Gen. Math. Notes, Vol. 22, No. 2, June 2014, pp.7-21
c
ISSN 2219-7184; Copyright ICSRS
Publication, 2014
www.i-csrs.org
Available free online at http://www.geman.in
On the Spectra of the Operator of the First
Difference on the Spaces Wτ and Wτ0 and
Application to Matrix Transformations
Bruno de Malafosse
Université du Havre
76610 Le Havre, France
E-mail: bdemalaf@wanadoo.fr
(Received: 10-2-14 / Accepted: 19-3-14)
Abstract
Given any sequence τ = (τn )n≥1 of positive real numbers and any set E
of complex sequences, we write Eτ for the set of all sequences x = (xn )n≥1
such that x/a = (xn /an )n≥1 ∈ E. We define the sets Wτ = (w∞P
)τ and Wτ0 =
(w0 )τ , where w∞ is the set of all sequences such that supn (n−1 P nm=1 |xm |) <
∞, and w0 is the set of all sequences such that limn→∞ (n−1 nm=1 |xm |) =
0. Then we explicitly calculate the spectra σ (∆, Wτ ) and σ (∆, Wτ0 ) of the
operator of the first difference on each of the sets Wτ and Wτ0 . We then
determinethe sets (E,
mapping E to F , with
F ) of allmatrix transformations
h
h
E = Wτ (∆ − λI) , or Wτ0 (∆ − λI)
and F = sξ , or s0ξ for complex
numbers λ and h and obtain simplifications of these sets for some values of λ.
Keywords: spectrum of an operator, operator of the first difference, matrix
transformations, sets of strongly C1 summable to zero and bounded sequences.
1
Introduction
In this paper we consider spaces that generalize the well known sets w0 and w∞
introduced and studied by Maddox [15]. Recall that w0 and w∞ are the sets of
strongly C1 summable to zero and bounded sequences. In [19] Malkowsky and
8
Bruno de Malafosse
Rakočević gave characterizations of matrix maps between w0 , w, or w∞ and
p
w∞
and between w0 , w, or w∞ and l1 . In [11, 4] were defined the spaces wα (λ),
(c)
wα (λ) and wα0 (λ) of all sequences that are α−strongly bounded, summable and
summable to zero respectively. For instance recall that wα (λ) is the set of all
sequences (xn )n such that
n
1 X
|xk | = αn O (1) (n → ∞) .
λn k=1
(c)
It was shown that these spaces can be written in the form sξ , sξ and s0ξ under
(c)
some conditions on α and λ, where sξ , sξ and s0ξ were defined for positive
sequences ξ by (1/ξ)−1 ∗ χ and χ = `∞ , c, c0 , respectively, (cf. [4]). More
recently in [18] it was shown that if λ is a sequence exponentially bounded then
(w∞ (λ) , w∞ (λ)) is a Banach algebra. This result led to consider bijective
operators mapping w∞ (λ) into itself.
In [8] de Malafosse and Malkowsky gave among other things properties
of the spectrum of the matrix of weighted means N q considered
as operator
in the set sa . In [12] were given simplifications of the set s0α (∆ − λI)h +
(c)
sβ (∆ − µI)l where h, l are complex numbers, α, β are given sequences,
using spectral properties of the operator of the first difference in the sets s0α and
(c)
sβ , then characterizations of matrix transformations in this set were stated.
Here we deal with the spectrum of the operator of the first difference over
0
the spaces Wτ = Dτ w∞ and
matrix trans
Wτ = Dτ w0 , and we characterize
h
h
formations in the sets Wτ (∆ − λI) and Wτ0 (∆ − λI) . We then obtain
simplifications for these sets under some conditions on λ, h and on the sequence
τ.
This paper is organized as follows. In Section 2 we recall some results on
matrix transformations and define the sets w0 and w∞ of strongly C1 summable
to zero and bounded sequences. In Section 3 we give some properties of the
sets Wτ and Wτ0 . In Section 4 we deal with the spectra of the operator of
the first difference on Wτ and Wτ0 . In Section 5 we determine the sets (E,
F)
h
of matrix transformations mapping E to F , with E = Wτ (∆ − λI) , or
h
0
Wτ (∆ − λI) and F = sξ , or s0ξ , for complex numbers λ and h and obtain
simplifications for these sets for some values of λ.
