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This book describes recent developments as well as some classical
results regarding holomorphic mappings. The book starts with a
brief survey of the theory of semigroups of linear operators
including the Hille-Yosida and the Lumer-Phillips theorems. The
numerical range and the spectrum of closed densely defined linear
operators are then discussed in more detail and an overview of
ergodic theory is presented. The analytic extension of semigroups
of linear operators is also discussed. The recent study of the
numerical range of composition operators on the unit disk is
mentioned. Then, the basic notions and facts in infinite
dimensional holomorphy and hyperbolic geometry in Banach and
Hilbert spaces are presented, L. A. Harris' theory of the numerical
range of holomorphic mappings is generalized, and the main
properties of the so-called quasi-dissipative mappings and their
growth estimates are studied. In addition, geometric and
quantitative analytic aspects of fixed point theory are discussed.
A special chapter is devoted to applications of the numerical range
to diverse geometric and analytic problems.
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