In this chapter we give a short introduction into spectral theory of abstract unbounded operators of a Hilbert space. In sec.1 we give a discussion of general facts on unbounded operators. In sec.2 we discuss the v.Neumann-Riesz theory of self-adjoint extension of hermitian operators. Sec.3 gives a general discussion of the abstract spectral theorem for unbounded self-adjoint operators. We discuss a proof of the spectral theorem in sec.4. Also, in sec.5 we discuss an extension of a result by Heinz and Loewner useful in the following. Finally an abstract result on Fredholm operators in a certain type of Frechet algebra related to a chain of Hilbert spaces generated by powers of a self-adjoint positive operator is discussed in sec.6. The typical ‘HS-chain’ is a chain of L2-Sobolev spaces.

The chapter is self-contained and elementary, and only requires some familiarity with general concepts of analysis and functional analysis of bounded linear operators.

Unbounded linear operators on Banach and Hilbert spaces.

The term “(unbounded) linear operator” (between Banach spaces X and Y) is commonly used to denote any linear map A:dom A → Y from a dense linear subspace dom A of X to Y. The space dom A ⊂ X then is called the domain of A. Here we distinguish between a linear map X → Y, and a linear operator: A linear map X → Y by definition has its domain equal to X.