit disk induces a bounded composition operator on the Hardy space and the standard weighted Bergman spaces. For a particular self-map, one might wonder whether there is any meaningful relationship between the norms of the corresponding operators acting on each of these different spaces. After discussing the context of the problem, we will prove an inequality that (at least to a certain degree) answers this question. We will also consider some related questions that are still open, and discuss possible connections between this problem and the issue of cosubnormality of composition operators. (This talk is based on ongoing research with Linda J. Patton.) |
January 22 |
NO SEMINAR - CONCURRENT COLLOQUIUM |
January 29 |
NO SEMINAR - DEPARTMENT MEETING |
February 5 |
Bill Ross, University of RichmondTruncated Toeplitz operatorsIn this talk I will make the case for truncated Toeplitz operators - compressions of classical Toeplitz operators on H^{2} to a model space (invariant subspace of the backward shift operator). I will present some recent work of Sarason and focus on the problem: When is a bounded operator on a model space a truncated Toeplitz operator?Notes |
February 12 |
Bill Ross, University of RichmondTruncated Toeplitz operators on finite dimensional spacesIn the previous talk I introduced the problem: When is a bounded operator on a model space a truncated Toeplitz operator? In this talk I, in some joint work with Warren Wogen and Joe Cima, will present a complete solution to this problem (in terms of matrix representations) when the model space is finite dimensional.Notes |
February 19 |
Tom Kriete, UVaToeplitz-composition C*-algebras with several generatorsSuppose S denotes the shift operator on the Hardy space H^{2} and let C*(S) denote the C*-algebra generated by S and the identity (otherwise known as the continuous Toeplitz algebra). An old but beautiful theorem of Coburn identifies the quotient of C*(S) by the ideal K of compact operators as the algebra of continuous functions on the unit circle. I'll describe a recent analogue of Coburn's theorem (joint work with Barbara MacCluer and Jennifer Moorhouse) in which C*(S) is replaced by the C*-algebra generated by S and a finite number of composition operators induced by linear fractional non-automorphisms of the disk having no boundary fixed point. |
February 26 |
David Sherman, UVaLocally inner automorphisms of operator algebrasThe first half of this talk will be a survey of automorphisms of operator algebras, beginning with the simplest possible cases. (Some participants may be surprised to hear that "inner" and "outer" have meaning outside Hardy spaces.) Then I will introduce locally inner automorphisms, which have two branches of ancestors, one from group theory and one from operator algebras. I will focus on the subtle line between "locally inner" and "inner." The relevant paper is here. |
March 4 |
NO SEMINAR - SPRING BREAK |
March 11 |
David Sherman, UVaDiagonal sums of automorphism orbitsI will start by summarizing some results from the previous talk, adding more background and tying up loose ends. Then I will consider a very natural question about automorphism orbits in C*-algebras: does it make sense to form (diagonal) direct sums? Yes for B(H), but no in general. The precise answer involves locally inner automorphisms. |
March 18 |
Chuck Akemann, UCSBThe Kadison-Singer problem and related questionsIn a 1959 paper, Dick Kadison and Iz Singer raised questions about the relationships between the algebra B(H) of all bounded linear operators on a separable (infinite dimensional, complex) Hilbert space and its maximal commutative C*-subalgebras. While they answered some of these questions, they left open several tantalizing problems. One such question, now known as the Kadison-Singer Problem, is as follows. If A is a maximal commutative C*-subalgebra of B(H) and f is a homomorphism of A into the complex numbers, does f have a unique, norm-preserving extension to all of B(H)? The algebra A of interest here is defined by taking an orthonormal basis for H and letting A be the algebra of all operators b in B(H) such that each element of the basis is an eigenvector for b. At first glance this problem seems somewhat obscure because the homomorphisms for which uniqueness is in doubt only exist by means of the axiom of choice. However this question has been reduced to a question about finite matrices; in fact, there are many equivalent matrix versions of this question. In the talk I will describe one of the matrix versions that is easy to understand and relate it to some combinatorial questions that have already been solved (and some that have not). Despite the original form of the Kadison-Singer Problem, understanding this talk will only require a basic undergraduate background in complex linear algebra. (Chuck will also give the department colloquium on Thursday, March 20th.) |
March 25 |
NO SEMINAR |
April 1 |
Katherine Heller, UVaAdjoints of rationally induced composition operators |
April 8 |
NO SEMINAR |
April 15 |
Katie Quertermous, UVaA lower bound on the essential norm of a difference of composition operators |
April 22 |
Barbara MacCluer, UVaJoining a compact operator to a non-compact operator by a continuous arc of composition operators on the Hardy space |
April 29 |
Jan Spakula, Vanderbilt UniversityOn uniform K-homologyAnalytic K-homology (the dual homology theory to K-theory) of a non-compact space maps to K-theory of the Roe C*-algebra of this space via the coarse assembly map. The Roe C*-algebras of spaces reflect their coarse (large-scale) geometry. Whether the coarse assembly map is an isomorphism or not is the subject of the coarse Baum-Connes conjecture. This conjecture has applications in geometry, most notably it implies the Novikov conjecture on homotopy invariance of higher signatures. In this talk, I will define a uniform version of analytic K-homology theory and prove a criterion for amenability of metric spaces in terms of their uniform K-homology. Furthermore, I will construct an assembly map from uniform K-homology of a metric space X to the K-theory of its uniform Roe C*-algebra. This is analogous to the classical (non-uniform) assembly map in the Coarse Baum-Connes conjecture. |