SOME PROPERTIES OF THE SUM OF LINEAR OPERATORS IN THE NON–DIFFERENTIAL CASE

Vol. 33, No. 1, 2003, 75–91

SOME PROPERTIES OF THE SUM OF LINEAR OPERATORS IN THE NON–DIFFERENTIAL CASE

Bruno de Malafosse¹

Abstract. In this paper we recall some results concerning an application of the sum of linear operators in the infinite matrix theory. Then, we give several extensions of these results in order to obtain new properties of infinite linear systems.

AMS Mathematics Subject Classification (2000): 46A15, 35A35

Key words and phrases: Infinite linear system, interpolation space, unital algebra, infinite matrix

1. Introduction

The theory of the sum of operators is well known, and has been studied by many authors such as Da Prato and Grisvard [1,2], Furman [3], R. Labbas and B. Terreni [7,8]. It can also be found in the work of de Malafosse [15], giving an application of the sum of operators in the infinite matrix theory in the commutative case. The aim is to study the equation

A.X+B.X−λX =Y, λ >0 (1)

in a Banach space E, where Y is given in E and A, B are two closed linear operators with domains D(A), D(B) included in E. This work extends the results obtained in the paper [6], entitled: ”An application of the sum of linear operators in infinite matrix theory”, whereE is not reflexive. Here, equation (1) is regarded as an infinite linear system in the space E = l^∞. A+B is considered as the sum of two particular infinite matrices defined respectively on D(A) = s_(1/an)

n and D(B) = s_1/β. In our case, it has been proved that the operators (−A) and (−B) are generators of analytic semigroups. The relative boundedness with respectAorBbeing not satisfied, the classical perturbation theory given by Kato [4] or Pazy [18], cannot be applied. The choice of the two infinite matrices is motived by the resolution of a class of non-symmetric linear infinite systems. Note that some results concerning the infinite matrices of operators are given in Maddox [9].

This work is organized as follows. In Section 2 are recalled the definitions and properties of some Banach spaces of infinite matrices used in R. Labbas and B. de Malafosse [5] and de Malafosse [11, 12, 14, 16]. We also give the main

1LMAH Universit´e du Havre, I.U.T., B.P 4006, Le Havre FRANCE.

results of the sum-strategy as in Labbas-Terreni [7] and define the two infinite matricesAandB regarded as two unbounded linear operators onE =l^∞ and study their sum. In Section 3 are used the regularity property ofAin order to give results on the sum of operators in the non-differential case. Further, we apply the previous results to new matrices obtained from the matrixA+B+λI.

Then, we deal with the linear infinite system (A+B+λI).X=Y forλ≥0.

Finally, we study the properties of^t(A+B+λI) when the trace of the kernull of the operatorA+B+λI onD(A)

D(B) is not reduced to {0}.

2. Recall of definitions and properties

2.1. SpacesS_c and s_c

For a sequencec= (c_n), wherec_n >0 for every integern≥1, we define the Banach algebra

S_c =

M = (a_nm)_n,m≥1| sup

n≥1

_∞

m=1

|anm|c_m c_n

<∞

, (2)

normed by

M_Sc = sup

n≥1

_∞

m=1

|a_nm|c_m c_n

,

see [10, 12, 14, 16, 17]. We also define the Banach spaces_c of one-row matrices by

s_c =

X= (x_n)_n | sup

n

|x_n| c_n <∞

, (3)

normed by

X_sc = sup

n

|xn| c_n . (4)

Ifc= (c_n)_n, andc= (c_n)_n are two sequences such that 0< c_n ≤c_n ∀n, then:

s_c⊂s_c.

A very useful particular case is the one when c_n =rⁿ, r >0. We denote then byS_rands_rthe spacesS_c ands_c. Whenr= 1 we obtain the space of the bounded sequencesl^∞=s₁.

IfI−M_Sc <1 we shall say that Asatisfies the condition Γ_c. Ifc= (rⁿ), Γ_c is replaced by Γ_r.

S_c being a unital algebra, we have the useful result:

ifM satisfies the condition Γ_c,M is invertible in the spaceS_cand for every B∈s_c the equation M X=B admits one and only one solution ins_c given by

X = ∞ n=0

(I−M)ⁿB.

