Abstract

We characterize operator-valued Riesz means via an algebraic law of composition and establish their functional calculus accordingly. With this aim, we give a new integral expression of the Leibniz derivation rule for smooth functions.

1. Introduction

Every self-adjoint operator on a Hilbert space admits a spectral decomposition (on Borel subsets of its spectrum) which allows to define -functional calculus. Spectral operators of scalar type (introduced in [1]) on arbitrary Banach spaces also have this functional calculus. If spectral projections (associated to a given operator) are required to be uniformly bounded only to closed intervals, these operators are said to be well-bounded. Roughly speaking, in reflexive Banach spaces , a well-bounded operator with admits a projection-valued function such that [1–3].

Well-bounded operators on have a functional calculus for absolutely continuous functions [2], and Riesz means, (for and ), are properly defined by this functional calculus; in some sense, is the -times integral of the decomposition of the identity associated to ; see [4, page 332].

The aim of this paper is to introduce the operator-valued Riesz means (for ) in an axiomatic way taking into account the algebraic law of the composition , the uniform boundedness, and the summability property; see Definition 5. The starting point is a multiplication identity for scalar Riesz means (Proposition 1) and led us to a new expression to the Leibniz formula, Proposition 3. To conclude, we show that certain holomorphic -semigroups, -functional calculus, and operator-valued Riesz means are equivalent concepts, essentially up some regularity; see Section 4.

There exist alternative approaches to operator-valued Riesz means , (for ) mainly closer to approximation theory and Fourier multipliers; see, for example, [5, page 193], [6, Section 2], [7, Section 3], and the references therein. Note that our point of view may be applied in some of these settings.

Notation. In this paper, ; is the characteristic function in the set ; we write . is a Banach space, and is the set of linear and bounded operators on ; is a closed operator on , and is the spectrum of .

2. Riesz Functions

For , we consider Riesz functions given by for and . Note that for and for .

Proposition 1. Let and . Then,

Proof. Take such that ; note that For , the equality holds trivially.

We consider functions defined by and now define functions by integration: for and . It is straightforward to prove that for , , and . Note that these functions, and , up some factors, are the remainders of the Taylor series of and , respectively.

Note that for and . We remind the reader that the Fourier transform of a function in is defined by It is well known that is continuous on and when (the Riemann-Lebesgue lemma). In the case that for some , the Fourier transform of is defined in terms of a limit in the norm of of truncated integrals: that is, and , where and is the characteristic function of the interval ; see, for example, [8, Volume 2, page 254]. Then, the existence of is guaranteed only at almost every , and may be noncontinuous and the Riemann-Lebesgue lemma could not hold (unlike the case when ).

Remark 2. For and , we have and for . To show this, note that a.e., and then we have that for , , and . We apply the Fourier transform and Proposition 1 to get equality (10).

3. Leibniz Formula via Integrals

In this section, we give a new expression of the Leibniz formula for functions , where the set is the Schwartz class of functions on .

Proposition 3. For and , one has that for .

Proof. We write by the right part of equality (12). We prove the formula by the induction method. For , we have that for . Now, we suppose that , and we will show that , where we conclude that . By the induction method, we conclude equality (12).
For , we have that Note that, for , we obtain that for , and then for and . Using these equalities and the combinatorial identity we obtain that Similarly, we get Taking into account that we conclude the proof.

Remark 4. Note that the Leibniz formula holds for functions which belong to larger sets than .

Given , we consider the norm defined by and the Banach space obtained as the completion of in the norm . A function if and only if there exists has all derivatives up to order satisfying , for , and the th derivative is absolutely continuous with ; see [4, Corollary 3.2]. In particular, and , for ; see more details in [4, page 319]. Consider functions and , where for and .

In fact, the space is a Banach algebra under pointwise multiplication for (see [4, Proposition 3.4]). Applying Proposition 3, a second proof may be given.

4. Functional Calculus for Operator-Valued Riesz Means

We introduce the operator-valued Riesz means in an axiomatic way and use them to define a functional calculus which domain is the Banach algebra . Then, we may associate a closed operator to operator-valued Riesz means.

Definition 5. For , one calls operator-valued Riesz mean of degree any strongly continuous family such that it satisfies the following conditions:(i);(ii) for ;(iii)the equality holds for and .

For , we get the Fejer family considered in [9, Section 4]. If is an operator-valued Riesz mean of degree , then is also an operator-valued Riesz mean of degree , where for . Note that the Riesz function is an operator-valued Riesz mean of degree in the space (Proposition 1) and is an operator-valued Riesz mean of degree in the space (Remark 2).

Theorem 6. Let , and let be a Banach space.(i)Given an operator-valued Riesz mean of degree , , the map defined by is a bounded functional calculus such that and for .(ii)Conversely, given a bounded functional calculus such that for any , there exists an operator-valued Riesz mean of degree such that is given by formula (24).

Proof. (i) It is clear that expression (24) defines a bounded homomorphism and . Now, we check that for . Due to condition (i) of Definition 5, we apply the Fubini theorem to obtain that for . Now, we apply condition (iii) of Definition 5 and the equality for to get
It follows that where we have applied formula (12) in the last equality.
We get directly that for . Now, take ; then, there exists such that for . Note that and we conclude that for .
Now, we check part (ii). We define . It is clear that is a strongly continuous family of operators and . Note that and the first condition of Definition 5 holds. To check the third condition, note that (for and ) where we have applied Proposition 1. The proof of part (ii) of Definition 5 is similar to the proof of part (i) above.

Recall that a holomorphic -semigroup is a family of operators such that the map is analytic, , and for and .

Now, we consider a holomorphic -semigroup such that for some . There exists a big amount of these semigroups; see, for example, [10], [4, Section 7], [3, Lemma 2.3], [11], and the references therein. Then, the family of operators defined by is an operator-valued Riesz mean of degree for . Conversely, given an operator-valued Riesz mean of degree , , then the family of operators defined by is a holomorphic -semigroup such that In this case, and where the operator is the infinitesimal generator of a holomorphic -semigroup . These results are essentially given in [12, Theorem 3.1] for and extended [4, Theorem 4.1] under more general conditions than those considered here.

Recall the family of functions considered in Section 2. The following result appears in [9, Corollary 4.5] for . The proof for may be done directly using formula (10) and is also a consequence of the case . Similar result for uniformly bounded cosine functions holds.

Corollary 7. Let be a uniformly bounded -group generated by on a Banach space . Then, the family of operators defined by is an operator-valued Riesz mean of degree associated to .

Conflict of Interests

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

Acknowledgments

The author thanks warmly the careful reading, interesting comments, and valuable suggestions made by an anonymous referee which have led to the improvement of the final version of this paper. The author also thanks Eva Fašangová, Ralph Chill, María Martínez, and José E. Galé for several comments, remarks, and helpful discussions. This research was done in the Laboratoire de Mathématiques at Université Paul Verlaine Metz (France) during a visit of the author. The author has been partially supported by Project MTM2010-16679, DGI-FEDER, of the MCYTS, Project E-64, D.G. Aragón, and JIUZ-2012-CIE-12, Universidad de Zaragoza, Spain.