Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 2.
WEYL THEORY
which tends to 0 with ϵ. It follows that T 0 = 0 and hence that
[ ]
[ ]
0 0
0 0
T = = T ST with S =
0 T 1 0 T1
−1
has a generalized inverse.
Corollary 2.2.16. If T ∈ B(X) is reguloid and has the SVEP then Weyl’s theorem
holds for T .
Proof. Immediate from Theorem 2.2.14.
Lemma 2.2.17. Let T ∈ B(X). If for any λ ∈ C, X T ({λ}) is closed then T has the
SVEP.
Proof. This follows from [Ai, Theorem 2.31] together with the fact that
X T ({λ}) = {x ∈ X : lim
n→∞ ||(T − λI)n x|| 1 n = 0}.
Corollary 2.2.18. If T ∈ B(X) satisfies
X T ({λ}) = N(T − λI) for every λ ∈ C, (2.23)
then T has the SVEP and both T and T ∗ are reguloid. Thus in particular if T satisfies
(2.23) then Weyl’s theorem holds for T .
Proof. If T satisfies the condition (2.23) then by Lemma 2.2.17, T has the SVEP. The
second assertion follows from [Ai, Theorem 3.96]. The last assertion follows at once
from Corollary 2.2.16.
An operator T ∈ B(X) is said to be paranormal if
||T x|| 2 ≤ ||T 2 x|| ||x|| for every x ∈ X.
It was well known that if T ∈ B(X) is paranormal then the following hold:
(a) T is normaloid;
(b) T has finite ascent;
(c) if x and y are nonzero eigenvectors corresponding to, respectively, distinct
nonzero eigenvalues of T , then ||x|| ≤ ||x + y|| ([ChR, Theorem 2,6])
In particular, p-hyponormal operators are paranormal (cf. [FIY]). An operator T ∈
B(X) is said to be totally paranormal if T − λI is paranormal for every λ ∈ C.
Evidently, every hyponormal operator is totally paranormal. On the other hand,
every totally paranormal operator satisfies (2.23): indeed, for every x ∈ X and λ ∈ C,
||(T − λI) n x|| 1 n ≥ ||(T − λI)x|| for every n ∈ N.
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