Normal Operator - Properties

Properties

Normal operators are characterized by the spectral theorem. A compact normal operator (in particular, a normal operator on a finite-dimensional linear space) is unitarily diagonalizable.

Let T be a bounded operator. The following are equivalent.

• T is normal.
• T* is normal.
• ||Tx|| = ||T*x|| for all x (use ).
• The selfadjoint and anti-selfadjoint parts of T (i.e., with rsp. commute .

If N is a normal operator, then N and N* have the same kernel and range. Consequently, the range of N is dense if and only if N is injective. Put in another way, the kernel of a normal operator is the orthogonal complement of its range; thus, the kernel of the operator coincides with that of for any . Every generalized eigenvalue of a normal operator is thus genuine. is an eigenvalue of a normal operator N if and only if its complex conjugate is an eigenvalue of Eigenvectors of a normal operator corresponding to different eigenvalues are orthogonal, and it stabilizes orthogonal complements to its eigenspaces . This implies the usual spectral theorem: every normal operator on a finite-dimensional space is diagonalizable by a unitary operator. There is also an infinite-dimensional generalization in terms of projection-valued measures. Residual spectrum of a normal operator is empty.

The product of normal operators that commute is again normal; this is nontrivial and follows from Fuglede's theorem, which states (in a form generalized by Putnam):

If and are normal operators and if A is a bounded linear operator such that, then .

The operator norm of a normal operator equals its numerical radius and spectral radius.

A normal operator coincides with its Aluthge transform.