Spectral Theory

To a bounded operator A on normed space X over $${\mathbb K}$$ , we associate a subset of $${\mathbb K}$$ , known as the spectrum of A. It is intimately related to the invertibility of a specific linear combination of the operator A and the identity operator. Eigenvalues and approximate eigenvalues of A form a part of the spectrum of A. Determining the spectrum of a bounded operator is one of the central problems in functional analysis. In case X is a Banach space, we show that the spectrum of a bounded operator A on X is a closed and bounded subset of $${\mathbb K}$$ . We explore special properties of the spectrum of a compact operator on a normed space. We find relationships between the spectrum of a bounded operator A and the spectra of the transpose $$A'$$ and the adjoint $$A^{*}$$ . They yield particularly interesting results when the operator A is ‘well behaved’ with respect to the adjoint operation. In the last section of this chapter, we show how a compact self-adjoint operator can be represented in terms of its eigenvalues and eigenvectors. This is used in obtaining explicit solutions of operator equations.