| Summary: | <p>This thesis presents and develops two tools which can be used to work with lower bounds of operators.</p> <p>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 <em>C</em><sub>0</sub>-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.</p> <p>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 <em>shape</em> a Banach space to attain properties that we impose. This idea is used to define <em>maximal parts</em> 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 <em>C</em><sub>0</sub>-semigroups, and a generalisation of a theorem by Goldberg and Smith leading to a characterisation of generators of <em>C</em><sub>0</sub>-semigroups which have an extending group with bounded inverses.</p>
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