Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 2.
WEYL THEORY
2.6 Comments and Problems
(a) Transaloid and SVEP. For an operator T ∈ B(X) for a Hilbert space X,
denote W (T ) = {(T x, x) : ||x|| = 1} for the numerical range of T and w(T ) =
sup {|λ| : λ ∈ W (T )} for the numerical radius of T . An operator T is called convexoid
if conv σ(T ) = cl W (T ) and is called spectraloid if w(T ) = r(T ) = the spectral radius.
We call an operator T ∈ B(X) transaloid if T − λI is normaloid for all λ ∈ C. It was
well known that
transaloid =⇒ convexoid =⇒ spectraloid,
(G 1 ) =⇒ convexoid and (G 1 ) =⇒ reguloid.
We would like to expect that Corollary 2.2.16 remains still true if “reguloid” is
replaced by “transaloid”
Problem 2.1. If T ∈ B(X) is transaloid and has the SVEP, does Weyl’s theorem
hold for T
The following question is a strategy to answer Problem 2.1.
Problem 2.2. Does it follow that
transaloid =⇒ reguloid
If the answer to Problem 2.2 is affirmative then the answer to Problem 2.1 is affirmative
by Corollary 2.2.16.
(b) ∗-paranormal operators.
said to be ∗-paranormal if
An operator T ∈ B(X) for a Hilbert space X is
||T ∗ x|| 2 ≤ ||T 2 x|| ||x|| for every x ∈ X.
It was [AT] known that if T ∈ B(X) is ∗-paranormal then the following hold:
T is normaloid; (2.54)
N(T − λI) ⊂ N((T − λI) ∗ ). (2.55)
So if T ∈ B(X) is ∗-paranormal then by (2.55), T − λI has finite ascent for every
λ ∈ C. Thus ∗-paranormal operators have the SVEP ([La]). On the other hand,
by the same argument as the proof of Corollary we can see that if T ∈ B(X) is
∗-paranormal then
σ(T ) \ ω(T ) ⊂ π 00 (T ). (2.56)
However we were unable to decide:
Problem 2.3. Does Weyl’s theorem hold for ∗-paranormal operators
The following question is a strategy to answer Problem 2.3.
73