• Aucun résultat trouvé

INTRODUCTION The notion of pseudo-resolvent has been introduced by E

N/A
N/A
Protected

Academic year: 2022

Partager "INTRODUCTION The notion of pseudo-resolvent has been introduced by E"

Copied!
9
0
0

Texte intégral

(1)

ABDUL SAMI AWAN and MIHAI VOICU

The present paper is divided into two sections. In the first section we recall the notions of resolvent and pseudo-resolvent, and a few remarkable properties. In the second section we introduce a new concept, theL-type pseudo-resolvent, and prove a characterization theorem. The generators ofL-type pseudo-resolvents are characterized in spectral terms. The connection between theL-type pseudo- resolvents andCo-equicontinuous semigroups is pointed out.

AMS 2000 Subject Classification: 47A56.

Key words: pseudo-resolvent, generator, semigroup.

1. INTRODUCTION

The notion of pseudo-resolvent has been introduced by E. Hille in his remarkable monograph [4]. In 1959, Kato [6] made a few interesting remarks on pseudo-resolvents and infinitesimal generators of semigroups.

Contraction pseudo-resolvents on Banach spaces have been considered by Cuculescu ([1], p. 261–281). He investigated excessive functions with respect to a pseudo-resolvent of integral operators.

Hirsch [5] defined in 1972 the concept of L-type pseudo-resolvent on Banach spaces. Voicu ([9], [11], [12], [14]) investigated pseudo-resolvents on locally convex spaces.

In the present paper we define a new concept, the L-type pseudo- resolvent, and prove a characterization theorem. We give necessary and suffi- cient conditions under which a linear operator is the generator of a L-type pseudo-resolvent.

We also prove that the class of generators ofL-type pseudo-resolvents generalizes the class of infinitesimal generators of Co-equicontinuous semi- groups acting on sequentially complete locally convex spaces.

Throughout the paper, X is a complex Hausdorff locally convex space, C the complex field, Cs(X) the continuous seminorms on X and L(X) the algebra of the continuous linear operators.

Definition 1.1. A mappingR: ∆⊂C→L(X) is calledpseudo-resolvent if (1) R(λ)−R(µ) = (µ−λ)R(λ)R(µ)

MATH. REPORTS10(60),1 (2008), 1–9

(2)

for allλ, µ∈∆. The resolvent equation (1) has a few interesting consequences stated below.

Proposition 1.1([3], p. 208). If R: ∆→ L(X) is a pseudo-resolvent then

1. R(µ)R(λ) =R(λ)R(µ) for all λ, µ∈∆;

2. kerR(λ), ker(I−λR(λ)), R(λ)(X)and (I−λR(λ))(X)do not depend on λ∈∆.

Definition 1.2. Let V :D→ X be a linear operator and λ∈C. We say that λ∈ρ(V) if

1. λI−V :D→X is one to one;

2. (λI−V)(D) =X and (λI−V)−1 : (λI−V)(D)→X is continuous.

The mapping λ → (λI −V)−1 = R(λ, V), λ ∈ ρ(V), is called the resolventof V.

Remark1.1. The mappingλ→R(λ, V) is not a pseudo-resolvent because R(λ, V)∈/ L(X)

Proposition 1.2 ([15], p. 211). Let V :D → X be a linear operator.

Suppose that ρ(V) 6= ∅ and (λI −V)(D) = X for all λ ∈ ρ(V). Then R :ρ(V)→L(X) defined by R(λ) = (λI−V)−1 is a pseudo-resolvent.

2. PSEUDO-RESOLVENTS

Proposition 2.1. The resolvent of a closed linear operator acting on a complete locally convex space is a pseudo-resolvent.

Proof. Let X be a complete locally convex space, V : D→ X a closed linear operator such that ρ(V)6=∅, and λ∈ρ(V).

Let y ∈ X and (xδ)δ∈∆ ⊂D a net such that (λI −V)(xδ) → y. Since (λI−V)−1is continuous, (xδ)δ∈∆is Cauchy, hence convergent. Letx= lim

δ∈∆xδ. Then lim

δ∈∆V(xδ) =λx−y. In addition, we have (xδ, V(xδ))→(x, λx−y). Since V is closed,x∈Dand V(x) =λx−y. It follows that (λI−D)(D) =X.

