Unbounded operators

Abstract In this chapter we develop the basic structure theory of unbounded operators, mostly on Hilbert spaces. We cover the following topics: the spectrum and the Hilbert adjoint of operators; the spectral theorem and the operational calculus for unbounded selfadjoint operators; the semi-simplicity space of unbounded operators with real spectrum on Banach spaces; the theory of symmetric (and in particular selfadjoint) extensions of symmetric operators on Hilbert spaces; and quadratic forms. A few long exercises are devoted to the generators of semigroups of operators and to their theory. Finally, the chapter offer exercises to challenge the reader, including the Friedrichs extension theorem.