The norm and functions of a linear operator

Verify that

$\operatorname {dim} \mathfrak {s l}(n, \mathfrak {K})=n^{2}-1, \quad \operatorname {dim} \mathfrak {s o}(n, \mathfrak {K})=n(n-1) / 2, \quad \operatorname {dim} \mathfrak {s u}(n)=n^{2}-1$.

Construct an isomorphism of the groups $U(1)$ and $S O(2, \mathbb {R})$.

The following matrices over $\mathbb {C}$ are called Pauli matrices:

Let us set

Obviously,

Verify that

Establish an isomorphism of the Lie algebras $\mathfrak {s o}(3, \mathbb {R})$ and $\mathfrak {s u}(2)$ with the Lie algebra of vectors in three-dimensional Euclidean space with respect to the vector product.

Prove that if $A, B$ are $n \times n$ matrices and $\operatorname {det} B \neq 0$, then

Find $\exp A$ for the matrix

Prove that every unitary operator $\mathcal{A}$ can be represented in the form $\mathcal{A}=e^{i \mathcal{B}}$, where $\mathcal{B}$ is a Hermitian operator.

Prove that $\left\| \mathcal{A B}\right\| =\left\| \mathcal{B}\right\|$ if $\mathcal{A}$ is unitary and $\mathcal{B}$ an arbitrary linear operator.

Prove that $\left\| \mathcal{A}^{k}\right\| =\left\| \mathcal{A}\right\|^{k}, k=1,2,3, \ldots$, for every normal operator $\mathcal{A}$.

Prove that $\lim_{k \rightarrow \infty } \mathcal{A}^{k}=\mathcal{O} \Longleftrightarrow r(\mathcal{A})<1$.