Normal Operators — Mathematics Knowledge System

Overview

A bounded operator N on a Hilbert space is normal if it commutes with its adjoint: NN* = N*N. Normal operators include self-adjoint operators (N = N*), unitary operators (N*N = NN* = I), and skew-adjoint operators (N* = -N). The spectral theorem for normal operators on a Hilbert space states that N is unitarily equivalent to a multiplication operator on some L^2 space, giving a complete diagonalisation.

Intuition

Normality is the condition that lets an operator be 'diagonalised' in an infinite-dimensional sense. A matrix is normal iff it has an orthonormal eigenbasis; the infinite-dimensional analogue is that the operator admits a spectral measure on its (complex) spectrum. The key property of normal operators is that their eigenspaces are reducing subspaces: if Nx = lambda x then N*x = bar(lambda) x, and the orthogonal complement of the eigenspace is also invariant.

Formal Definition

Definition

N in B(H) is normal if NN* = N*N. Equivalently, ||Nx|| = ||N*x|| for all x in H. The spectrum sigma(N) is a compact subset of C, and the spectral theorem provides a unique spectral measure E on the Borel subsets of sigma(N) such that N = integral_{sigma(N)} lambda dE(lambda) and N* = integral_{sigma(N)} bar(lambda) dE(lambda).

N N^* = N^* N

Normality condition

\|Nx\| = \|N^*x\| \quad \forall x \in H

Equivalent norm condition

N = \int_{\sigma(N)} \lambda\, dE(\lambda),\quad N^* = \int_{\sigma(N)} \bar{\lambda}\, dE(\lambda)

Spectral decomposition

Notation

Notation	Meaning
$NN^* = N^*N$	N is normal
$E$	Spectral measure of N
$W^*$	Von Neumann algebra generated by N and N*

Theorems

Theorem 6.1: Theorem 6.1 (Spectral Theorem for Normal Operators)

Every bounded normal operator N on a separable Hilbert space H is unitarily equivalent to a multiplication operator M_f on L^2(mu) for some measure mu and bounded measurable function f: (U*NU)(g) = f * g.

Theorem 6.2: Theorem 6.2

N in B(H) is normal if and only if ||Nx|| = ||N*x|| for all x in H. Self-adjoint operators (N = N*) and unitary operators (N*N = NN* = I) are normal.

Theorem 6.3: Theorem 6.3 (Eigenspace reduction)

If N is normal and Nx = lambda x for some lambda in C and x != 0, then N*x = bar(lambda) x. Consequently, eigenspaces of distinct eigenvalues are orthogonal.

Worked Examples

1
By definition, UU* = U*U = I, so U commutes with its adjoint trivially.
2
Hence U is normal.
3
The spectrum of U lies on the unit circle: if lambda in sigma(U) then |lambda| <= ||U|| = 1, and if |lambda| < 1 then U - lambda I = U(I - lambda U*) is invertible since ||lambda U*|| = |lambda| < 1 (Neumann series). So sigma(U) subset {|lambda| = 1}.

✓ Answer

Unitary operators are normal because UU* = U*U = I implies normality directly, with spectrum on the unit circle.

Practice Problems

Mediumproof writing

Show that if N is normal, eigenvectors for distinct eigenvalues are orthogonal.

MediumMultiple choice

Which class of operators is NOT always normal?

Self-adjoint operatorsUnitary operatorsThe sum of two normal operatorsSkew-adjoint operators (T* = -T)

Quiz

Summary

N is normal if NN* = N*N, equivalently ||Nx|| = ||N*x|| for all x.
Normal operators include self-adjoint, unitary, and skew-adjoint operators.
The spectral theorem: normal operators are unitarily equivalent to multiplication operators on L^2.
Eigenvectors for distinct eigenvalues are orthogonal; if Nx = lambda x then N*x = bar(lambda)x.
Normality is the precise condition for infinite-dimensional 'diagonalisability'.

References

BookConway, J. — A Course in Functional Analysis (2nd ed.), Chapter IX
BookRudin, W. — Functional Analysis (2nd ed.), Chapter 12
Websiteen.wikipedia.org