(5)

Similarly, we define the Banach algebraS_r with unit element I= (δ_nm)_n,m≥1, (withδ_nm= 0 ifn=m, andδ_nn= 1) by

S_r=

M = (a_nm)_n,m≥1| sup

n

m

|a_nm|r^m−n

<∞ (6)

normed by

M_Sr = sup

n

m

|anm|r^m−n

, (7)

see [10-13]. Let us recall that the product of two matrices ofS_rbelongs to this space and

∀M ∈S_r ∀X∈s_r:M X∈s_r withM X_sr ≤ M_SrX_sr.

2.2. Sum of linear operators

We recall here some results given in Da Prato-Grisvard [1] and Labbas- Terreni [7]. E being a Banach space, we consider two closed linear operators A andB, whose domains are D(A) and D(B) included in E. For every X ∈ D(A)

D(B) we then define their sum SX =AX+BX. The spectral properties ofA andB are:

(H₁)

























∃CA, C_B>0, ε_A, ε_B ∈]0, π[ such that i)ρ(A)⊃

A={z∈C/|Arg(z)|< π−ε_A} (A−zI)⁻¹

L(E)≤ C_A

|z| ,∀z∈

A− {0}, ii)ρ(B)⊃

B={z∈C/|Arg(z)|< π−ε_B} (B−zI)⁻¹

L(E)≤ C_B

|z| ,∀z∈

B− {0}, iii)ε_A+ε_B< π

Here, we are not in the commutative case, that is:

(A−zI)⁻¹(B−zI)⁻¹−(B−zI)⁻¹(A−zI)⁻¹=

(A−zI)⁻¹,(B−zI)⁻¹

, is not equal to zero for all z∈ρ(A) and z ∈ρ(B). Furthermore, the density ofD(A) andD(B) being not true, we must assume that (see Labbas-Terreni [7], [8])

(H₂)













∃C >0, k∈N, (ξ_i)_1≤i≤k, (η_i)_1≤i≤k such that

∀i≥1 : 0≤1−ξ_i< η_i≤2 and µA(A−λI)⁻¹.

A⁻¹; (B+µI)⁻¹

L(E)≤K(ε_A, ε_B)^k

i=1|λ|^−ξi|µ|^−ηi for |λ|, |µ| → ∞; λ∈ρ(A), µ∈ρ(B).

Now introduce the real interpolation space between D(A) and E, defined for allσ∈]0,1[ by

D_A(σ,∞) =

X∈E /sup

Z∈Γ

z^σA(A−zI)⁻¹X

E<∞

, (8)

Γ being a simple infinite sectorial curve lying inρ(A−λI)

ρ(−B). Here we have

D_A(σ,∞) =

X ∈E /sup

t>0

t^σA(A+tI)⁻¹X

E<∞

.

In the same wayD_B(σ,∞) denotes the interpolation space betweenD(B) and E. Now we can express the main result given in [8], where

δ= min

1≤i≤k(ξ_i+η_i−1)>0.

Theorem 1. Suppose that (H₁) and (H₂) are satisfied. There exists λ^∗ such that∀λ≥λ^∗ and∀Y ∈D_A(σ,∞)equation [A+B−λI].X=Y has a unique solutionX₀ in the space D(A)∩D(B)such that

i)(A−λI)X₀∈D_A(θ,∞) ∀θ∈]0,min (σ, δ) [, ii) BX₀∈D_A(θ,∞) ∀θ∈]0,min (σ, δ) [, iii)(A−λI)X₀∈D_B(θ,∞) ∀θ∈]0,min (σ, δ) [.

2.3. Definition of operators A and B

As in [6], we apply the results of the previous subsections to particular matrices.

LetAbe the infinite matrix:







a₁ b₁ O . . O a_n b_n

.





, (9)

where (a_n) and (b_n) satisfy

i)a_n >0 ∀n, (an) is strictly increasing, and lim

n→∞a_n =∞ ii)∃MA>0 such that: |bn| ≤M_Afor alln.

(10)

In the same way we denote byB the lower triangular matrix







β₁ O

. . γ_n β_n

O .





