The Maximum Principle for Holomorphic Operator Functions

摘要

We show that if an operator-valued analytic function f of a complex variable attains its maximum modulus at z0, then the coefficients of the nonconstant terms in the power series expansion about z_0 cannot be invertible, provided a complex uniform convexity condition holds. One application is that the norm of the resolvent of an operator on a complex uniformly convex space cannot have a local maximum.