Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 3.
HYPONORMAL AND SUBNORMAL THEORY
projection of K onto H, then S ∗n f = P N ∗n f, f ∈ H. If f 0 , · · · , f n ∈ H then
∑
⟨S j f k , S k f j ⟩ = ∑ ⟨N j f k , N k f j ⟩
j,k
j,k
= ∑ ⟨N ∗k N j f k , f j ⟩
j,k
= ∑ ⟨N j N ∗k f k , f j ⟩
j,k
= ∑ ⟨N ∗k f k , N ∗j f j ⟩
j,k
∥ ∑ ∥∥∥∥
2
=
N ∗k f
∥
k .
So (3.3) holds.
(b) ⇒ (c): Put g k = S k f k . Then (3.3) implies
∑
⟨S j g k , S k g j ⟩ = ∑ ⟨S j+k f k , S j+k f j ⟩.
j,k
j,k
k
So (3.4) holds.
(c) ⇒ (a): See [Con2].
(b) ⇒ (d): If B 0 , · · · , B n ∈ C ∗ (S), let f k = B k f. Then
(3.3) ⇐⇒
⟨ ∑
j,k
B ∗ j S ∗k S j B k f, f
⟩
≥ 0.
(d) ⇒ (b): By Zorn’s lemma,
any operator = ⊕ star-cyclic operator.
So we may assume that S has a star-cyclic vector e 0 , i.e., assume H = cl [C ∗ (S)e 0 ].
If B 0 , · · · , B n ∈ C ∗ (S) then (3.3) holds for f k = B k e 0 . Since (3.3) holds for a dense
set of vector, (3.3) holds for all vectors.
(a) ⇒ (e): Let N = ∫ zdE(z) be the spectral decomposition of N, a normal
extension of S acting on K ⊇ H. Let P be the orthogonal projection of K onto H.
Define
Q(△) := P E(△)| H for every Borel subset △ of C.
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