Ph.D Student | Alexander Goldvard |
---|---|

Subject | Semigroups and Geometric Function Theory in J*-Algebras |

Department | Department of Applied Mathematics |

Supervisors | Professor Emeritus Reich Simeon |

Full Professor Shoikhet David |

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.