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 C_{o}-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, C_{s}(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

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 (x_{δ})δ∈∆ ⊂D a net such that (λI −V)(x_{δ}) → y. Since
(λI−V)^{−1}is continuous, (x_{δ})δ∈∆is Cauchy, hence convergent. Letx= lim

δ∈∆x_{δ}.
Then lim

δ∈∆V(x_{δ}) =λx−y. In addition, we have (x_{δ}, V(x_{δ}))→(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:

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

ρ_{K}(x) = sup

t∈K

|x(t)|.

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

T(x)(t) =

x(t) fort∈K_{0}
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 T^{2} =T, so T is a projection. Let ∆ = C\{0} and
R : ∆→L(X) defined by R(λ) = _{λ}^{1}T. 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

the norm

kfk= sup

t∈[0,1]

|f(t)|, f ∈X.

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

The first order differential equation λu+u^{0} =f has a unique solutionu∈D
and u(x) = Rx

0 e^{λ(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

e^{λ(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^{−λ}kfk 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∈C_{s}(X), ∃ q∈C_{s}(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).

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−x_{j}) + limsup

n

p(nR(n)(x_{j})−x_{j})+p(x_{j}−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

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∈C_{s}(X), ∃q ∈C_{s}(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∈C_{s}(X)∃q ∈C_{s}(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. IfR^{0}: ∆^{0} →L(X) andR: ∆→L(X) are pseudo-resolvents
with a common generator V :D→X, then

R^{0}(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∈C_{s}(X), ∃q ∈C_{s}(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.

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 seminormp_{n}:X →
R defined by

p_{n}(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) =−i^{2}u, i.e.,

V(u)(x) =−x^{2}u(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+x^{2})u(x) = 0, ∀x∈[0,∞), u(x) = 0, ∀x∈[0,∞), u= 0_{X},
hence λI−V is one to one.

Let now w ∈ X be an arbitrary function and u = _{λ+i}^{w}2. 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+x^{2})u(x) and |(λI−V)(u)(x)|>|λ| |u(x)|.
Hence

p_{n}(λI−V)(u)>|λ|p_{n}(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+i^{2} 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.

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 C_{o}-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^{−λt}T(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 C_{o}-
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 e^{tW(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.

[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