AA 2024-25, Fabrizio Colombo, Antonino De Martino

The H-infinity-functional calculus allows one to define meaningful functions for the class sectorial operators on a Banach space, similar to how functions of bounded linear operators are defined via the Riesz-Dunford functional calculus based on the Cauchy formula of holomorphic functions. As a general fact to generate functions of unbounded operators for function with polynomial growth at infinity is a very delicate issue that requires a two steps procedure, where in the first step the holomorphic functional calculus is used to generate functions of sectorial operators for functions that suitably decay at the origin and at infinity; then, in a second step, a regularization procedure allows to generate the functional calculus for functions with polynomial growth. This calculus is called the H-infinity-functional calculus.

Contents of the course:

1. Preliminaries on holomorphic functions.
2. Holomorphic functional calculus for bounded operators and its properties.
3. Compact operators and Fredholm operators.
4. Closed operators, sectorial operators and their functional calculus (H-infinity).
5. Semigroups and fractional powers of sectorial operators.
6. Sectorial differential operators, applications to mathematical physics.
7. Generalizations

Semigroups generated by sectorial operators are a crucial tool in the study of evolution equations. Given a sectorial operator A defined on a Banach space, the associated semigroup is a family of operators depending on a parameter t that maps the solution of an evolution equation u'(t)=Au(t), at time t_0 to the solution at time t>t_0. Sectorial operators A also admit fractional powers A^{alpha}, for alphain (0,1). Fractional powers A^{alpha} have several applications. For example, the fractional powers of the Laplace operator generate nonlocal operators that take into account nonlocal effects in heat propagation. Properties of sectorial operators and the associated semigroups are extensively studied in functional analysis, and they are widely used in various fields including partial differential equations, mathematical physics, and control theory.