This paper deals with the study of some operator inequalities involving the power -bounded operators along with the most known properties and results, in the more general framework of locally convex vector spaces.

1. Introduction

Let be a Hausdorff locally convex vector space over the complex field . By calibration for the locally convex space we understand a family of seminorms generating the topology of , in the sense that this topology is the coarsest with respect to the fact that all the seminorms in are continuous. Such a family of seminorms was used by the author and Wu [1] and many others in different contexts (see [25]).

It is well known that calibration is characterized by the property that the set is a neighborhood subbase at . Denote by the locally convex space endowed with calibration .

Recall that a locally convex algebra is an algebra with a locally convex topology in which the multiplication is separately continuous. Such an algebra is said to be locally -convex (l.m.c.) if it has a neighborhood base at such that each is convex and balanced (i.e., for ) and satisfies the property .

Any algebra with identity will be called unital. It is well known that unital locally -convex algebra is characterized by the existence of calibration such that each is submultiplicative (i.e., , for all ) and satisfies , where is the unit element.

An element of locally convex algebra is said to be bounded in if there exists such that the set is bounded in (see [6]). The set of all bounded elements in will be denoted by .

Let be the Alexandroff one-point compactification of . Following Waelbroeck [7, 8], we introduce the following.

Definition 1. We call resolvent set in the Waelbroeck sense of an element from a locally convex unital algebra the set of all elements for which there exists such that the following conditions hold:(a)the element is invertible in , for any ;(b)the set is bounded in .

The resolvent set in Waelbroeck sense of an element will be denoted by . The Waelbroeck spectrum of will be defined as

2. -Bounded Operators

Following Michael [9] (see also [2, 10]), we introduce the following.

Definition 2. We say that a linear operator is -bounded (quotient-bounded) with respect to if for any there exists such that

Denote by the set which consists of all -bounded operators with respect to calibration .

For a seminorm , the application defined as is also a seminorm. Note that We denote by the family of seminorms . The space will be endowed with a topology generated by . Remark that [9, Proposition 2.4(j)] implies that under this topology becomes a Hausdorff locally -convex topological algebra (in the sense of [9, Definition 2.1]).

If , the -spectral radius, denoted by , is considered as the boundedness radius in the sense of Allan [6] (see also [1113]), where, by common consent, .

The set of all bounded elements in will be denoted by (see [12]). It easily follows from [6, Proposition 2.14(ii)] that

For we denote by the Waelbroeck resolvent set of and by the Waelbroeck spectrum of . The function is called the resolvent function of . It is well known that

In this paper we evaluate the behaviour of the power of a -bounded operator from the algebra by some type of approximations. The main results have been announced in [14].

3. The Main Results

We continue to employ the notations from the previous sections and we will introduce two types of operatorial approximations for operators from the algebra which approximate a given operator on a convergent power bounded series. The power boundedness problem for operators acting on Banach spaces was largely developed in various frameworks by many authors (see [1517]).

In the following, using the functional calculus from the algebra (see [7, 8]), some important boundedness properties are obtained. Denote . First we have the following.

Theorem 3. If satisfies for , then for and for all with .

Proof. Assume that for . Since for , then, by using the generalized binomial formula, we get from where we deduce for any and any . Therefore, the conclusion is verified.

Conversely, we have the following.

Theorem 4. If and for all with , then for .

Proof. Let us suppose condition is true for all , for any and . For fixed, by choosing the integration path , with the aid of the functional calculus from the algebra , we obtain Thus, for all , we have which implies the desired result.

Moreover, we can formulate the following.

Theorem 5. If and for and for all with , then

Proof. Integrating by parts times, for , we obtain
Now choosing the circle of radius and by using the hypothesis, for , we get The last inequality was obtained by using Stirling’s approximation.

Now, for we introduce (see [18]) the following.

Definition 6. The Yosida approximation of , for , is defined as

Next theorem shows how an operator from the algebra is related to its Yosida approximation.

Theorem 7. The Yosida approximation is analytic for and the series representation converges for . Moreover, (1);(2)if there exists such that , then for ;(3).

Proof. By evaluating in terms of the resolvent , for we obtain from where it follows that the assertion of the theorem is true. Moreover, so (1) is true.
To prove (2) one can observe that, from it follows that on a set for which . Moreover, for .
A simple reasoning shows that ; then it follows .
From [19, Theorem 3.1.14], for , we have for all , and on , which could be written as , for any , so (3) is proved.

Below we state an equivalence between a power bounded operator from the algebra and the power of its Yosida approximation.

Theorem 8. Let and its Yosida approximation. Then the following assertions are equivalent:(i), for any ;(ii), for any and for all with .

Proof. Property (i) implies so that the argumentation given in the proof of Theorem 7 implies that any with belongs to the resolvent set of . Hence, using the generalized binomial formula, we get Now, by applying (i) again we obtain for any , whence by passing to supremum, the inequality (ii) holds.
Conversely, (i) is a direct consequence of (ii).

For , consider now the following Möbius transformation (see [20]):

Definition 9. The Möbius approximation of is defined as

Proposition 10. is holomorphic in and satisfies

Proof. Let . By evaluating the right member of the above equality, we get successively for . If , then from Definition 9 we have . On the other side converges to , when .

A similar result as in Theorem 8 is given below.

Theorem 11. Let and its approximation as above. Then the following assertions are equivalent:(i), for any ;(ii), for any and for every with .

Proof. From Theorem 8, for , is equivalent to The conclusion follows taking into account that for .

4. Application

For let be the space of continuous functions on endowed with the norm .

Consider , given by Following [19], we see that the resolvent of is given by the Yosida approximation of is and the Möbius approximation of is Remark that, for all , we have The above implies that is a contraction for .

If , then we can introduce for each the following norm on : Then a simple computation gives that On the other hand,

Remark that, by Theorem 11, for all , we get if and only if .

It is clear that for estimating the powers of it seems to be better to use the Yosida approximation or Möbius approximation than the resolvent approximation.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.


The author is grateful to the anonymous referees for their very careful reading and for useful suggestions that helped in better exposing this material.