, (11)

where (β_n)_n and (γ_n)_n satisfy:





i)β_n>0 ∀n, andβ_2n→L, ii) (β_2n+1/a_2n+1)_n→ ∞,

iii)∃MB >0 such that |γn| ≤M_B for alln.

(12)

Ais defined onD(A) =s_(1/an)

nandBis defined onD(B) =s_(1/βn)

n, these spaces being included inE=l^∞=s₁. We deduce from (12) i), ii) thatD(A) is not included inD(B) andD(B) is not included inD(A). In [6], the following results are proved:

Proposition 2. In the Banach spaceE the two linear operatorsA andB are closed and satisfy

i)D(A) =s_(1/an)

n={X = (x_n)/ a_nx_n=O(1) (n→ ∞)}, ii) D(B) =s_(1/βn)

n, iii)D(A)=s₁,D(B)=s₁.

iv) There exist numbersε_A,ε_B>0 (withε_A+ε_B < π ) such that (A−λI)⁻¹

L(s1)≤ M

|λ|, ∀λ= 0 and |Arg(λ)| ≥ε_A, (B+µI)⁻¹

L(s1)≤ M

|µ|, ∀µ= 0 and |Arg(µ)| ≤π−ε_B. Now let us consider the following additional assumption onA

sup

n≥1

|b_n−1|β_n a_n

<∞. (13)

Then we have [6]

Proposition 3. Under (10), (12), and (13) there exists a constantK(ε_A, ε_B)

>0such that µA(A−λI)⁻¹.

A⁻¹; (B+µI)⁻¹

L(s1)≤K(ε_A, ε_B) 1

|λ| |µ| + 1

|µ|²

,

∀λ= 0 such that|Arg(λ)| ≥ε_A and∀µ= 0such that |Arg(µ)| ≤π−ε_B. As we deal with the non-commutative case, we must use the interpolation space defined in (8). It has been proved in [6] that for allθ∈]0,1[

D_A(θ,∞) =s_(1/aθ n)=

X = (x_n)/sup

t>0

t^θA(A+tI)⁻¹X

s1

<∞

. Then we can assert the following result:

Theorem 4. AandB satisfy the hypotheses (10), (12), (13). For anyθ∈]0,1[

there exists λ^∗ such that ∀λ≥λ^∗ and ∀Y ∈s_(1/aθ

n), the linear infinite system (−A−B−λI).X=Y has a unique solutionX₀ in the spaceD(A)∩D(B) = s_(1/an)∩s_(1/βn) such that

i)(A+λI)X₀∈s_(1/aθ n)_n, ii) BX₀∈s_(1/aθ

n)_n.

Remark 1. It is easy to see thats_(1/an)

n∩s_(1/βn)

n =s_d, where d= (d_n)_n is defined by

d_n=





 1

a_2k if n= 2k, 1

β_2k+1 otherwise.

Corollary 5. Conditions i) and ii) in Theorem 4 are equivalent to the condition AX₀∈s_(1/aθ

n)_n.

Proof. First we see that for every θ ∈]0,1[, s_d ⊂s_(1/aθ

n)_n. In fact, takeX = (x_n)_n∈s_d. Since we have (10) i),x_2n =O(1/a_2n) impliesx_2n =O

1/a^θ_2n as n→ ∞. And from (12) ii),x_2n+1 =O(1/β_2n+1) impliesx_2n+1 =O

1/a^θ_2n+1 asn → ∞. ThenX ∈s_(1/aθ

n). We deduce thatZ₀= (A+λI)X₀∈ s_(1/aθ n) is equivalent to

AX₀=Z₀−λX₀∈s_(1/aθ n).

