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Elliptic operators arise naturally in several different
mathematical settings, notably in the representation theory of Lie
groups, the study of evolution equations, and the examination of
Riemannian manifolds. This book develops the basic theory of
elliptic operators on Lie groups and thereby extends the
conventional theory of parabolic evolution equations to a natural
non-commutative context. In order to achieve this goal, the author
presents a synthesis of ideas from partial differential equations,
harmonic analysis, functional analysis, and the theory of Lie
groups. He begins by discussing the abstract theory of general
operators with complex coefficients before concentrating on the
central case of second-order operators with real coefficients. A
full discussion of second-order subellilptic operators is also
given. Prerequisites are a familiarity with basic semigroup theory,
the elementary theory of Lie groups, and a firm grounding in
functional analysis as might be gained from the first year of a
graduate course.
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