T. Diagana INTEGER POWERS OF SOME UNBOUNDED LINEAR ...

214 T. Diagana
Similarly, from Eq. (18), one has:
|x m | ‖ f m ‖ = |x m | |̟m| 1/2
∣ ∑ ∣∣∣∣
ω
∣ n y n a nm
n∈N
=
|̟m| 1/2 . |µ m − µ|
⎡
sup
(|a nm | |ω n | 1/2)
⎤
≤ ⎢ n∈N
⎥
⎣ |µ m − µ| . |̟m| 1/2 ⎦ . ‖y‖
=
≤
⎡
( ) sup
(|a
1
nm | |ω n | 1/2)
⎤
. ⎢ n∈N
⎥
|µ m − µ| ⎣ |̟m| 1/2 ⎦ . ‖y‖
⎡
sup
(|a
1
inf m∈N |µ m − µ| . nm | |ω n | 1/2)
⎤
⎢ n∈N
⎥
⎣ |̟m| 1/2 ⎦ . ‖y‖.
Therefore
‖(A − µ) −1 γ
‖ ≤
inf |µ m − µ| ,
m∈N
⎡
sup
(|a nm | |ω n | 1/2)
⎤
where γ = sup ⎢ n∈N
⎥
⎣ |̟m| 1/2 ⎦ < ∞.
DEFINITION 7. Let z ∈ J(µ). Define integer powers A z of the diagonal operator
A by:
(19)
m∈N
In summary, ρ(A) = {λ ∈ K : λ ̸= µ i , ∀i ∈ N}.
Under previous assumptions, one defines integer powers of A by:
⎧
⎪⎨
⎪⎩
D(A z ) = {x = (x i ) ⊂ K : lim
i
|µ i | z |x i |‖ f i ‖ = 0},
A z x = ∑ i∈N
µ i z x i f i , for each x = (x i ) i∈N ∈ D(A z ).
Similar results as in Section 5 hold for those type of diagonal operators.
REMARK 3. We complete this paper by mentioning two challenging questions
related to powers of linear operators on E ω .