A densely defined closed operator is sectorial of angle if its spectrum lies in and, for every larger angle , its resolvent outside that larger sector satisfies . If , then generates an analytic semigroup. A real shift of the generator allows exponential growth without changing the short-time smoothing estimates.
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A sectorial operator is a type of linear operator in functional analysis that generalizes the concept of self-adjoint operators. Sectorial operators arise in the study of partial differential equations and the theory of semigroups of operators. They are particularly important in the context of evolution equations and their solutions. An operator \( A \) on a Banach space \( X \) is said to be sectorial if it has a sector in the complex plane where its spectrum lies.