2
Preliminaries and Well Known Results
For a given infinite matrix AP
= (anm )n,m≥1 we define the operators An for any
integer n ≥ 1, by An (x) = ∞
m=1 anm xm , where x = (xn )n≥1 and the series
On the Spectra of the Operator of the First...
9
are assumed to be convergent. So we are led to the study of the infinite linear
system An (x) = bn with n = 1, 2, ... where b = (bn )n≥1 is a one-column matrix
and x is the unknown, see for instance [4-8]. The equations An (x) = bn for
n = 1, 2, ... can be written in the form Ax = b, where Ax = (An (x))n≥1 . Let
E and F be two sets of sequences, then (E, F ) denotes the set of all operators
mapping E to F , [15]. We write s for the set of all complex sequences, `∞
and c0 for the sets of all bounded and null sequences. It is well known that
A ∈ (`∞ , `∞ ) if and only if
sup
n
∞
X
|anm | < ∞;
(1)
m=1
and A ∈ (c0 , c0 ) if and only if (1) holds and limn→∞ anm = 0 for all m ≥ 1.
A Banach space E of complex sequences with the norm kkE is a BK space
if each projection Pn : x 7→ Pn x = xn is continuous. We will write e =
(1, ..., 1, ...), and define by e(m) the sequence with 1 in the m − th position and
0 otherwise. A BK space E ⊂ s is said to have
every sequence x =
P AK if(m)
x
e
. The set B (E) of
(xm )m≥1 ∈ E has a unique representation x = ∞
m=1 m
∗
all operators L : E −→ E with the norm kLkB(E) = supx6=0 (kL (x)kE / kxkE )
is a Banach algebra and it is well known that if E is a BK space with AK, then
B (E) = (E, E). In all what follows we will use the set U + of all sequences
(un )n≥1 with un > 0 for all n. For any given sequence τ = (τn )n≥1 ∈ U + , we
write Dτ for the infinite diagonal matrix defined by [Dτ ]nn = τn . For any subset
E of s, Dτ E is the set of all sequences x = (xn )n such that (xn /τn )n≥1 ∈ E.
(c)
Note that have Dτ E = Eτ . Then we put Dτ c0 = s0τ , Dτ `∞ = sτ and Dτ c = sτ .
(c)
It is well known that each of the spaces s0τ , sτ and sτ is a BK space normed
by kxksτ = supn (|xn | /τn ), (cf. [6]). Recall the next elementary and useful
result.
Lemma 1 Let τ , ξ ∈ U + , and E, F ⊂ ω. Then A ∈ (Dτ E, Dξ F ) if and only
if D1/b ADξ ∈ (E, F ).
For λ = (λn )n≥1 ∈ U + define the triangle C (λ) by [C (λ)]nm = 1/λn for
m ≤ n. It can be easily shown that the matrix ∆ (λ) defined by

if m = n,
 λn
−λn−1 if m = n − 1 and n ≥ 2,
[∆ (λ)]nm =

0
otherwise,
is the
of C (λ). Using the notation |x| = (|xn |)n , we have [C (λ) |x|]n =
Pinverse
n
−1
λn
m=1 |xm |. In this way we consider the spaces of strongly bounded and
summable sequences w∞ (λ) and w0 (λ) defined by
w∞ (λ) = x = (xn )n≥1 ∈ s : C (λ) |x| ∈ `∞ ,
w0 (λ) = {x ∈ s : C (λ) |x| ∈ c0 } .
10
Bruno de Malafosse
These spaces were studied by Malkowsky, with the concept of exponentially
bounded sequences, see for instance [19]. Recall that Maddox [16] defined and
studied the previous sets where λn = n for all n and it is written w∞ (λ) = w∞
and w0 (λ) = w0 .
3
The Sets Wτ and Wτ0
In this section we state some results on the sets Wτ = Dτ w∞ and Wτ0 = Dτ w0
and deal with triangles ∆ρ and ∆Tρ mapping from Wτ to itself.
3.1
Some Properties of the Sets Wτ and Wτ0
Here we consider the sets Wτ = Dτ w∞ and Wτ0 = Dτ w0 , (see [17, 9]), which
can be written as
!
