Woo Young Lee Lecture Notes on Operator Theory

CHAPTER 1.
FREDHOLM THEORY
1.9 The Riesz-Schauder (or Browder) Theory
An operator T ∈ B(X) is said to be quasinilpotent if
and is said to be nilpotent if
∥T n ∥ 1 n −→ 0
T n = 0 for some n.
An example for quasinilpotent but not nilpotent:
T : l 2 → l 2
T (x 1 , x 2 , x 3 , . . .) ↦−→ (0, x 1 , x 2
2 , x 3
3 , . . .).
An example for quasinilpotent but neither nilpotent nor compact:
T = T 1 ⊕ T 2 : l 2 ⊕ l 2 −→ l 2 ⊕ l 2 ,
where
T 1 : (x 1 , x 2 , x 3 , . . .) ↦−→ (0, x 1 , 0, x 3 , 0, x 5 , . . .)
T 2 : (x 1 , x 2 , x 3 , . . .) ↦−→ (0, x 1 , x 2
2 , x 3
3 , . . .).
Remember that if T ∈ B(X) we define L T , R T ∈ B(B(X)) by
L T (S) := T S and R T (S) := ST for S ∈ B(X).
Lemma 1.9.1. We have:
(a) L T is 1-1 ⇐⇒ T is 1-1;
(b) R T is 1-1 ⇐⇒ T is dense;
(c) L T is bounded below ⇐⇒ T is bounded below;
(d) R T is bounded below ⇐⇒ T is open.
Proof. See [Be3].
Theorem 1.9.2. If T ∈ B(X), then
(a) T is nilpotent =⇒ T is neither 1-1 nor dense;
(b) T is quasinilpotent =⇒ T is neither bounded below nor open.
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