Spectrum and related sets: a survey

Abstract

In order to understand the behaviour of a square matrix or a bounded linear operator on a Banach space or more generally an element of a Banach algebra, some subsets of the complex plane are associated with such an object. Most popular among these sets is the spectrum $\sigma (a)$ of an element a in a complex unital Banach algebra A with unit 1 defined as follows:

$$\begin{aligned} \sigma (a) := \{\lambda \in \mathbb {C}: \lambda - a \,\, \text{ is } \text{ not } \text{ invertible } \text{ in }\, A\}. \end{aligned}$$

Here and also in what follows, we identify $\lambda .1$ with $\lambda$. Also quite popular is Numerical range V(a) of a. This is defined as follows:

$$\begin{aligned} V(a) := \{ \phi (a): \phi \,\, \text{ is } \text{ a } \text{ continuous } \text{ linear } \text{ functional } \text{ on } \,\, A \, \, \text{ satisfying } \, \, \Vert \phi \Vert = 1 = \phi (1) \}. \end{aligned}$$

Then there are many generalizations, modifications, approximations etc. of the spectrum. Let $\epsilon > 0$ and n a nonnegative integer. These include $\epsilon -$ condition spectrum $\sigma _{\epsilon }(a)$, $\epsilon -$pseudospectrum $\Lambda _{\epsilon }(a)$ and $(n, \epsilon )-$pseudospectrum $\Lambda _{n, \epsilon }(a)$. These are defined as follows:

$$\begin{aligned} \sigma _{\epsilon }(a) := \left\{ \lambda \in \mathbb {C}: \Vert \lambda - a\Vert \Vert (\lambda -a)^{-1}\Vert \ge \frac{1}{\epsilon } \right\} \end{aligned}$$

In this and the following definitions we follow the convention : $\Vert (\lambda -a)^{-1}\Vert = \infty$ if $\lambda - a$ is not invertible.

$$\begin{aligned}&\Lambda _{\epsilon }(a) := \left\{ \lambda \in \mathbb {C}: \Vert (\lambda -a)^{-1}\Vert \ge \frac{1}{\epsilon } \right\} \\&\Lambda _{n, \epsilon }(a) := \left\{ \lambda \in \mathbb {C}: \Vert (\lambda -a)^{-2^n}\Vert ^{ 1/2^n } \ge \frac{1}{\epsilon } \right\}. \end{aligned}$$

In this survey article, we shall review some basic properties of these sets, relations among these sets and also discuss the effects of perturbations on these sets and the question of determining the properties of the element a from the knowledge of these sets.