Existence Families, Functional Calculi and Evolution Equations

Intuition and elementary examples.- Existence families.- Regularized semigroups.- The solution space of an operator and automatic well-posedness.- Exponentially bounded (Banach) solution spaces.- Well-posedness on a larger space; Generalized solutions.- Entire vectors and entire existence families.- Reversibility of parabolic problems.- The cauchy problem for the Laplace equation.- Boundary values of holomorphic semigroups.- The Schrödinger equation.- Functional calculus for commuting generators of bounded strongly continuous groups.- Petrovsky correct matrices of generators of bounded strongly continuous groups.- Arbitrary matrices of generators of bounded strongly continuous groups.- More examples of regularized semigroups.- Existence and uniqueness families.- C-resolvents and Hille-Yosida type theorems.- Relationship to integrated semigroups.- Perturbations.- Type of an operator.- Holomorphic C-existence families.- Unbounded holomorphic functional calculus for operators with polynomially bounded resolvents.- Spectral conditions guaranteeing solutions of the abstract Cauchy problem.- Polynomials of generators.- Iterated abstract Cauchy problems.- Equipartition of energy.- Simultaneous solution space.- Exponentially bounded simultaneous solution space.- Simultaneous existence families.- Simultaneous existence families for matrices of operators.- Time dependent evolution equations.