Algebra Structure of Operator-Valued Riesz Means

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