Normal Operators

Prove that self-adjoint, orthogonal (unitary), skew-symmetric (skew-Hermitian) operators are normal operators.

Prove that an operator A is normal if and only if for any vector $x$ we have

Prove that a linear operator A is normal if and only if in its polar decomposition $\mathbf{A} = \mathbf{B U}$, where B is a nonnegative self-adjoint operator and U is a unitary (orthogonal) operator, the operators B and U commute.

Let A be a normal operator, and let the operator B commute with A, i.e. $\mathbf{A B} = \mathbf{B A}$. Prove that B also commutes with $A^*$.

Let A, B be normal operators, and let the operator C satisfy the relation $\mathbf{A C} = \mathbf{C B}$. Prove that $\mathbf{A}^{*} \mathbf{C} = \mathbf{C B}^{*}$.

Prove that an operator A in a Hermitian space is normal if and only if in its Hermitian decomposition $\mathbf{A} = \mathbf{H}_{1}+i \mathbf{H}_{2}$ the self-adjoint operators $\mathbf{H}_{1}$ and $\mathbf{H}_{2}$ (the Hermitian components of the operator A) commute.

Prove that if the operator A is normal, then for any scalar $\lambda$ the operator $\mathbf{A}-\lambda \mathbf{E}$ is also normal.

Prove that an operator A in a Euclidean (Hermitian) space is normal if and only if in its polar decompositions $\mathbf{A} = \mathbf{P}_{1} \mathbf{U} = \mathbf{U P}_{2}$ the operators $\mathbf{P}_{1}$ and $\mathbf{P}_{2}$ coincide.

Prove that an operator A in a Euclidean (Hermitian) space is normal if and only if there exists a common canonical orthonormal basis for the nonnegative operator $\mathbf{P}_{1}$ and the orthogonal (unitary) operator $\mathbf{U}$ of its polar decomposition $\mathrm{A} = \mathbf{P}_{1} \mathbf{U}$. An analogous statement holds for the polar decomposition $\mathbf{A} = \mathbf{U P}_{2}$.

Prove that an operator A in a Euclidean (Hermitian) space is normal if and only if the eigenvalues of the nonnegative operator $\mathbf{P}_{1}$ in its polar decomposition $\mathrm{A} = \mathbf{P}_{1} \mathbf{U}$ are equal to the moduli of the eigenvalues of $\mathbf{A}$. An analogous statement holds for the polar decomposition $\mathbf{A} = \mathbf{U P}_{2}$.

Prove that an eigenvector $x$ of a normal operator A in a Euclidean (Hermitian) space with eigenvalue $\lambda$ is also an eigenvector of the operator $\mathbf{A}^{*}$ with eigenvalue $\bar{\lambda }$.

Prove that the orthogonal complement to an eigenvector of a normal operator A in a Euclidean (Hermitian) space is invariant under A and $A^*$.

Prove that eigenvectors of a normal operator in a Euclidean (Hermitian) space corresponding to different eigenvalues are orthogonal.

Prove that an operator A in a Hermitian space is normal if and only if there exists an orthonormal basis of the space consisting of eigenvectors. Is this statement true in a Euclidean space (compare with problem 1447)?

Prove that for any (finite or infinite) set of pairwise commuting normal operators in a Hermitian space there exists a common orthonormal canonical basis.

Prove that an operator A in a Hermitian space is normal if and only if every eigenvector of A is also an eigenvector of $A^*$. Is this statement true in a Euclidean space?

Show that in a Hermitian space invariant subspaces of a normal operator A are also invariant under $A^*$.

Prove that an operator A in a Hermitian space is normal if and only if the invariance of a subspace $W$ under A implies the invariance of $W^{\perp }$ under A. Is this statement true in a Euclidean space?

Prove that the restriction of a normal operator to an invariant subspace in a Hermitian space is a normal operator.

Prove that an operator A in a Hermitian space is normal if and only if the adjoint operator $A^*$ is a polynomial in A.

Prove that if $W$ is an invariant subspace of a normal operator A in a Euclidean space, then $W^{\perp }$ is also invariant.

Prove that a normal operator A in a Euclidean (Hermitian) space is:

Prove that an operator A in a Euclidean space is normal if and only if in some orthonormal basis it has a block-diagonal matrix consisting of blocks of size 1 and of size 2 of the form $\left(\begin{smallmatrix} \mu & -\nu \\ \nu & \mu \end{smallmatrix}\right)$. This matrix is called the canonical form of the normal operator, and the corresponding orthonormal basis is called the canonical basis of the normal operator.

Prove that in the canonical form of a normal operator A the two-dimensional blocks of the form

and the vectors $e_{2 k-1}, e_{2 k}$ of the canonical basis corresponding to such blocks, i.e.

can be found as follows.

Consider the complexification $\mathbf{A}_{\mathbb {C}}$ of our operator. The operator $\mathbf{A}_{\mathbb {C}}$ has the same matrix as $\mathbf{A}$. Let $\lambda = \mu +i \nu$ be a root of the characteristic polynomial of this matrix. Let us find a basis of the corresponding eigenspace $V_{\lambda } \subset V$ and orthogonalize it (with respect to the Hermitian scalar product): let $z_{1}, z_{2}, \ldots , z_{l}$ be the resulting vectors, where $x_{j}, y_{j}$ are their real and imaginary parts: $z_{j} = x_{j}+i y_{j}, j = 1,2, \ldots , l$.

Then the system of real vectors $x_{1}, y_{1}, x_{2}, y_{2}, \ldots , x_{l}, y_{l}$ is orthogonal, and $\left|x_{j}\right| = \left|y_{j}\right|, j = 1, \ldots , l$ (with respect to the Euclidean scalar product), while the operator A acts on the vectors of the system in the required way: $\mathbf{A}\left(x_{j}\right) = \mu x_{j}+\nu y_{j}, \mathbf{A}\left(y_{j}\right) = -\nu x_{j}+\mu y_{j}$. Normalize the vectors

and include them in the canonical basis: each such pair of vectors $x_{j}^{\prime }, y_{j}^{\prime }$ corresponds to the block $\left(\begin{smallmatrix} \mu & -\nu \\ \nu & \mu \end{smallmatrix}\right)$.

Prove that the minimal polynomial of a normal operator in a Euclidean (Hermitian) space is a product of pairwise distinct irreducible factors.

Find the canonical form and the corresponding orthonormal basis of the normal operator given (in some orthonormal basis) by the matrix

Find the canonical form and the corresponding orthonormal basis of the normal operator given (in some orthonormal basis) by the matrix

Prove that the operator given in some orthonormal basis of $\mathbb {R}^{3}$ by the matrix

is normal for any real values of $\lambda_{0}$. Find the canonical form and the corresponding orthonormal basis.