Page 325 - DMTH502_LINEAR_ALGEBRA

Unit 29: Spectral Theory and Properties of Normal Operators

(iii) each V has an orthonormal basis { , } with the property that Notes
j j j
T = a + b
j j j
T = – b + a .
j j j
2
In other words, if r = a 2 b and is chosen so that a = r cos and b = r sin , then V is an
orthogonal direct sum of two-dimensional subspaces V on each of which T acts as ‘r times
j
rotation through the angle .
The proof of Theorem 10 will be based on the following result.

2
Lemma: Let V be a real inner product space and S a normal operator on V such that S + I = 0. Let
be any vector in V and = S . Then
S *
S * … (1)

( | ) = 0, and = .
2
2
2
2
Proof: We have S = and S = S = – . Therefore 0 = S – + S + = S – 2(S | )
2
2
+ + S + 2(S | ) + 2 .
Since S is normal, it follows that
2
2
2
2
2
0 = S* – 2(S* | ) + + S* + 2(S* | ) + = S* + + S* – . 2
This implies (1); hence
( | ) = (S* | ) = ( |S )
= ( | – )
= – ( | )
and ( | ) = 0. Similarly
2
2 = (S* | ) = ( |S ) = .
Proof of Theorem 10: Let V , …, V be a maximal collection of two-dimensional subspaces
1 s
satisfying (i) and (ii), and the additional conditions.
T* = a – b ,
j j j
1 j s. … (2)

T* = b – a
j j j
Let W = V + … + V . Then W is the orthogonal direct sum of V , …, V . We shall show that
1 s 1 s
W = V. Suppose that this is not the case. Then W {0}. Moreover, since (iii) and (2) imply that W
–1
is invariant under T and T*, it follows that W is invariant under T* and T = T**. Let S = b (T – aI).
2
–1
2
Then S* = b (T* – aI), S*S = SS*, and W is invariant under S and S*. Since (T – aI) + b I = 0, it
2
follows that S + I = 0. Let be any vector of norm 1 in W and set = S . Then is in W and
S = – . Since T = aI + bS, this implies
T = a + b
T = – b + a .
By the lemma, S* = – , S* = , ( | ) = 0, and = 1. Because T* = aI + bS*, it follows that
T* = a – b
T* = b + a .

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