Functional calculus

Summary

In mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. It is now a branch (more accurately, several related areas) of the field of functional analysis, connected with spectral theory. (Historically, the term was also used synonymously with calculus of variations; this usage is obsolete, except for functional derivative. Sometimes it is used in relation to types of functional equations, or in logic for systems of predicate calculus.) If f is a function, say a numerical function of a real number, and M is an operator, there is no particular reason why the expression f(M) should make sense. If it does, then we are no longer using f on its original function domain. In the tradition of operational calculus, algebraic expressions in operators are handled irrespective of their meaning. This passes nearly unnoticed if we talk about 'squaring a matrix', though, which

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Spectral Properties and Stability of Self-Similar Wave Maps

Roland Donninger

In this thesis the Cauchy problem and in particular the question of singularity formation for co--rotational wave maps from 3+1 Minkowski space to the three--sphere

S^3

is studied. Numerics indicate that self--similar solutions of this model play a crucial role in dynamical time evolution. In particular, it is conjectured that a certain solution

f_0

defines a universal blow up pattern in the sense that the future development of a large set of generic blow up initial data approaches

f_0

. Thus, singularity formation is closely related to stability properties of self--similar solutions. In this work, the problem of linear stability is studied by functional analytic methods. In particular, a complete spectral analysis of the perturbation operators is given and well--posedness of the linearized Cauchy problem is proved by means of semigroup theory and, alternatively, the functional calculus for self--adjoint operators. These results lead to growth estimates which provide information on the stability of self--similar wave maps. Finally, convergence properties of

f_n

for large

n

and the spectra of the corresponding perturbation operators are investigated. The thesis is intended to be self--contained as far as possible, i.e. all the mathematical requirements are carefully introduced, including proofs for many results which could be found elsewhere.

University of Vienna2007