)
(
n
1 X |xm |
<∞
Wτ = x ∈ s : kxkWτ = sup
n m=1 τm
n
and
(
Wτ0 =
x ∈ s : lim
n→∞
n
1 X |xm |
n m=1 τm
!
)
=0 .
For τ ∈ U + it was shown in [9] that the sets Wτ and Wτ0 are BK spaces
normed by kkWτ and Wτ0 has AK, [9, Proposition 3.1, p. 54]. So We = w∞
and We0 = w0 . It was shown in [18, 7] that the class (w∞, w∞ ) is a Banach
algebra normed by kAk∗(w∞ ,w∞ ) = supx6=0 kAxkw∞ / kxkw∞ . In the following
we will write Dr = D(rn )n for any given r > 0, and define the sets
(
Wr = Dr w∞ =
x : sup
n
and
n
1 X |xm |
n m=1 rm
!
)
<∞
(
Wr0 = Dr w0 =
)
n
1 X |xm |
x : lim
=0 .
m
n→∞ n
r
m=1
Note that we have W1 = w∞ and W10 = w0 .
3.2
On the Operators ∆ρ and ∆+
ρ Considered as Maps
in Wτ and Wτ0
On the operators ∆ρ and ∆+
ρ considered as operators in Wτ . In all what
follows we use the convention x0 = 0. For given ρ = (ρn )n≥1 we will consider
On the Spectra of the Operator of the First...
11
the operator ∆ρ defined by [∆ρ x]n = xn −ρn−1 xn−1 for all n ≥ 1. Then putting
+ T
∆+
ρ = (∆ρ ) we obtain ∆ρ x n = xn − ρn xn+1 for all n ≥ 1. To state the next
Lemma we will put for τ ∈ U + and any integer k
ρ−
n (τ ) = ρn
τn−1
τn+1
and ρ+
for all n.
n (τ ) = ρn
τn
τn
Now recall the next lemma which is a direct consequence of [9, Proposition
3.3, pp. 56-57]
Lemma 2 Let ρ, τ ∈ U + .
i) Let χ be any of the symbols W or W 0 .
a) If ρ− (τ ) ∈ `∞ , then ∆ρ ∈ (χτ , χτ ) and
k∆ρ k∗(Wτ ,Wτ ) ≤ 1 + ρ− (τ )l∞ .
b) If limn→∞ |ρ−
n (τ )| < 1, then the operator ∆ρ is a bijection from χτ to
itself and
χτ (∆ρ ) = χτ .
ii) a) If ρ+ (τ ) ∈ `∞ , then ∆+
ρ ∈ (Wτ , Wτ ) and
+ ∗
∆ρ (W
τ ,Wτ )
≤ 1 + 2 ρ+ (τ )l∞ .
+
b) If limn→∞ |ρ+
n (τ )| < 1, then the operator ∆ρ is a bijection from Wτ
to itself.
Remark 3 The proof of i) b) for χ = W 0 comes from the fact that Wτ0 is a
BK space with AK which implies B (Wτ0 ) = (Wτ0 , Wτ0 ) is a Banach algebra.
4
On the Spectra of the Operator of the First
Difference on Wτ0 and Wτ
In this section we deal with the spectra of the operator of the first difference
∆ defined by ∆xn = ∆e xn = xn − xn−1 for all n, considered as an operator
from Wτ0 to itself and from Wτ to itself.
Let E be a BK space and A be an operator mapping E to itself, (note that
A is continuous since E is a BK space). We denote by σ (A, E) the set of all
complex numbers λ such that A − λI considered as an operator from E to
itself is not invertible. Then we write ρ (A, E) = [σ (A, E)]c for the resolvent
set, which is the set of all complex numbers λ such that λI − A considered
as an operator from E to itself is bijective. Recall that the resolvent set of a
linear operator on E is an open subset of the complex plane C. We use the
12
Bruno de Malafosse
notation D (λ0 , r) = {λ ∈ C : |λ − λ0 | ≤ r} for λ0 ∈ C and r > 0. Recently
the fine spectra of the operator of the first difference over the sequence spaces
`p and bvp , were studied in [1], where bvp is the space of p-bounded variation
sequences, with 1 ≤ p < ∞. In [2] there is a study on the fine spectrum of the
generalized difference operator B (r, s) on each of the sets `p and bvp . In [14]
there is a study of the spectrum of the operator of the first difference on the
(c)
sets sα , s0α , sα and `p (α) (1 ≤ p < ∞). In [13], among other things there is
a study of the spectrum of the operator B (r, s) on the sets sα and s0α . In the
following we deal with the spectra of ∆ in the sets Wτ and Wτ0 . For this we
need the next lemmas.
