I recently learned something about spectrum in functional analysis and saw some examples. However I struggling with this when trying to understand how it can be used for the following example regarding Hilbert spaces. I will write the example below.
Let $H_1,H_2$ be Hilbert spaces, then the direct sum $H_1\oplus H_2$ is given by the following $$\langle (x_1,y_1)|(x_2,y_2)\rangle:=\langle x_1|x_2 \rangle_{H_1}+\langle y_1|y_2 \rangle_{H_2}.$$ Let $T_1\oplus T_2$ be given where $T_1\in B(H_1)$ and $T_2\in B(H_2)$. How can one find the spectrum of $T_1\oplus T_2$?
I have the definition of the spectrum as follows:
Definition: Let $A\in \mathscr{A}$, where $\mathscr{A}$ is a unital Banach algebra. The spectrum of $A$ denoted $\sigma(A)$ is $\{\lambda \in \mathbb{C}:A-\lambda I \text{ is not invertible in } \mathscr{A}\}$. Of course we have $I$ as our unit.
Notice that I now know that $\sigma(T_1\oplus T_2)=\sigma(T_1)\cup \sigma(T_2)$ so I have to compute each $\sigma(T_1)$ and $\sigma(T_2)$. I have to compute $\det(\lambda I_1-T_1)$ and $\det(\lambda I_2-T_2)$ however what matrices should I use?