Examples of Spectrums of Elements in an Algebra

Recall from The Spectrum of an Element in an Algebra over C page that if $\mathfrak{A}$ is an algebra with unit and $x \in \mathfrak{A}$ then the spectrum of $x$ in $\mathfrak{A}$ is:

(1)

\begin{align} \quad \mathrm{Sp}(\mathfrak{A}, x) = \{ \lambda \in \mathbb{C} : x - \lambda \in \mathrm{Sing}(\mathfrak{A}) \} \end{align}

And if $\mathfrak{A}$ is an algebra without unit and $x \in \mathfrak{A}$ then the spectrum of $x$ in $\mathfrak{A}$ is:

(2)

\begin{align} \quad \mathrm{Sp}(\mathfrak{A}, x) = \{ 0 \} \cup \left \{ \lambda \in \mathbb{C} \setminus \{ 0 \} : \frac{1}{\lambda} x \in \mathrm{q-Sing}(\mathfrak{A}) \right \} \end{align}

We will now look at some examples of spectra.

Example 1

Let $M_n(\mathbb{C})$ denote the set of all $n \times n$ matrices with complex entries. It is clear that $M_n(\mathbb{C})$ is a Banach algebra with the operations of matrix addition, scalar multiplication, and matrix multiplication. Furthermore, $M_n(\mathbb{C})$ has unit $I_n$ where $I_n$ denotes the $n \times n$ identity matrix.

Let $A \in M_n(\mathbb{C})$. Note that $A - \lambda I_n$ is singular if and only if $\lambda \in \mathbb{C}$ is an eigenvalue of $A$. Thus:

(3)

\begin{align} \quad \mathrm{Sp}(M_n(\mathbb{C}), A) = \{ \lambda : \lambda \mathrm{\: is \: an \: eigenvalue \: of \:} A \} \end{align}

Example 2

Let $C[a, b]$ denote the set of all continuous complex-valued functions defined on the closed bounded interval $[a, b]$. Again, it is clear that $C[a, b]$ is a Banach algebra with the operations of pointwise function addition, pointwise scalar multiplication, and pointwise function multiplication. Furthermore, $C[a, b]$ has unit $1$ where $1$ denotes the constant function which only takes on the value $1$.

Let $f \in C[a, b]$. Note that $f - \lambda 1$ is singular if and only if $[f - \lambda1](x) = 0$ for some $x \in [a, b]$, i.e., if and only if $f(x) = \lambda$ for some $x \in X$. Thus:

(4)

\begin{align} \quad \mathrm{Sp}(C[a, b], f) = \{ \lambda \in \mathbb{C} : f - \lambda 1 \in \mathrm{Sing}(X) \} = \mathrm{range}(f) \end{align}