Unbounded Symmetric Homogeneous Domains in Spaces of Operators

We define the domain of a linear fractional transformation in a space of operators and show that both the affine automorphisms and the compositions of symmetries act transitively on these domains. Further, we show that Liouville's theorem holds for domains of linear fractional transformations, and, with an additional trace class condition, so does the Riemann removable singularities theorem. We also show that every biholomorphic mapping of the operator domain $I < Z^*Z$ is a linear isometry when the space of operators is a complex Jordan subalgebra of ${\cal L}(H)$ with the removable singularity property and that every biholomorphic mapping of the operator domain $I + Z_1^*Z_1 < Z_2^*Z_2$ is a linear map obtained by multiplication on the left and right by J-unitary and unitary operators, respectively. Readers interested only in the finite dimensional case may identify our spaces of operators with spaces of square and rectangular matrices

Computation of functions of certain operator matrices

AbstractThis note gives a simple method to compute the entries of holomorphic functions of a 2×2 block or operator matrix which can be written as a product. To illustrate this method, the entries are given for the exponential, fractional powers, and inverse of such operator matrices

Removable Singularities in C*-Algebras of Real Rank Zero

Let be a C*-algebra with identity and real rank zero. Suppose a complex- valued function is holomorphic and bounded on the intersection of the open unit ball of and the identity component of the set of invertible elements of . We give a short transparent proof that the function has a holomorphic extension to the entire open unit ball of . The author previously deduced this from a more general fact about Banach algebras

Fixed Points of Holomorphic Mappings for Domains in Banach Spaces

We discuss the Earle-Hamilton fixed-point theorem and show how it can be applied when restrictions are known on the numerical range of a holomorphic function. In particular, we extend the Earle-Hamilton theorem to holomorphic functions with numerical range having real part strictly less than 1. We also extend the Lumer-Phillips theorem estimating resolvents to dissipative holomorphic functions

Fixed Point Theorems for Infinite Dimensional Holomorphic Functions

This talk discusses conditions on the numerical range of a holomorphic function defined on a bounded convex domain in a complex Banach space that imply that the function has a unique fixed point. In particular, extensions of the Earle-Hamilton Theorem are given for such domains. The theorems are applied to obtain a quantitative version of the inverse function theorem for holomorphic functions and a distortion form of Cartan\u27s uniqueness theorem

Industrial life in a north Indian village

Thesis (M.A.)--Boston Universit