How do you compare the spectral radius and the operator norm of a bounded linear operator?

1 Spectral radius

The spectral radius of a bounded linear operator A is the supremum of the absolute values of its eigenvalues. In other words, it is the largest magnitude of an eigenvalue of A. The eigenvalues of A are the complex numbers λ such that there exists a nonzero vector x satisfying Ax = λx. The spectral radius of A is denoted by ρ(A) and it is always a nonnegative real number. For example, if A is the 2x2 matrix [1 2; 0 1], then its eigenvalues are 1 and 1, and its spectral radius is ρ(A) = 1.

2 Operator norm

The operator norm of a bounded linear operator A is the smallest number M such that the inequality ||Ax|| ≤ M||x|| holds for all vectors x in the domain of A. Here, ||x|| denotes the norm of x, which is a measure of its length or magnitude. The operator norm of A is also denoted by ||A|| and it is always a positive real number. For example, if A is the same 2x2 matrix as before, then its operator norm is ||A|| = √5, which is the largest singular value of A.

3 Comparison of spectral radius and operator norm

One of the basic results in spectral theory is that the spectral radius of a bounded linear operator is always less than or equal to its operator norm. That is, ρ(A) ≤ ||A|| for any A. This can be proved by using the Gelfand formula, which states that ρ(A) is equal to the limit of ||A^n||^(1/n) as n goes to infinity. Since ||A^n|| ≤ ||A||^n for any n, it follows that ρ(A) ≤ ||A||. The equality ρ(A) = ||A|| holds if and only if A is normal, which means that it commutes with its adjoint A*. For example, if A is a diagonal matrix, then it is normal and ρ(A) = ||A||.

4 Applications of spectral radius and operator norm

The spectral radius and the operator norm of a bounded linear operator can be used to study various aspects of its behavior and performance. For instance, the spectral radius can be used to determine the convergence of power series involving operators, such as the exponential function e^A or the resolvent (I - A)^(-1). The operator norm can be used to measure the sensitivity of the output of an operator to the input, or the stability of a dynamical system governed by an operator. For example, if A is the transition matrix of a Markov chain, then ||A|| ≤ 1 and ρ(A) = 1 if and only if the chain is ergodic.

5 Further reading

Spectral theory of operators is an incredibly interesting and complex topic with many connections and applications in mathematics and beyond. If you want to learn more about it, consider consulting books such as Introductory Functional Analysis with Applications by Kreyszig, Perturbation Theory for Linear Operators by Kato, Methods of Modern Mathematical Physics I: Functional Analysis by Reed and Simon, or Partial Differential Equations I: Basic Theory by Taylor.