Elsewhere i)⇔ii), sinceBX₀= (A+B+λI).X₀−Z₀∈s_(1/aθ

n). 2

3. New properties of the operator A + B, in the non–differential case

3.1. Consequence of the regularity property More precisely, from i) in Theorem 4 we have:

Corollary 6. The unique solutionX₀= x⁰_n

satisfies the following property

∀σ∈]0, θ[ x⁰_n= o(1)

a^σ+1_n (n→ ∞). Proof. From Corollary 5 we deduce thatAX₀∈s_(1/aθ

n)implies that there exists a realK >0 such that∀σ∈]0, θ[, ∀n

a^σ_na_nx⁰_n+b_nx⁰_n+1≤ K a^θ−σ_n . Then

a^σ_n

a_nx⁰_n+b_nx⁰_n+1

=o(1) (n→ ∞). (14)

On the other hand, (10) i) implies that there existsK >0 such that b_na^σ_nx⁰_n+1≤





 Ka^σ_2k

β_2k+1 ifn= 2k, Ka^σ_2k+1

a_2k+2 if n= 2k+ 1.

(15)

Using (10) i) and (12) ii) we deduce thatb_na^σ_nx⁰_n+1=o(1) (n→ ∞), and from (15) we conclude thata^σ+1_n x⁰_n=o(1) asntends to infinity. 2 3.1.1. Numerical application

Assume here that a_n = αⁿ with α > 1 and consider the matrix M_λ(t₁) = (A+B+λI) (t₁) obtained fromA+B+λI, by adding the supplementary row t₁ =

0, α^2ω,0, α^4ω, ...

with 1≤ω <2. Y(u) is the matrix obtained fromY by adding the numberu. From the regularity property we get

∀σ∈]0, θ[ x⁰_2n = o(1)

α^2n(σ+1) (n→ ∞),

for anyθ∈]0,1[. We deduce that∃C >0 such thatα^2ωmx⁰_2m≤C/α^{2m(σ+1−ω)}. Thus, the series

mα^2ωmx⁰_2m is convergent, since for a given ω ∈[1,2[ one can associate θ ∈]0,1[ and σ ∈]0, θ[, such that σ+ 1−ω > 0. Then the product M_λ(t₁)X₀ exists and belongs to the space s_(1/αnθ). The equation M_λ(t₁)X = Y(u), whereY ∈ s_(1/αnθ) admits a unique solution in s_d if and only ifu=

mα^2ωmx⁰_2m. Notice that the property: X₀∈D(A)∩D(B) =s_d is not sufficient to assure the convergence of the series

mα^2ωmx⁰_2m. 3.2. Expression of the solution in the Banach space s_d

In this section, we use only the hypotheses (10), (12) and the following supplementary condition

a_2n a_2n−1

∈s₁. (16)

Then we obtain the expression of the solutionX₀ in Theorem 4 for any second member Y ∈s₁. So we shall see that there exists Y ∈s₁−s_(1/aθ

n) such that the equation (A+B+λI)X =Y admits a solution in the spaceD(A)∩D(B) satisfying the propertyAX₀ ∈/ s_(1/aθ

n). In the case we shall study, this means that (A+B+λI) is a bijection fromD(A)∩D(B) intol^∞.

For the following results we shall putν_n=a_n+β_n for short.

Proposition 7. There existsλ^∗ >0 such that λ≥ λ^∗ implies that for every Y ∈s₁ equation

(A+B+λI)X =Y (17)

admits in D(A)∩D(B) = s_d a unique solution which can be written in the form

X₀= ∞ n=0

[I−D_λ(A+B+λI)]ⁿD_λY, (18)

whereD_λ= δnm

νn+λ

n,m≥1. Proof. (17) is equivalent to

D_λ[(A+B+λI)X] =D_λY itself equivalent to

[D_λ(A+B+λI)]X =D_λY.

(19) We see that

D_λ(A+B+λI) =









1 _ν^b1

1+λ γ2

ν2+λ 1 _ν^b2

2+λ O

. . .

O _ν^γn

n+λ 1 _ν^bn

n+λ

. .







.

