Computing spectra in Banach algebras

Question

In general, computing the spectrum of a specific element in a Banach algebra can be very difficult. What are some of the less obvious tricks that you've encountered?

Answer 1

Some important tricks/theorems:

• The spectrum is non-empty for Banach algebras (over $\mathbb{C}$)
• The spectral radius formula $$ r(a) = \lim \|a^n\|^{1/n} $$ tells you that if $a$ is nilpotent, then $\sigma(a) = \{0\}$
• If $A$ is commutative, then $$ \sigma(a) = \{\tau(a) : \tau \in \Omega(A)\} $$ where $\Omega(A)$ denotes the set of non-zero multiplicative linear functionals on $A$
• If $B$ is a unital subalgebra of $A$ with the same unit as $A$, and $a\in B$, then $$ \sigma_A(a) \subset \sigma_B(a) $$ and $$ \partial \sigma_B(a) \subset \sigma_A(b) $$ where $\sigma_A(a)$ and $\sigma_B(a)$ denote the spectra in relative to those algebras.
• In particular, if $\sigma_A(b)$ has no "holes" (bounded connected components of $\mathbb{C}\setminus \sigma_A(b)$), then $$ \sigma_B(a) = \sigma_A(b) $$ This is often handy when $B$ can be chosen to be commutative.
• If $A$ is a $C^{\ast}$-algebra, then we get $$ \sigma_B(a) = \sigma_A(a) $$ for any element $a$ (regardless of whether $\sigma_A(a)$ has holes or not)
• Furthermore, if $A$ is a $C^{\ast}$-algebra, then knowing the relationship between $a$ and $a^{\ast}$ is often helpful. For instance, if $a = a^{\ast}$, then $$ \sigma(a) \subset \mathbb{R} $$ If $aa^{\ast} = a^{\ast}a = 1_A$, then $$ \sigma(a) \subset \mathbb{T} $$ and so on.