resolvent set

The resolvent set is the 'good region' in the complex plane — every parameter where the operator can be cleanly inverted. It is simply everything the spectrum is not. If the spectrum is the list of trouble spots, the resolvent set is the open sea of safe values where life is easy.

Formally, for a bounded operator T on a complex Banach space, the resolvent set rho(T) is the set of complex lambda for which T - lambda*I is a bijection with bounded inverse. By the open mapping theorem the inverse is automatically bounded once T - lambda*I is a bounded bijection of a Banach space onto itself. The resolvent set is precisely the complement of the spectrum: rho(T) = C \ sigma(T).

The resolvent set is always open, and the resolvent R(lambda) is analytic on it; correspondingly the spectrum is always closed. Since the spectrum is also bounded by the operator norm, the spectrum is compact, so the resolvent set always contains the entire exterior region |lambda| > ||T||. For an unbounded operator the spectrum need not be bounded, so the resolvent set can be far from co-compact.