By Proposition 1.2, R:ρ(V)→ L(X) defined by R(λ) = (λI−V)−1 is a pseudo-resolvent.

Definition 2.1. A pseudo-resolventR: ∆→L(X) is called generated if∃ a linear operatorV :D=R(λ)(X)→Xsuch that ∆⊂ρ(V), (λI−V)(D) = X and R(λ) = (λI−V)−1 =R(λ, V), λ∈∆.

Actually, if such aV exists, it is unique and is called thegeneratorofR.

Proposition 2.2. Let R : ∆ → L(X) be a pseudo-resolvent. The fol- lowing statements are equivalent:

(3)

1. R is generated;

2. kerR(λ) ={0}, λ∈∆.

Proof. 1 ⇒ 2 is a consequence of Definition 2.1. If kerR(λ) ={0} then λI−R−1(λ) does not depend onλ∈∆ andV =λI−R−1(λ) is the generator of R. For details see ([15], p. 216).

Example 2.1. Let K(R) the family of compact subsets of R and X = {x:R→C:xis bounded on the compact subsets ofR}. For eachK∈ K(R) consider the functional defined by

ρK(x) = sup

t∈K

|x(t)|.

Then (X, ρK)K∈K(R) is a Hausdorff locally convex space. LetK0 ∈ K(R) and T :X→X defined by

T(x)(t) =

x(t) fort∈K0 0 fort /∈K0.

It is clear that T is linear and for anyK ∈ K(R) we have ρK(T(x))6ρK(x).

Moreover, it follows that T2 =T, so T is a projection. Let ∆ = C\{0} and R : ∆→L(X) defined by R(λ) = λ1T. Let λ, µ∈∆. Then we have

(2) R(λ)−R(µ) = µ−λ

λµ T

(3) (µ−λ)R(λ)R(µ) = µ−λ λµ T.

Thus, we get

(4) R(λ)−R(µ) = (µ−λ)R(λ)R(µ), that is R is a pseudo-resolvent.

Remark 2.1. R is not generated because T is not one to one.

We now introduce a remarkable class of generated pseudo-resolvents, closely connected with semigroups.

Definition 2.2. A pseudo-resolvent R : ∆→ L(X) is called of L-type if∃ ω∈Rsuch that (ω,∞)⊂∆ and lim

n nR(n)(x) =x, ∀x∈X.

Remark 2.2. It is clear that aL-type pseudo-resolvent is generated and R(λ)(X) =X.

Example2.2. LetX =C([0,1]) be the set of all continuous real functions defined on [0,1]. It is well known that X is a Banach space with respect to

(4)

the norm

kfk= sup

t∈[0,1]

|f(t)|, f ∈X.

Consider also the subspace D={u :u ∈C1([0,1]), u(0) = 0} of X and the linear operator V : D → X defined by V(u) = −u0. Let λ∈ R and f ∈ X.

The first order differential equation λu+u0 =f has a unique solutionu∈D and u(x) = Rx

0 eλ(t−x)f(t)dt. This is equivalent to saying that ∀λ ∈ R, the linear operator λI−V :D→X is a bijection and (λI−V)(u) =f. For any x∈[0,1] the inequalities

(5)

(λI−V)−1(f)(x)

=|u(x)|6 Z x

0

eλ(t−x)|f(t)|dt61

λ(1−e−λx)kfk, ∀λ >0, do hold. Hence

(6)

(λI−V)−1(f)(x) 6 1

λkfk, ∀λ >0,

(7)

V−1(f)(x)

6kfk if λ= 0,

(8)

(λI−V)−1(f)(x) 6 1

|λ|e−λkfk if λ <0.

Hence ρ(V) = Rand, by Definition 2.1, R : R → L(X) defined by R(λ) = R(λ, V) = (λI−V)−1 is a generated pseudo-resolvent. On the other hand, one can remark that R is not a L-type pseudo-resolvent because D is not dense in X.

Remark 2.3. The class ofL-type pseudo-resolvents is strictly included in the class of generated pseudo-resolvents.

Theorem 2.1. Let R : ∆ → L(X) be a pseudo-resolvent and ω ∈ R such that (ω,∞)⊂∆. Then the two statements below are equivalent:

1. R is of L-type; 2.





a) R(λ) (X) =X b)

∀p∈Cs(X), ∃ q∈Cs(X) such that lim sup

n

p(nR(n) (x))≤q(x) ∀x∈X.

