What is the spectrum of a tensor product $A_1\otimes\operatorname{id}_{H_2}$?

Question

Let $H_i$ be a $\mathbb C$-Hilbert space, $A_1\in\mathfrak L(H_1)$, $H_1\otimes H_2$ denote the Hilbert tensor product of $H_1$ and $H_2$ and $A_1\otimes\operatorname{id}_{H_2}$ denote the tensor product of $A_1$ and $\operatorname{id}_{H_2}$.

How can we determine the spectrum $\sigma(A_1\otimes\operatorname{id}_{H_2})$? Is it equal to $\sigma(A)$?

We clearly know that the point spectrum $\sigma_p(\operatorname{id}_{H_2})=\{1\}$. And, if I'm not missing something, the continuous and residual spectrum of $\operatorname{id}_{H_2}$ should be empty. So, $\sigma(\operatorname{id}_{H_2})=\sigma_p(\operatorname{id}_{H_2})$.

Now maybe there is a general formula for $\sigma(A_1\otimes A_2)$ when $A_2\in\mathfrak L(H_2)$?

Answer 1

The map $$\Psi: B(H_1) \to B(H_1 \otimes H_2): T \mapsto T \otimes \operatorname{id}_{H_2}$$ is an injective $*$-homomorphism. Therefore,

$$\sigma(A_1) = \sigma(\Psi(A_1)) = \sigma(A_1 \otimes \operatorname{id}_{H_2}).$$

I don't think there is a general formula for $\sigma(A_1 \otimes A_2).$