We get

Dλ(A+B+λI)−I_Sδ = sup

supn≥1(µ_n),sup

n≥1(µ_n) , where

µ_n= γ_2n

ν_2n+λ a_2n

β_2n−1 + b_2n

ν_2n+λ a_2n

β_2n+1, (20)

and

µ_n=

γ_2n+1 ν_2n+1+λ

β_2n+1 a_2n +

b_2n+1 ν_2n+1+λ

β_2n+1 a_2n+2. (21)

We see that the sequence defined by ρ_n=|γ_2n| a_2n

β_2n−1 +|b_2n| a_2n β_2n+1,

is bounded. Indeed, from (12) and (16) ii) the sequence defined by a_2n

β_2n−1 = a_2n a_2n−1

a_2n−1 β_2n−1 is bounded. It is the same for the sequence

a_2n

β_2n+1 , since (a_n) is strictly increasing. Takingλ > ξ = sup_n≥1(ρ_n), we have sup_n≥1(µ_n)<1. Further, we deduce from (10) and (12) that there exists an integerN such that

n≥Nsup+1

|γ_2n+1|

a_2n +|b_2n+1| a_2n+2 <1,

which implies that sup_n≥N+1(µ_n)<1. Let now ξ_N =M_Bsup

n≤N

β_2n+1

a_2n +M_Asup

n≤N

β_2n+1 a_2n+2 . Ifλ≥ξ_N we have sup_n≤N(µ_n)<1 and

sup

n

(µ_n) = sup

sup

n≤N(µ_n), sup

n≥N+1(µ_n) <1.

Put nowλ^∗= sup (ξ, ξ_N ). Ifλ > λ^∗ sup

supn≥1(µ_n),sup

n≥1(µ_n) <1

and the matrixD_λ(A+B+λI) satisfies the condition Γ_d. So, (17) admits a unique solutionX₀ inD(A)∩D(B) =s_d for everyY such that

D_λY = y_n

ν_n+λ ∈s_d.

The previous property is also satisfieded for allY ∈s₁, since a_2n

y_2n ν_2n+1+λ

=O(1), β_2n+1

y_2n+1 ν_2n+1+λ

=O(1),

asntends to infinity. Then we can writeX₀ in the form (18). 2 3.3. Resolution of systems obtained from the preceding

In this subsection we suppose thatAandBsatisfied (10), (12) and (13). We are going to generalize the results of Theorem 4, in whichA+B+λI was an infinite tridiagonal matrix. So, we shall use an infinite upper triangular matrix P = (p_nm)_n,m≥1 ∈S₁ such thatp_nn = 1 ∀n, and consider the infinite matrix C= (c_nm)_n,m≥1 defined by

c_nm=

















ν₁+λ+p₁₂γ₂ ifm=n= 1,

p_1m−1b_m−1+p_1m(ν_m+λ) +p_1m+1γ_m+1 ifn= 1, m≥2,

p_nn−1γ_n−1 ifn≥3, m=n−2,

p_nn−1(ν_n−1+λ) +γ_n ifn≥2, m=n−1, p_nn−1b_n−1+ν_n+λ+p_nn+1γ_n+1 ifn≥2, m=n , p_nmb_m+p_nm+1(ν_m+1+λ) +p_nm+2γ_m+2 ifn≥2, m≥n+ 1, the other elements corresponding to 1≤m≤n−2 withn≥3 being equal to zero.

Proposition 8. Let θ∈]0,1[, and assume that the sequence(p_nm)_n,m≥1 satis- fies

sup

n

_∞

m=n+1

|pnm| a_n

a_m

θ

<1.

(22)

There existsλ^∗ > 0 such that for every λ≥ λ^∗ and Y ∈ s_(1/aθ

n)_n the system

defined by

m≥1

c_nmx_m=y_n (n= 1,2, ...) admits a unique solutionX₀ ins_d such thatAX₀∈s_(1/aθ

n)_n. Proof. We deduce from (22) thatP satisfies the condition Γ_(1/aθ

n), i.e.

I−P_S

(^1/aθn)<1.

(23)

We have C =P(A+B+λI). And since P ∈ S₁ the series of general terms p_nmγ_m−1x_m−1, p_nm(a_m+β_m)x_m and p_nmb_m+1x_m+1, are absolutely conver- gent. We deduce that the matrix equationCX =Y is equivalent to

P[(A+B+λI)X] =Y and to

(A+B+λI)X =P⁻¹Y.

