Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 3.
HYPONORMAL AND SUBNORMAL THEORY
3.2 The Berger-Shaw Theorem
If A is a selfadjoint operator then A is said to be absolutely continuous if its scalarvalued
spectral measure is absolutely continuous with respect to the Lebesgue measure
on the line.
Let N = ∫ zdE(z) be the spectral decomposition of N. A scalar-valued spectral
measure for N is a positive Borel measure µ on σ(N) such that
µ(△) = 0 ⇐⇒ E(△) = 0.
Since W ∗ (N) is an abelian von Neumann algebra, W ∗ (N) has a separating vector e 0 ,
i.e.,
Ae 0 = 0 =⇒ A = 0 for A ∈ W ∗ (N).
Define µ on σ(N) by
µ(△) = ∥E(△)e 0 ∥ 2 .
In fact, this µ is the unique scalar-valued spectral measure for N.
Theorem 3.2.1. (Putnam, 1963) If S is a pure hyponormal operator and S = A+iB,
where A and B are selfadjoint then A and B are absolutely continuous.
Proof. See [Con2, p.150].
Definition 3.2.2. An operator T ∈ B(H) is said to be finitely multicyclic if there
exist a finite number of vectors g 1 , · · · , g m ∈ H such that
H = ∨ {
f(T )gj : 1 ≤ j ≤ m and f ∈ Rat σ(T ) } .
The vectors g 1 , · · · , g m are called generating vectors. If T is finitely multicyclic and
m is the smallest number of generating vectors then T is said to be m-multicyclic.
Theorem 3.2.3. (The Berger-Shaw Theorem) If T is an m-multicyclic hyponormal
operator then [T ∗ , T ] is a trace class operator and
tr [T ∗ , T ] ≤ m Area (σ(T )).
π
This inequality is sharp: indeed, consider the unilateral shift T :
⎡ ⎤
1
[T ∗ ⎢
, T ] = ⎣
0 ⎥
⎦ , σ(T ) = cl D, m = 1,
. ..
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