Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 3.
HYPONORMAL AND SUBNORMAL THEORY
3.4 p-Hyponormal Operators
Recall that the numerical range of T ∈ B(H) is defined by
{
}
W (T ) := ⟨T x, x⟩ : ||x|| = 1
and the numerical radius of T is defined by
{
}
w(T ) := sup |λ| : λ ∈ W (T ) .
It was well-known (cf. [Ha3]) that
(a) W (T ) is convex (Toeplitz-Haussdorff theorem);
(b) conv σ(T ) ⊂ cl W (T );
(c) r(T ) ≤ w(T ) ≤ ||T ||;
1
(d)
dist ≤ ||(T − (λ,σ(T )) λ)−1 1
|| ≤
dist (λ,cl . W (T ))
Definition 3.4.1. (a) T is called normaloid if ||T || = r(T );
(b) T is called spectraloid if w(T ) = r(T );
(c) T is called convexoid if conv σ(T ) = cl W (T );
(d) T is called transaloid if T − λ is normaloid for any λ;
(e) T is siad to satisfy (G 1 )-condition if
||(T − λ) −1 || ≤
1
dist (λ, σ(T )) , in fact, ||(T − 1
λ)−1 || =
dist (λ, σ(T )) .
(f) T is called paranormal if ||T 2 x|| ≥ ||T x|| 2 for any x with ||x|| = 1.
It was well-known that it T is paranormal then
(i) T n is paranormal for any n;
(ii) T is normaloid;
(iii) T −1 is paranormal if it exists;
and that
hyponormal ⊂ paranormal ⊂ normaloid ⊂ spectraloid.
Theorem 3.4.2. If T ∈ B(H) then
(a) T is convexoid ⇐⇒ T − λ is spectraloid for any λ, i.e., w(T − λ) = r(T − λ);
(b) T is convexoid ⇐⇒ ||(T − λ) −1 || ≤
1
dist (λ, conv σ(T ))
for any λ /∈ conv σ(T ).
97