|Ph.D Student||Goldvard Alexander|
|Subject||Semigroups and Geometric Function Theory in J*-Algebras|
|Department||Department of Applied Mathematics||Supervisors||Professor Emeritus Simeon Reich|
|Professor David Shoikhet|
We study star-like and spiral-like functions defined by the Riesz-Dunford integral on the open unit ball of a J*-algebra. These functions are also called l-analytic. A J*-algebra is a norm closed subspace S of the space of all bounded linear operators which transform one complex Hilbert space to another, such that AA*A belongs to S whenever A belongs to S. Our motivation for considering J*-algebras is that the open unit balls of J*-algebras include all the classical Cartan domains, their infinite dimensional analogues and all finite and infinite products of these. It is known that an l-analytic spiral-like function h defined on the open unit ball D of a J*-algebra satisfies the equation kh(A)=h'(A)f(A), Re k >0, where f is an l-analytic generator of a one-parameter continuous semigroup of l-analytic self-mappings of D. We consider those perturbations of the above-mentioned differential equation which define l-analytic functions which are spiral-like with respect to interior points of the unit ball. We examine the dynamics and stability of the process which results when these points are pushed back to the boundary. In this way, we show, in particular, that each l-analytic function spiral-like with respect to a boundary point is, in fact, a complex power of an l-analytic function star-like with respect to the same boundary point. This enables us to get some new representations of such functions as well as to point out some estimates of their growth.
Another direction in our study of star-like l-analytic functions defined on the open unit ball of a J*-algebra is the asymptotic representations of these functions via continuous semigroups of l-analytic self-mappings of the open unit ball. We provide several formulas for these representations and use them to obtain a distortion theorem for l-analytic functions which are star-like with respect to a boundary point. We also consider some aspects of these representations in general Banach spaces.