Lemma 4 Let u be a sequence with un 6= 0 for all n, and assume (un /un−1 )n ∈
c. We have
un ≤ 1.
u ∈ `∞ implies lim n→∞ un−1 Proof. Assume limn→∞ |un /un−1 | = L > 1. Then for 0 < ε < L − 1, there is
an integer N such that
un un−1 ≥ L − ε > 1 for all n ≥ N .
So we obtain
un un−1 uN ... |uN −1 | ≥ (L − ε)n−N +1 |uN −1 | for all n ≥ N .
|un | = un−1 un−2
uN −1 Since (L − ε)n−N +1 → ∞ (n → ∞) we conclude u ∈
/ `∞ .
Lemma 5 We have
(w0 , w0 ) ⊂ c0 , s0(n)n and (w∞ , w∞ ) ⊂ `∞ , s(n)n .
(2)
Pn
−1
Proof. Trivially we have c0 ⊂ w0 , and since
|x
|
/n
≤
n
n
k=1 |xk | we
0
0
deduce w0 ⊂ s(n) . Thus we have (w0 , w0 ) ⊂ c0 , s(n) . By similar arguments
n
n
we obtain (w∞ , w∞ ) ⊂ `∞ , s(n)n .
In the next result we put τ • = (τn−1 /τn )n≥2 .
Theorem 6 Let χ be any of the symbols W , or W 0 . Then
(i) If τ • ∈ `∞ , then we have
σ (∆, χτ ) ⊂ D 1, lim τn• .
n→∞
(3)
13
On the Spectra of the Operator of the First...
(ii) If τ • ∈ c, then we have
σ (∆, χτ ) = D 1, lim τn• .
n→∞
.
(iii) For any given r > 0, we have
σ (∆, χr ) = D (1, 1/r) .
0
Proof. (i) We only consider
the case χ = •W
,c the case χ = W can be obtained
in a similar way. Let λ ∈ D 1, limn→∞ τn , that is, λ 6= 1 and
lim τn• < |λ − 1| .
(4)
n→∞
Putting ρn = 1/ |λ − 1| for all n we have
1
•
−
τ
∈ `∞
ρn (τ ) =
|λ − 1| n n≥1
and inequality (4) means that limn→∞ |ρ−
n (τ )| < 1. By Lemma 2 where
∆ρ =
1
(∆ − λI)
1−λ
we deduce ∆ − λI is bijective from Wτ to itself. This shows that
ic
h D 1, lim τn•
⊂ ρ (∆, Wτ )
n→∞
and (3) in (i) is satisfied for χ = W . This concludes the proof of (i).
(ii) Case χ = W 0 . First we show
D 1, lim τn• ⊂ σ ∆, Wτ0 ⊂ D 1, lim τn• .
n→∞
n→∞
The inclusion σ (∆, Wτ0 ) ⊂ D (1, limn→∞ τn• ) is a direct consequence of (i), since
we have τ • ∈ c. Now we show
(5)
D 1, lim τn• ⊂ σ ∆, Wτ0 .
n→∞
Since the inclusion ρ (∆, Wτ0 ) ⊂ [D (1, limn→∞ τn• )]c is equivalent to (5), we will
show if λI − ∆ considered as an operator from Wτ0 to itself is invertible, then
λ 6= 1 and
|λ − 1| ≥ lim τn• .
n→∞
−1
We have (λI − ∆)
∈ (Wτ0 , Wτ0 ) if and only if
D1/τ (λI − ∆)−1 Dτ ∈ (w0 , w0 ) .
14
Bruno de Malafosse
Then by Lemma 5 we have (w0 , w0 ) ⊂ c0 , s0(n) , and
n
D(1/nτn )n (λI − ∆)−1 Dτ ∈ (c0 , c0 ) .
(6)
Now it is well known that (λI − ∆)−1 is the triangle defined for λ 6= 1, by
(λI − ∆)−1 nm =
(−1)n−m
for m ≤ n.
