Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 3.
HYPONORMAL AND SUBNORMAL THEORY
Definition 3.3.9. If S is a subnormal operator on H and N is a normal extension
of S to K then N is called a minimal normal extension of S if
K = ∨ }
{N ∗k h : h ∈ H, k ≥ 0 .
Proposition 3.3.10. If S is a subnormal operator then any two minimal normal
extensions are unitarily equivalent.
Proof. For p = 1, 2 let N p be a minimal normal extension of S acting on K p ⊇ H.
Define U : K 1 −→ K 2 by
U (N ∗ 1 n h) = N ∗ 2 n h (h ∈ H).
We want to show that U is an isomorphism. If h 1 , · · · , h m ∈ H and n 1 , · · · , n m ≥ 0
then
∥ ∑ ∥∥∥∥
2 ⟨ N2
∗ n ∑
k
h
∥
k = N ∗ n k
2 h k , ∑ ⟩
N ∗ n j
2 h j
k
k
j
= ∑ j,k
⟨N 2
n j
h k , N 2
n k
h j ⟩
= ∑ j,k
= ∑ j,k
⟨S n j
h k , S n k
h j ⟩
⟨N 1
n j
h k , N 1
n k
h j ⟩
∥ ∑ ∥∥∥∥
2
=
N1
∗ n k
h
∥
k ,
k
which shows that
U
]
N1
∗ n k
h k = ∑ k
N ∗ 2
n k
h k
[ ∑
k
is a well defined linear operator from a dense linear manifold in K 1 onto a dense linear
manifold in K 2 and U is an isometry. Also for all h ∈ H, Uh = h. Thus for h ∈ H
and n ≥ 0,
UN 1 N ∗ 1 n h = UN ∗ 1 n Sh = N ∗ 2 n Sh = N 2 N ∗ 2 n h = N 2 UN ∗ 1 n h,
i.e., UN 1 = N 2 U, so that N 1 and N 2 are unitarily equivalent.
Now it is legitimate to speak of the minimal normal extension of a subnormal
operator. Therefore it is unambiguous to define the normal spectrum of a subnormal
operator S, σ n (S), as the spectrum of its minimal normal extension.
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