Functional Analysis/Operator Algebras Seminar, Winter 2000

• Date: 1/21/00
  • Speaker: Eric Reolon, Universitaet Saarbruecken (visiting UCSB)
  • Title: Topological K-theory and Clifford modules, Part I
    

• Date: 1/28/00
  • Speaker: Eric Reolon, Universitaet Saarbruecken (visiting UCSB)
  • Title: Topological K-theory and Clifford modules, Part II
    

• Date: 2/4/00
  • Speaker: Stephen Simons, UCSB
  • Title: Hahn-Banach and minimax theorems
  • Abstract: We introduce a generalized form of the Hahn--Banach theorem, which we will use to prove various classical results on the existence of linear functionals, and also to prove a minimax theorem.
    
    We will also explain why one cannot generalize the minimax theorem too much, in the sense that any reasonable attempt to generalize the minimax theorem to a pair of noncompact sets is probably doomed to failure.
    
    We will, however, mention an unreasonable generalization that is true, hard and related to R. C. James's ``sup theorem''.

• Date: 2/11/00. Note: The talk will start at 4:30 pm.
  • Speaker: Jonathan Shapiro, Cal Poly San Luis Obispo
  • Title: Kernels of Hankel Operators and Hyponormality of Toeplitz Operators
  • Abstract: We give a formula for the kernel of the product of Hankel operators. We use this to explore the hyponormality of Toeplitz operators whose symbols are of circulant type and some more general types. In addition, we discuss formulas for and estimates of the rank of the self-commutator of a hyponormal Toeplitz operator.
    

• Date: 2/18/00
  • Speaker: Laszlo Zsido, Universita Roma II ``Tor Vergata '' (visiting UCSB)
  • Title: On Weyl-von Neumann theorems, Part I
    

• Date: 2/25/00, Different time: 2:15-3:30 pm, SH 6635
  • Speaker: Laszlo Zsido, Universita Roma II ``Tor Vergata '' (visiting UCSB)
  • Title: On Weyl-von Neumann theorems, Part II
    

• Date: 3/3/00
  • no meeting
    

• Date: 3/10/00
  • Speaker: Philippe Biane, ENS (Paris)
  • Title: Quantum random walks and their boundaries
  • Abstract: We define quantum analogues of random walks and study their potential theory. We give the analogue of the Choquet Deny theorem, and of the Martin compactification. We give applications to dimensions of irreducible representations of compact Lie groups.
    

• Date: 3/17/00
  • Speaker: Harold Shapiro, Royal Institute of Technology, Stockholm
  • Title: Spectral properties of some differential operators
  • Abstract: Recently, Boris Shapiro (at the University of Stockholm) initiated the study of the following differential operators:
    
    For a given monic polynomial Q, Q(z) = z^k + ...(lower degree terms) with complex coefficients, T_Q denotes the operator f -> (d/dz)^k (Qf) say, from the vector space of entire analytic functions to itself.
    
    It is easily seen that, for each Q, T_Q admits for every nonnegative integer m, a unique (up to normalization) eigenfunction f_m that is a polynomial of precise degree m. He proved some remarkable properties of the zeroes of the f_m, and also conjectured a number of others on the basis of computer studies, which have not been proven. I'll give a brief survey of this, and follow it with new results of my own that go in a different direction, and are directed to the following questions:
    
    (i) Are there other eigenfunctions of T_Q than these polynomial ones, in the larger space of entire functions? (Answer: No.)
    
    (ii) The T_Q belonging to all Q of fixed degree k have the same eigenvalues. Are they in fact s i m i l a r as operators on some natural Hilbert spaces of entire functions? (Answer: Yes, but it is not clear how "big" Hilbert spaces we can take where this remains true).
    
    (iii) Are there multidimensional generalizations of these operators? (Answer: Yes, with very rich and interesting properties, so far however largely uncharted.)