(λ − 1)n−m+1
We have
un = D(1/nτn )n (λI − ∆)−1 Dτ n1 =
τ1
for n ≥ 2,
nτn |λ − 1|n
and by Lemma 4 we obtain
un
n−1 1
= lim
τ • ≤ 1,
n→∞ un−1
n→∞
n |λ − 1| n
lim
and
1
lim τ • ≤ 1.
|λ − 1| n→∞ n
We conclude
o
n
ρ ∆, Wτ0 ⊂ λ ∈ C : |λ − 1| ≥ lim τn• and λ 6= 1
n→∞
and
D 1, lim τn• ⊂ σ ∆, Wτ0 .
n→∞
We then have
D 1, lim τn• ⊂ σ ∆, Wτ0 ⊂ D 1, lim τn•
n→∞
n→∞
and since σ (∆, Wτ0 ) is a closed subset of C, and D (1, limn→∞ τn• ) is the smallest
closed set containing D (1, limn→∞ τn• ), we conclude σ (∆, Wτ0 ) = D (1, limn→∞ τn• ) .
Case χ = W . The proof follows exactly the same lines that above. It
is enough to notice that by Lemma 5, the condition D1/τ (λI − ∆)−1 Dτ ∈
(w∞ , w∞ ) implies
D1/τ (λI − ∆)−1 Dτ ∈ `∞ , s(n)n ,
and
D(1/nτn )n (λI − ∆)−1 Dτ ∈ (`∞ , `∞ ) = S1 .
This completes the proof of (ii).
(iii) is an immediate consequence of (ii) with τn = rn .
15
On the Spectra of the Operator of the First...
5
Matrix Transformations in
Wτ0
(∆ − λI)
h
In this section we recall results on the sets (E, F ) where E is either w0 and
w∞ and F = `∞ , or c0 . Then we applythe resultsof Section
4 to determine
h
0
0
0
0
the sets (E , F ) where E is either Wτ (∆ − λI) , or Wτ (∆ − λI)h and
F 0 = sξ , or s0ξ .
5.1
Matrix Transformations in the Sets w0 and w∞
Here we recall some results that are direct consequence of [3, Theorem 2.4],
where it is written
∞
X
(an ) =
2ν
n≥1 M
ν=1
max
2ν ≤m≤2ν+1 −1
|am | .
(7)
sing the notation An = (anm )m≥1 we obtain the following.
Lemma 7 [3] (i) We have (w0 , `∞ ) = (w∞ , `∞ ) and A ∈ (w∞ , `∞ ) if and only
if
!
∞
X
2ν ν max
|anm | < ∞,
(8)
sup (kAn kM ) = sup
ν+1
n
n
ν=1
2 ≤m≤2
−1
(ii) A ∈ (w∞ , c0 ) if and only if
lim kAn kM = lim
n→∞
n→∞
∞
X
ν=1
!
2ν
max
2ν ≤m≤2ν+1 −1
|anm |
= 0.
(iii) A ∈ (w0 , c0 ) if and only if (8) holds and
lim anm = 0 for all m.
n→∞
5.2
Matrix Transformations in the Sets w0 (T ) and w∞ (T )
To state the next result we consider the matrix Σ+ , by [Σ+ ]nm = 1 for m ≥ n
and [Σ+ ]nm = 0 otherwise, and from any matrix A = (anm )n,m≥1 we define for
any integer i, the triangle W (i) by
(i) W nm = Σ+ D(ain )n T −1 nm for m ≤ n.
So an elementary calculations yield
∞
X
(i) W nm =
aik skm for m ≤ n,
k=n
(9)
16
Bruno de Malafosse
where T −1 is the triangle whose nonzero entries are defined by [T −1 ]nm = snm .
From [3, Lemma 4.1 and Theorem 4.2], we obtain the following.
Lemma 8 Let χ be any of the sets w∞ or w0 and Y be an arbitrary subset of
s. Then A ∈ (χ (T ) , Y ) if and only if
(i) AT −1 ∈ (χ, Y ),
(ii) W (i) ∈ (χ, c0 ) for all i ≥ 1.
