An Introduction to Abstract Linear Operator Theory
In this chapter you will read about the beginning material of operator theory. The chapter is written with the aim of getting to spectral theory as quickly as possible. Matrices are examples of linear operators. They transform one linear space into another and do so linearly. “Spectral values” are the infinite-dimensional analogues of eigenvalues in the finite-dimensional situation. Spectral values can be used to decompose operators, in much the same way that eigenvalues can be used to decompose matrices. You will see an example of this sort of decomposition in the last section of this chapter, where we prove the spectral theorem for compact Hermitian operators. One of the most important open problems in operator theory at the start of the twenty-first century is the “invariant subspace problem.” In the penultimate section of this chapter we give a description of this problem and discuss some partial solutions to it. We also let the invariant subspace problem serve as our motivation for learning a bit about operators on Hilbert space. The material found at the end of Section 3 (from Theorem 5.7 onwards) through the last section (Section 5) of the chapter is not usually covered in an undergraduate course. This material is sophisticated, and will probably seem more difficult than other topics we cover.
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