Improving properties of operators by extensions and reductions

This thesis presents and develops two tools which can be used to work with lower bounds of operators. One tool in working with lower bounds is invertible extensions. They allow one to turn a lower bound of an operator into the norm of an inverse operator. Some results giving extensions are known for single operators and certain other semigroup representations. Chapter 3 includes some positive new results for operators on Hilbert space and also certain unbounded operators. An example shows that a uniform lower bound for the powers of an operator does not give an extension with power bounded inverse. Variations are given for generators of C0-semigroups, and for operators on Hilbert space. A result by Read, which gives an extension with minimal spectrum raises the question how the lower bounds of the original operator are related to the resolvent bounds of the extension. Another tool which is developed in this thesis is a reduction using semi- norms. A seminorm can place a different emphasis on elements and even neglect some. In this way, we can shape a Banach space to attain properties that we impose. This idea is used to define maximal parts in Chapter 4. They are identified in the context of contractivity and expansiveness of a bounded operator, and in the context of dissipativity and accretivity for certain unbounded operators. Applications are an improvement of a theorem by Batty and Tomilov which characterises embeddings into hyperbolic C0-semigroups, and a generalisation of a theorem by Goldberg and Smith leading to a characterisation of generators of C0-semigroups which have an extending group with bounded inverses.