Laura Smithies

Harnack's theorem for harmonic compact operator-valued functions

Abstract In this paper we show that harmonic compact operator-valued functions are characterized by having harmonic diagonal matrix coefficients in any choice of basis. We also give an example which shows that an operator-valued function with values outside the compact operators can have harmonic diagonal matrix coefficients in any choice of basis without being a harmonic operator-valued function. We use our harmonic matrix coefficients characterization to establish a Harnacks theorem for an increasing sequence of harmonic compact self-adjoint operator-valued functions and we show that this Harnacks theorem need not hold when the compactness restriction is dropped.

Lipschitz-type bounds for the map A→|A| on L(H)

It is well known that the absolute value map on the self-adjoint operators on an infinite dimensional Hilbert spaces is not Lipschitz continuous, although Lipschitz continuity holds on certain subsets of operators. In this note, we provide an elementary proof that the absolute value map is Lipschitz continuous on the set of all operators which are bounded below in norm by any fixed positive constant. Applications are indicated.

Finite rank harmonic operator-valued functions

Abstract The purpose of this paper is to characterize when a harmonic function with values in the finite rank operators on a Hilbert space is expressible as a harmonic matrix-valued function. We show that harmonic function with values in the rank 1 normal operators is expressible as a harmonic matrix-valued function. We also prove that for any natural number, n , a harmonic function with values in the rank n non-negative operators is expressible as a matrix-valued function and we give examples showing that these decomposition theorems fail when various hypotheses are relaxed.

An analytic Riemann-Hilbert correspondence for semi-simple Lie groups

Geometric Representation Theory for semi-simple Lie groups has two main sheaf theoretic models. Namely, through Beilinson-Bernstein localization theory, Harish-Chandra modules are related to holonomic sheaves of D modules on the flag variety. Then the (algebraic) Riemann-Hilbert correspondence relates these sheaves to constructible sheaves of complex vector spaces. On the other hand, there is a parallel localization theory for globalized Harish-Chandra modules—i.e., modules over the full semi-simple group which are completions of Harish-Chandra modules. In particular, Hecht-Taylor and Smithies have developed a localization theory relating minimal globalizations of Harish-Chandra modules to group equivariant sheaves of D modules on the flag variety. The main purpose of this paper is to develop an analytic Riemann-Hilbert correspondence relating these sheaves to constructible sheaves of complex vector spaces and to discuss the relationship between this “analytic” study of global modules and the preceding “algebraic” study of the underlying Harish-Chandra modules.