Elements of spectral theory without the spectral theorem

This chapter is mainly devoted to a collection of basic facts from the spectral theory of operators in Hilbert spaces. It summarizes some efficient methods how to construct a closed operator with nonempty resolvent set. The chapter also talks about operators that are similar to self-adjoint (or more generally normal) operators. It recalls the notion of pseudospectra as more reliable information about non-self-adjoint operators than the spectrum itself and collects some abstract methods that can be effectively used to construct a quasi-m-accretive operator from a formal expression. Symmetric forms are familiar in quantum mechanics, where they have a physical interpretation of expectation values. For non-self-adjoint operators, a more general class of sectorial forms is needed. The theory of compact operators in Hilbert spaces is reminiscent of the theory of operators in finite-dimensional spaces. Highly non-self-adjoint operators have properties very different from self-adjoint or normal operators.