Using (23), we haveP⁻¹∈S_(1/aθ

n)_n, thenP⁻¹Y ∈s_(1/aθ

n)_nand Theorem 4 can

be applied. 2

This method allows us to consider systems having a zero on the main diago- nal. In this case, there exists no sequencec= (c_n), (c_n>0,∀n), such that the matrix of the coefficients satisfies the condition Γ_c. Takingλ₀> λ^∗, we deduce from the preceding, the following result:

Corollary 9. Let (α_n)_n≥2 be any sequence andθ∈]0,1[. Consider the system

(S₁)







(b₁γ₂−ν₁ν₂)x₂−ν₁b₂x₃=y₁, γ_nx_n−1+ (ν_n +α_nγ_n+1)x_n+

b_n+α_nν_n+1

x_n+1+α_nb_n+1x_n+2=y_n, n≥2, whereν_n =ν_n+λ₀, andγ₂= 0. Assume that

ν₁

|γ₂| a₁

a₂

θ

<1 and sup

n≥2

|αn| a_n

a_n+1

θ

<1.

(24)

Then for allY ∈s_(1/aθ

n)_n, the system(S₁)admits a unique solutionX₀= x⁰_n ins_d such that n

n≥1sup

a^θ_na_nx⁰_n+b_nx⁰_n+1<∞.

Proof. Define here the infinite matrixP = (p_nm)_n,mbyp_nm= 0 for everyn,m such thatm=n,n+ 1;p_nn= 1 for alln≥1;p₁₂=−^ν_γ¹₂, andp_nn+1 =α_n∀n≥ 2. (24) implies that I−P_S

(^1/aθn) <1. Doing the product P(A+B+λ₀I) we get the matrix:











0 b₁−^ν1_γ^ν₂² −^ν_γ¹₂b₂ 0

γ₂ ν₂ +α₂γ₃ b₂+α₂ν₃ α₂b₃ O .

. . . . . .

γ_n ν_n +α_nγ_n+1 b_n+α_nν_n+1 α_nb_n+1

. . .

O . . .









 .

We conclude reasoning as above. 2

3.4. Study of equation(A+B+λI)X =Y, for λ≥0

In this subsection we suppose that (10), (12) and (16) hold. Let κ be an integer and denote heret₁=

1, b₁

ν₁+λ,0, ... and t_n=

0, ..., γ_n

ν_n+λ,1, b_n

ν_n+λ,0, ... ,

where 2 ≤ n ≤ κ. t₁,...,t_κ are the κ first rows of D_λ(A+B+λI). Let Q_κ be the matrix obtained fromD_λ(A+B+λI) by replacing its κfirst rows t₁, t₂,...t_κ, bye₁, e₂,...e_κ, where e_n = (...0,1,0, ...), (1 being in the nthposition).

We haveQ_κ= (q_nm)_n,m≥1 withq_nn= 1 for allnand for everyn > κ q_nn−1= γ_n

ν_n+λ andq_nn+1= b_n

ν_n+λ, the other terms being equal to zero. Consider now the determinant

∆ (λ) =

1 b₁

ν₁+λ 0 . . t₁X_κ

γ₂

ν₂+λ 1 b₂

ν₂+λ 0 . t₂X_κ

0 γ₃

ν₃+λ 1 b₃

ν₃+λ 0 .

. . .

. 1 t_κ−1X_κ

O γ_κ

ν_κ+λ t_κX_κ ,

and recall some definitions and results given in [16, 12, 13]. LetM be an infinite matrix. M(t₁, t₂, ...t_k),wherekis an integer, is the infinite matrix obtained from M by addition of the following rows

t₁= (t_1,m)_m≥1, t₂= (t_2,m)_m≥1, ... t_k = (t_k,m)_m≥1, t_ii = 0 (i= 1,2, ...k),

wheret_ij is any scalar. In the same way, set

tY(u₁, u₂, ...u_k) = (u₁, ...u_k, b₁, b₂, ...),

and let D(k) be the diagonal matrix whose elements are the inverses of the diagonal elements ofM(t₁, t₂, ...t_k) = (a_nm), that is, D(k) = (a⁻¹_nnδ_nm). Then we have the following result:

Proposition 10. Let c= (c_n) withc_n>0for all nbe a sequence such that I−D(k)M(t₁, t₂, ...t_k)Sc<1,

(25) and

D(k)Y(u₁, u₂, ..., u_k)∈s_c, (26)

then

i) solutions ofM X=Y in the spaces_c are

X = [D(k)M(t₁, t₂, ...t_k)]⁻¹D(k)Y(u₁, u₂, ...u_k) u₁, u₂, ....u_k ∈C.

ii) The linear space KerM∩s_c of the solutions ofM X= 0 in the spaces_c is of dimensionk and is given by

(KerM)∩s_c =span(X₁, X₂, ...X_k) where

X_i= [M(t₁, t₂, ...t_k)]⁻¹.^te_i, i= 1,2, ....k.