Proof. 1⇒2 is a simple consequence of Definition 2.2.

2⇒1.Letµ∈∆ and x∈X. By the resolvent equation we have (9) nR(n)R(µ)(x) = 1

µ−nnR(n)(x) + n

n−µR(µ)(x), n6=µ

By 2 b), the sequence (nR(n)(x))n∈N, n>ω is bounded inX and lettingn→ ∞ in (9) we get

limn nR(n)R(µ)(x) =R(µ)(x).

(5)

Let now a net xj ∈R(µ)(X), j ∈J, such that xj →x. For n > ω and j ∈J we have

(10) nR(n)(x)−x=nR(n)(x−xj) +nR(n)(xj)−xj+xj−x.

Let p, q∈Cs(X) given by 2 b). Then by (10) we have

p(nR(n)(x)−x)6p(nR(n)(x−xj)) +p(nR(n)(xj)−xj) +p(xj−x).

Moreover, (11) limsup

n

p(nR(n)(x)−x)6q(x−xj) + limsup

n

p(nR(n)(xj)−xj)+p(xj−x).

Hence limsup

n

p(nR(n)(x)−x) = 0. Consequently, limn nR(n)(x) =x In conclusion, R is aL-type pseudo-resolvent.

Theorem 2.2. Let R : ∆→ L(X) be a L-type pseudo-resolvent, V : D→X its generator, and W : ∆→ L(X) defined by W(λ) =λ(λR(λ)−I).

Then the following assertions hold.

1. D= n

x∈X:∃lim

n W(n)(x) o

; 2. V(x) = lim

n W(n)(x), x∈D.

Proof. LetR : ∆→L(X) be a L-type pseudo-resolvent. Put M =

n

x∈X:∃lim

n W(n)(x) o

.

Letµ∈∆, x∈X and n∈N∩∆. By the resolvent equation, n(nR(n)R(µ)(x)−R(µ)(x)) =µnR(n)R(µ)(x)−nR(n)(x) Lettingn→ ∞we get

limn W(n)R(µ)(x) =µR(µ)(x)−x=V R(µ)(x).

Therefore,

R(µ)(X) =D⊂M.

Let now x∈M and z= lim

n W(n)(x). Then R(µ)(z) = lim

n R(µ)W(n)(x) = lim

n W(n)R(µ)(x) =µR(µ)(x)−x.

Thus, we have

R(µ)(z) =µR(µ)(x)−x and x=R(µ)(µx−z).

In conclusion, x∈Dand the proof is complete.

Theorem 2.3. A linear operator V : D → X is the generator of a L-type pseudo-resolvent if and only

(6)

1. V is closed and D=X.

2. ∃ω∈Rsuch that (ω,∞)⊂ρ(V)and (λI−V)(D) =X for all λ > ω.

3. ∀p∈Cs(X), ∃q ∈Cs(X) such that limsup

n

p(n(nI−V)−1(x))6q(x), ∀x∈X.

In this case, the mapping R : (ω,∞) →L(X) defined by R(λ) = (λI− V)−1 is a pseudo-resolvent of L-type whose generator is V.

Proof. IfR: ∆→L(X) is aL-type pseudo-resolvent whose generator is V, then the three conditions on V are consequences of Definition 2.2 and Theorem 2.1. If V fulfills the three conditions, one can defineR : (ω,∞) → L(X) byR(λ) = (λI−V)−1.

It follows from the last condition that∀p∈Cs(X)∃q ∈Cs(X) such that limsup

n

p(nR(n)(x))6q(x), ∀x∈X.

Since R(λ)(X) = D, R fulfills the conditions of Theorem 2.1. Conse- quently, R is a pseudo-resolvent ofL-type andV is its generator.

Remark 2.4. IfR0: ∆0 →L(X) andR: ∆→L(X) are pseudo-resolvents with a common generator V :D→X, then

R0(λ) =R(λ) = (λI−V)−1 for all λ∈∆0∩∆.

Corollary 2.1.Let X be a complete locally convex space. A linear ope- rator V :D→X is the generator of a L-type pseudo-resolvent if and only if it fulfills the three conditions below.

1. V is closed and D=X.

2. ∃ω∈R such that (ω,∞)⊂ρ(V).

