Page 321 - DMTH502_LINEAR_ALGEBRA

Unit 29: Spectral Theory and Properties of Normal Operators

The theorem shows that the algebra  is commutative and that each element of  is a Notes
diagonalizable normal operator. We show next that  has a single generator.
Corollary: Under the assumptions of the theorem, there is an operator T in  such that every
member of  is a polynomial in T.

k
Proof: Let T = t P j where t , . . . , t are distinct scalars. Then
j
k
1
j 1
k
n
T = t P j
n
j
j 1
for n = 1, 2, . . . If
8
f = a x n
n
n 1
it follows that
8 8 k
n
f(T) = a T n a t P
n n j j
n 1 n 1 j 1
k 8
= a t n j P j
n
j 1 n 1
k
= f ( )P j
t
j
j 1
Given an arbitrary

k
U = c P j
j
j 1
in , there is a polynomial f such that f(t ) = c (1 j k), and for any such f, U = f(T).
j j
29.2 Properties of Normal Operators

In unit 26 we developed the basic properties of self-adjoint and normal operators, using the
simplest and most direct methods possible. In last section we considered various aspects of
spectral theory. Here we prove some results of a more technical nature which are mainly about
normal operators on real spaces.
We shall begin by proving a sharper version of the primary decomposition theorem of unit 18
for normal operators. It applies to both the real and complex cases.

Theorem 9: Let T be a normal operator on a finite-dimensional inner product space V. Let p be the
minimal polynomial for T and p , . . ., p its distinct monic prime factors. Then each p occurs with
1 k j
multiplicity 1 in the factorization of p and has degree 1 or 2. Suppose W is the null space of p (T).
j j
Then
(i) W is orthogonal to W when i j;
j i
(ii) V = W . . . W ;
1 k
(iii) W is invariant under T, and p is the minimal polynomial for the restriction of T to W ;
j j j

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