Remark 2. The solutions given in i) can be also written asX =X₀+^k

i=1u_iX_i where X₀ = [D(k)M(t₁, t₂, ...t_k)]⁻¹D(k)Y(0,0, ..,0) is a particular solution of M X=Y.

From the preceding we can deduce the following result:

Proposition 11. i) For all λ≥ 0 such that ∆ (λ) = 0,(17) admits a unique solution ins_d for allY ∈s₁.

ii) If ∆ (λ) = 0, equation (17) where Y ∈ s₁, either does not admit any solution ins_d, or admits infinitely many solutions in s_d.

Proof. From (20) and (21) we see thatµ_n tends to 0 asntends to infinity and it is the same forµ_n, since

0≤µ_n≤ M_B

a_2n + M_A a_2n+2.

We deduce that there existsκsuch that I−Q_κSd <1. Denote now byP_κ,λ^∗ the matrix obtained fromQ_κby deleting itsκfirst rows and byY_κ ∈s₁ the one

column matrix^tY_κ=

y_κ+1 , y_κ+2, ...

withy_n=y_n/(ν_n+λ). Applying Propo- sition 10 we see that equation P_κ,λ^∗ X = Y_κ admits infinitely many solutions defined for all scalarsu₁, u₂,...,u_κ by

X = (Q_κ)⁻¹.^t

u₁, u₂, ...u_κ, y_κ+1, y_κ+2 , ...

. (27)

Let now

X₀= (Q_κ)⁻¹.^t

0,0, ...0, y_κ+1 , y_κ+2 , ...

. (28)

It is easy to see that theκ−1 first rows of (Q_κ)⁻¹aree₁,e₂,...e_κ−1, and if we denote byX_κ itsκ−th column we deduce, using (27) and (28), that

X=X₀+

κ−1

i=1

u_ie_i+u_κX_κ. (29)

Replacing now these solutions in theκfirst equations of the system D_λ[(A+B+λI)X] =D_λY,

(D_λ defined in Proposition 7), we obtain the finite linear system t_nX = y_n, n= 1, 2, ...κ. This one is equivalent to

(S)

κ−1

i=1

u_it_ne_i+u_κt_nX_ν =y_n−t_nX₀ n= 1,2, ..., κ,

whereu₁,u₂,...u_κare the unknowns. Doing the calculations oft_ne_i, (1≤n≤κ, 1 ≤i ≤ κ−1) we deduce that ∆ (λ) is the determinant of the coefficients of (S). One can apply the well-known results on finite linear systems and conclude, considering the cases where ∆ (λ) is equal to 0 or not. This completes the proof.

2

In the case when ∆ (λ) = 0, we have the following property: ifb_n= 0,∀n, the rank of the system (S) is equal toκ−1. In fact, it suffices to adapt Proposition 10 and apply this one to the matrix triangle obtained from D_λ(A+B+λI), by adding the rowe₁. Thus,

dim [KerD_λ(A+B+λI)]&

s_d= 1.

