Woo Young Lee Lecture Notes on Operator Theory

Chapter 3
Hyponormal and Subnormal
Theory
3.1 Hyponormal Operators
An operator A ∈ B(H) is called hyponormal if
[A ∗ , A] ≡ A ∗ A − AA ∗ ≥ 0.
Thus if A ∈ B(H) then
A is hyponormal ⇐⇒ ∥Ah∥ ≥ ∥A ∗ h∥ for all h ∈ H.
If A ∗ A ≤ AA ∗ , or equivalently, ∥A ∗ h∥ ≥ ∥Ah∥ for all h, then A is called a cohyponormal
operator. Operators that are either hyponormal or cohyponormal are
called seminormal.
Proposition 3.1.1. Let A ∈ B(H) be a hyponormal operator. Then we have:
(a) If A is invertible then A −1 is hyponormal.
(b) A − λ is hyponormal for every λ ∈ C.
(c) If λ ∈ π 0 (A) and Af = λf then A ∗ f = λf, i.e., ker (A − λ) ⊆ ker (A − λ) ∗ .
(d) If f and g are eigenvectors corresponding to distinct eigenvalues of A then
f ⊥ g.
(e) If M ∈ Lat A then A| M is hyponormal.
Proof. (a) Recall that if T is positive and invertible then
T ≥ 1 =⇒ T −1 ≤ 1 :
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