From (9) and (7) we easily see that for every n ∈ N, there is ν (n) uniquely
defined with 2ν(n) ≤ n ≤ 2ν(n)+1 − 1, and for any i ≥ 1 we have
∞
∞
ν(n)−1
X
X
X
(i) ν(n)
ν
Wn =
aik skm + 2
max aik skm .
2 ν max
M
ν+1
ν(n)
2 ≤m≤2
−1 2
≤m≤n ν=0
k=n
k=n
Now we state the next lemma which isP
a direct consequence of Lemma 7 and
Lemma 8, where we have [AT −1 ]nm = ∞
k=m ank skm for all n, m.
Lemma 9 (i) A ∈ (w∞ (T ) , `∞ ) if and
a)
∞
X
sup
2ν ν max
ν+1
n
2 ≤m≤2
ν=0
only if
!
∞
X
a
s
< ∞.
nk km −1 (10)
k=m
b) For every i ≥ 1 we have
lim Wn(i) M = 0.
n→∞
(ii) A ∈ (w∞ (T ) , c0 ) if and only if (11) holds for all i, and
!
∞
∞
X
X
lim
2ν ν max
a
s
= 0.
nk km n→∞
2 ≤m≤2ν+1 −1 ν=0
(11)
(12)
k=m
(iii) A ∈ (w0 (T ) , `∞ ) if and only if (10) holds, and for each i we have
!
∞
∞
X
X
ν
sup
2 ν max
aik skm < ∞,
(13)
ν+1
2 ≤m≤2
−1 n
ν=0
k=n
and
lim
∞
X
n→∞
aik skm = 0 for all m
(14)
k=n
(iv) A ∈ (w0 (T ) , c0 ) if and only if (10) holds, (14) and (13) hold for all i,
and
∞
X
lim
ank skm = 0 for all m.
n→∞
k=m
17
On the Spectra of the Operator of the First...
The Operator (∆ − λI)h , where h ∈ C
5.3
For any given h ∈ C, we put
( −h (−h + 1) ... (−h + k − 1)
−h + k − 1
=
k!
k
1
if k > 0,
if k = 0 ,
(cf. [10]). To simplify we will write
−h + k − 1
[−h, k] =
.
k
It is known that (∆ − λI)h with λ 6= 1 is the triangle defined by
h
i
[−h, n − m]
(∆ − λI)h
=
for m ≤ n,
nm
(1 − λ)−h+n−m
see [10, Theorem 8, pp. 295-296].
5.4
The Sets (E, F ) where E is either
h
Wτ (∆ − λI) and F = sξ , or s0ξ .
Wτ0
h
(∆ − λI)
, or
In the following we consider for λ 6= 1, matrix transformations mapping in the
sets
!
(
)
n
1 X 1 [−h, n − m]
h
xm < ∞
Wτ (∆ − λI) = x ∈ s : sup
n m=1 τm (1 − λ)−h+n−m n
and
Wτ0 (∆ − λI)h =
(
x ∈ s : lim
n→∞
n
1X 1
n m=1 τm
!
)
[−h, n − m]
xm = 0 .
(1 − λ)−h+n−m To state the next result we put
χnm (i) =
∞
X
k=n
[h, k − m]
aik
(1 − λ)h+k−m
for n, m, i ≥ 1 integers.
Theorem 10 (i)
Let λ 6= 1. Then
h
a) A ∈ Wτ (∆ − λI) , sξ if and only if
sup
n
!
∞
∞
X
X
1
ank
ν
2 ν max
[h, k − m]
τm < ∞
h+k−m
ν+1
ξn ν=0 2 ≤m≤2 −1 k=m
(1 − λ)
(15)
18
Bruno de Malafosse
and for every n ∈ N, there is ν (n) uniquely defined with 2ν(n) ≤ n ≤ 2ν(n)+1 −1,
and for any i ≥ 1 we have
 

 1 ν(n)−1

X
ν
ν(n)


lim
2 ν max
|χnm (i)| τm + 2
max |χnm (i)| τm
= 0.
n→∞  ξn

2 ≤m≤2ν+1 −1
2ν(n) ≤m≤n
ν=0
(16)
b) A ∈ Wτ (∆ − λI)
if and only if (16) holds for every i ≥ 1 and
!
