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Spectrum, resolvent, numerical range, functional calculus, operator semigroups. Special classes of operators: compact, Fredholm, dissipative, differential, integral, pseudodifferential, etc.
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Residues of analytic operators
Suppose we have analytic operators $P_{z}: C^1[0,1]\to C^1[0,1]$, where $z \in \mathbb{C}$, and the spectrum of $P_{z_0}$ possesses an isolated eigenvalue $1$ (assuming multiplicity is 1 and $I-P_z$ n …