3. ∀p∈Cs(X), ∃q ∈Cs(X)such that limsup

n

p(n(nI−V)−1(x))6q(x), ∀x∈X.

Proof. If λ∈ρ(V) then, according to Proposition 2.1, (λI−V)(D) =X.

Corollary 2.2.Let X be a Banach space. A linear operator V :D→ X is the generator of a L-type pseudo-resolvent if and only if the three conditions below hold.

1. V is closed and D=X.

2. ∃ω∈R such that (ω,∞)⊂ρ(V).

3. The sequence (n(nI−V)−1)n>ω is equicontinuous.

(7)

Proof. Necessity: By Theorem 2.3, the sequence (n(nI −V)−1)n>ω is pointwise bounded and, consequently, uniformly bounded, i.e., equicontinuous.

Sufficiency: By Theorem 2.3 the three conditions are also sufficient.

Remark 2.5. Corollary 2.3 covers the similar result given by F. Hirsch in [5].

Example2.3. LetX=C([0,∞)) denote the set of all complex continuous functions defined on [0,∞). For each n∈N consider the seminormpn:X → R defined by

pn(u) = sup

x∈[0,n]

|u(x)|.

Thus,Xis a Fr`echet space whose topology is given by the family of seminorms (pn)n∈N. Let i∈X be the identity function (i.e., i(x) =x, ∀x ∈[0,∞)) and the linear operator V :X→X defined byV(u) =−i2u, i.e.,

V(u)(x) =−x2u(x), ∀x∈[0,∞).

Let λ∈Cwith the property that Re(λ)>0 andu∈X such that (λI−V)(u) = 0X, λI(u)−V(u) = 0X,

(λ+x2)u(x) = 0, ∀x∈[0,∞), u(x) = 0, ∀x∈[0,∞), u= 0X, hence λI−V is one to one.

Let now w ∈ X be an arbitrary function and u = λ+iw2. Then (λI− V)(u) = w which shows that (λI −V)(X) = X. Let now u ∈ X, x ∈ [0, n]

and n∈N. Then we have

((λI−V)(u)) (x) = (λ+x2)u(x) and |(λI−V)(u)(x)|>|λ| |u(x)|. Hence

pn(λI−V)(u)>|λ|pn(u) which is equivalent to

pn(λ(λI−V)−1(w))6pn(w), ∀w∈X.

Therefore, V fulfills the conditions of Theorem 2.3 and, consequently, is the generator of aL-type pseudo-resolventR:{λ∈C: Re(λ)>0} →L(X) defined by

R(λ)(w) = (λI−V)−1(w) = w

λ+i2 for all w∈X.

Theorem 2.3 suggests a connection between theL-type pseudo-resolvents and semigroups. We have in this sense the following result.

Theorem2.4. Let X be a sequentially complete locally convex space and V :D→X a linear operator. The statements below are equivalent.

1. V is the infinitesimal generator of a Co-equicontinuous semigroup.

(8)

2. V is the generator of a L-type pseudo-resolvent R: (0,∞)→L(X) such that (nR(n))k, n>1, k∈N, be equicontinuous.

Proof. 1⇒ 2. LetT : [0,∞)→L(X) be a semigroup whose generator is V. By the characterization theorem for Co-equicontinuous semigroups ([15], p. 246), D =X, (0,∞) ⊂ρ(V) and (λI−V)(D) = X, ∀λ > 0. Moreover, we have

R(λ, V)(x) = (λI−V)−1(x) = Z

0

e−λtT(t)(x)dx, ∀λ >0 and x∈X.

Consequently, the family (λR(λ, V))k, k ∈N, λ > 0, is equicontinuous. By Theorem 2.3, the mapping R : (0,∞) → L(X) given by R(λ) = R(λ, V) is a L-type pseudo-resolvent whose generator isV.

2⇒ 1. By Theorem 2.3, D=X, (0,∞)⊂ρ(V) and (λI−V)(D) =X,

∀λ > 0. Now, it is sufficient to use the characterization theorem for Co- equicontinuous semigroups mentioned above. Therefore, V is the generator of a Co-equicontinuous semigroup S : [0,∞) → L(X). Let W : (0,∞) → L(X) introduced above byW(λ) =λ(λR(λ)−I).In this context, we haveS(t)(x) = limn etW(n)(x), ∀t>0 and x∈X.