∞
∞
X
X
ank
1
ν
2 ν max
[h, k − m]
(17)
τm = 0.
lim
h+k−m
ν+1
n→∞
ξn ν=0 2 ≤m≤2 −1 k=m
(1 − λ)
c) A ∈ Wτ0 (∆ − λI)h , sξ if and only if (15) holds and for every n ∈ N,
h
, s0ξ
there is ν (n) uniquely defined with 2ν(n) ≤ n ≤ 2ν(n)+1 − 1, and for any i ≥ 1
condition (17) holds and
 

 1 ν(n)−1

X
ν
ν(n)


sup
2 ν max
|χnm (i)| τm + 2
max |χnm (i)| τm
< ∞.

2 ≤m≤2ν+1 −1
2ν(n) ≤m≤n
n  ξn
ν=0
Wτ0
d) A ∈
for all i, and
h
(∆ − λI)
, s0ξ
(18)
if and only if (15) holds, (17) and (18) hold
∞
1 X
ank
lim
[h, k − m]
τ = 0 for all m.
h+k−m m
n→∞ ξn
(1
−
λ)
k=m
(ii) Let h ∈ N. Assume that τ • = (τn−1 /τn )n≥2 ∈ `∞ and let λ such that
|λ − 1| > lim τn• .
n→∞
(19)
a) We have Wτ (∆ − λI)h , sξ = Wτ0 (∆ − λI)h , sξ and A ∈ Wτ (∆ − λI)h , sξ
if and only if
)
(
∞
1 X ν
sup
2
max
|anm | τm < ∞.
(20)
ξn ν=0 2ν ≤m≤2ν+1 −1
n
b) We have A ∈ Wτ0 (∆ − λI)h , s0ξ if and only if (20) holds and limn→∞ anm /ξn =
0 for all m ≥ 1.
c) We have A ∈ Wτ (∆ − λI)h , s0ξ if and only if
)
(
∞
1 X ν
2
max
|anm | τm = 0.
lim
n→∞
ξn ν=0 2ν ≤m≤2ν+1 −1
19
On the Spectra of the Operator of the First...
Proof. (i) By Lemma 9 with T = (∆ − λI)h , we have T −1 = (∆ − λI)−h
which is defined by
h
i
[h, n − m]
(∆ − λI)−h
=
for m ≤ n.
nm
(1 − λ)h+n−m
Then we have
∞
h
i
h
i
X
A (∆ − λI)−h
=
ank (∆ − λI)−h
nm
=
k=m
∞
X
km
[h, k − m]
k=m
We also have
∞
i
h
X
(i) W nm =
aik (∆ − λI)−h
k=n
km
ank
(1 − λ)h+k−m
for all n, m ≥ 1.
= χnm (i) for m ≤ n and for all i ≥ 1.
Then by Lemma 8 we have A ∈ Wτ (∆ − λI)h , sξ if and only if
D1/ξ A (∆ − λI)−h Dτ ∈ (w∞ , `∞ )
and
D1/ξ W (i) Dτ ∈ (w∞ , c0 ) for all i ≥ 1.
So there is ν (n) uniquely defined with 2ν(n) ≤ n ≤ 2ν(n)+1 − 1, and for any
i ≥ 1 we have
ν(n)−1
X
1
D1/ξ W (i) Dτ = 1
2ν ν max
|χnm (i)| τm + 2ν(n) max |χnm (i)| τm .
n M
ν+1
2 ≤m≤2
−1
ξn ν=0
ξn
2ν(n) ≤m≤n
Applying Lemma 7 with anm replaced by
∞
ank
1 X
[h, k − m]
τm
ξn k=m
(1 − λ)h+k−m
we obtain (i) a). Then replacing anm by
1
χnm (i) τm for m ≤ n and for i ≥ 1,
ξn
we obtain (i) b). The statements (i) c) and (i) d) can be shown in the same
way.
(ii) Since (19) holds, by Theorem
/ σ (∆, Wτ0 ), and ∆ − λI is
6, we have λ ∈
bijective from Wτ0 to itself and Wτ0 (∆ − λI)h = Wτ0 . Then
h
0
A ∈ Wτ (∆ − λI) , sξ = Wτ0 , sξ
if and only if D1/ξ ADτ ∈ (w0 , `∞ ), and we conclude applying Lemma 7. The
cases b) and c) are direct consequences of Theorem 6 and Lemma 7.
20
Bruno de Malafosse
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