Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 3.
HYPONORMAL AND SUBNORMAL THEORY
Then Q is a POM with ∥Q(△)∥ ≤ 1 for all △. But Q is a spectral measure if and
only if P commutes with E(△) for any △.
If Q is a POM and ϕ is a bounded Borel function on X then ∫ ϕdQ denotes the
unique operator T defined by the bounded quadratic form
∫
⟨T f, f⟩ = ϕ(x)d⟨Q(x)f, f⟩.
Theorem 3.3.7. If S ∈ B(H), the following are equivalent:
(a) S is subnormal.
(b) (Bram-Halmos, 1955/1950) If f 0 , · · · , f n ∈ H then
∑
⟨S j f k , S k f j ⟩ ≥ 0. (3.3)
j,k
(c) (Embry, 1973) For any f 0 , · · · , f n ∈ H
∑
⟨S j+k f j , S j+k f k ⟩ ≥ 0. (3.4)
j,k
(d) (Bunce and Deddens, 1977) If B 0 , · · · , B n ∈ C ∗ (S) then
∑
Bj ∗ S ∗k S j B k ≥ 0
j,k
(e) (Bram, 1955) There is a POM Q supported on a compact subset of C such that
∫
S ∗n S m = z n z m dQ(z) for all m, n ≥ 0. (3.5)
(f) (Embry, 1973) There is a POM Q on some interval [0, a] ⊆ R such that
∫
S ∗n S n = t 2n dQ(t) for all n ≥ 0.
Proof. (a) ⇒ (b): Let N =
[ ] S ∗ H
0 ∗ H′
be a normal operator on K. If P is the
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