Then we can suppose for instance that the determinant ∆_κ(λ), obtained from

∆ (λ) by striking out the κthrow and theκthcolumn, is not equal to 0. Con- sidering then the determinant ∆_Y (λ) obtained from ∆ (λ) by replacing theκth column byy₁,...y_κwithy_n=y_n −t_nX₀, (1≤n≤κ) we have

Corollary 12. Assume that∆ (λ) = 0.

i) IfY ∈s₁ satisfies∆_Y (λ)= 0, equation (17) does not admit any solution.

ii) If ∆_Y (λ) = 0, there exist scalars α₁, α₂,...,α_κ−1, µ₁, µ₂, ...µ_κ−1, u_κ and a vectorX_κ ∈s_d such that∀Y ∈s₁ equation (17) admits infinitely many solutions ins_d which can be written

X =X₀+

X_κ+

κ−1

i=1

α_ie_i

u_κ+

κ−1

i=1

µ_ie_i. (30)

Proof. i) is obvious. Assertion ii). (17) being equivalent to (S), u_κ is the variable and there existα₁,α₂,...,α_κ−1,µ₁,µ₂, ...µ_κ−1 such that

u_i=α_iu_κ+µ_i i= 1,2, ..., κ−1.

Then using (29), the solutions can be written as

X=X₀+

κ−1

i=1

(α_iu_κ+µ_i)e_i+u_κX_κ,

which permits us to make the conclusion. 2

Remark 3. Let ^tY_κ =

y_κ+1 , y_κ+2, ...

. Then we see that for every Y_κ ∈ s₁, one can associate a unique X₀ ∈ s_d and a subspace V of C^κ, such that if (y₁, ...y_κ)∈V equation (17) admits ins_d the solutions X =X₀+xW₁+W₂, whereW₁=X_N +^κ−1

i=1α_ie_i,W₂=^κ−1

i=1µ_ie_i for all scalars x.

Remark 4. If we assume that (10), (12) and (13) hold, then Proposition 11 and Corollary 12 remain true if we replace the conditionY ∈s₁byY ∈s_(1/aθ

n)_n. Note that we do not have necessarily the property of regularity.

3.5. Property of the operator ^t(A+B+λI), for λ≥0

In this partAandBsatisfy (10) and (12). DenoteA+B+λI= (a_nm)_n,m≥1 for short and recall thatl¹={X = (x_n)/

n|x_n|<∞},then we have Proposition 12. Assume that for a real λ≥0

[Ker(A+B+λI)]&

D(A)&

D(B)={0}, (31)

there exists a non-empty setI⊂N^∗ such that∀b= 0, the equation

t(A+B+λI)X =b^te_n0 n₀∈I, (32)

does not admit any solution inl¹.

Proof. LetZ= (z_n) be a non-zero element of [Ker(A+B+λI)]

D(A) D(B) and denoteI={n∈N^∗/ z_n = 0}. Then for all χ= (χ_n)∈l¹ we have

∞ n=1

χ_{n ∞}

m=1

a_nmz_m

= 0,

and using the fact thatZ∈s_d we deduce that there existsK >0 such that for everyn≥2

∞ m=1

|a_nm| |z_m| |χ_n| ≤K(|γ_n|d_n−1+ν_nd_n+|b_n|d_n+1)|χ_n|;

and, since the series ^∞

n=2|γn|d_n−1|χn|, ^∞

n=2ν_nd_n|χn|, ^∞

n=2|bn|d_n+1|χn|are con- vergent we deduce that

∞ n=1

∞ m=1

|anm| |zm| |χn|<∞.

Thus ∞

n=1

χ_{n ∞}

m=1

a_nmz_m

= ∞ m=1

z_{m ∞}

n=1

a_nmχ_n

= 0.

Now let (τ_n) be a sequence such that for an integern₀ ∈I, τ_n0 = 0, the other terms being equal to 0. If the system

∞ n=1

a_nmχ_n=τ_m m= 1,2, ...

would admit a solutionχ= (χ_n) in the spacel¹we should have ∞

m=1

z_{m ∞}

n=1

a_nmχ_n

=z_n0τ_n0= 0,

which is contradictory. This completes the proof. 2 Remark 5. We see applying the proposition that if equation

t(A+B+λI)X =Y, (∀Y ∈s₁),

admits a solution in l¹ it has not a solution any more when the n−th term of Y, n ∈ I, is modified. Indeed, let n₀ ∈ I and denote by Y the ma- trix obtained from Y by replacing the n₀−th coefficient by another one. If the equation ^t(A+B+λI)X = Y admitted a solution in l¹ the equation

t(A+B+λI)X = Y would not admit any solution, since Y−Y is of the formb^te_n0,b= 0.