Remark 2.6. The class of generators of L-type pseudo-resolvents is a proper extension of the class of the infinitesimal generators ofCo-equicontinuous semigroups acting on sequentially complete locally convex spaces.

REFERENCES

[1] I. Cuculescu, Markov Processes and Excessive Functions. Ed. Academiei, Bucure¸sti, 1968. (Romanian)

[2] E.B. Davies,One Parameter Semigroups. Academic Press, London, 1980.

[3] K. Engel and R. Nagel, One Parameter Semigroups for Linear Evolution Equations.

Springer-Verlag, Berlin, 1999.

[4] E. Hille, Functional Analysis and Semigroups.Amer. Math. Soc., Colloq. Publ. 31., Providence, RI, 1948.

[5] F. Hirsch,Families r´esolvants, gen´erateurs, cogen´erateurs, potentiels.Ann. Inst. Fourier 22(1972), 89–210.

[6] T. Kato, Remarks on pseudo-resolvents and infinitesimal generators of semigroups.

Proc. Japan Acad.35(1959), 467–468.

[7] W. Arendt, A. Garabosch, G. Greiner, U. Groh, H.P. Lotz, U. Moustakas, R. Nagel, F. Neubrander and U Schlotterbeck,One-parameter Semigroups of Positive Operators.

Lecture Notes in Math.1184. Springer-Verlag, Berlin, 1986.

[8] A. Pazy,Semigroups of Linear Operators and Application to Partial Differential Equa- tions.Appl. Math. Sci.44. Springer-Verlag, New York, 1983.

[9] M. Voicu,Positive resolvents and the property of dispersiveness in locally convex lattices.

Rev. Roumaine Math. Pures Appl.33(1988),5, 471–477.

(9)

[10] M. Voicu,Dissipative operators and resolvents. Rev. Roumaine Math. Pures Appl.42 (1997),5-6, 449–459.

[11] M. Voicu,Dissipative and accretive operators on locally convex spaces.Rend. Circ. Mat.

Palermo (2) Suppl. No.52, Vol. II (1998), 805–815.

[12] M. Voicu,Resolvents on locally convex spaces. In: Order Structures in Functional Analy- sis,4, pp. 204–258. Ed. Acad. Romˆane, Bucure¸sti, 2001.

[13] M. Voicu,Pseudo-resolvents and semigroups on locally convex spaces.In: Order Struc- tures in Functional Analysis,5, pp. 198–232. Ed. Acad. Romˆane, Bucure¸sti, 2006.

[14] M. Voicu, Pseudo-resolvents on locally convex spaces. Rend. Circ. Mat. Palermo (2) Suppl. No.76(2005), 655–665.

[15] K. Yosida, Functional Analysis. Springer-Verlag, Berlin, 1980.

Received 21 May 2007 School of Mathematical Sciences 68-B, New Muslim Town

Lahore, Pakistan abdulsamiawan@gmail.com

and

Technical University of Civil Engineering Department of Mathematics

B-dul Lacul Tei 124 020396 Bucharest, Romania

mvoicuy@yahoo.com

Références

Documents relatifs

We consider families of Darboux first integrals unfolding H ε (and its cuspidal point) and pseudo- Abelian integrals associated to these unfolding.. integrable systems,

It has been one important feature of the analysis of abstract symmetric semigroups in the eighties to show that functional Sobolev inequalities are equivalent to

We prove here our main result, Theorem 2: a square-free Weierstrass polynomial F ∈ K [[x]][y] is pseudo-irreducible if and only if it is balanced, in which case we

The main purpose of this paper is to construct a toy class of Feller semigroups, hence, Feller processes, with state space R n × Z m using pseudo- dierential operators not

Evans–Hudson flows are constructed for a class of quantum dynamical semigroups with unbounded generator on UHF algebras, which appeared in [Rev. It is shown that these flows are

http://www.numdam.org/.. Let G be a real linear algebraic group operating on the finite dimen- sional real vector space V. If Go is the connected component of the neutral element in

RHANDI, On the essential spectral radius of semigroups generated by perturbations of Hille-Yosida operators, Differential Integral Equations (to appear). [Pr]

SIMONENKO, On the question of the solvability of bisingular and poly- singular equations,