Celso Martínez

A functional calculus and fractional powers for multivalued linear operators

The class of univalent linear operators is unstable under the operations closure, inverse and adjoint. This is not the case if we consider the more general class of multivalued linear operators. On the other hand, Favini and Yagi [7] and Yagi [15] have proved existence and uniqueness theorems of the strict solutions of degenerate evolution equations by means of this class of operators. For these reasons, it is interesting to extend some results of functional calculus to the multivalued linear case as well as obtaining a theory of fractional powers for multivalued linear operators. This problem has already been studied by Alaarabiou [1, 2]. He extended the well-known Hirsch functional calculus (see [8, 9]) to the set M. of multivalued nonnegative linear operators in a Banach space. His main idea was to endow Λ4 with an appropriate topology so that if / e T+ (that is, f(l/z) is a Stieltjes transform of a non-negative Radon measure), then / : Λ4 —>• Λ4 is continuous. The basic properties of a functional calculus were proved in [1] by the above mentioned continuity. Nevertheless, this kind of reasoning does not allow us to obtain two fundamental properties: the product formula and the spectral mapping theorem. Moreover, this functional calculus does not generate interesting operators such as the fractional powers of complex exponent, or the semigroup generated by the fractional powers either, because the functions z, 0 < Reα < 1, and e~, 0 < a < 1/2 and t > 0, do not belong to the class T+. Sections 3 and 4 of this paper are devoted to improving a functional calculus valid for a wider class of functions T which contains the earlier mentioned functions. This process is not trivial because we have neither f(Λ4) C Λ4 nor continuity of / . So we have developed an original method to obtain the main properties of a functional calculus. First of all, in Theorem 3.2 we study the inverse operator of /(A). Then, in Proposition 3.5, we relate /(A) and / (A + ε) for ε > 0. These results enable us to

An Extension of the Hirsch Symbolic Calculus

The symbolic calculus developed by Francis Hirsch (in several papers, between 1972 and 1976) is an already classical theory that introduces and studies the operators f(A) associated to a non-negative linear operator A on a Banach space and to the Stieltjes transform f of a Radon measure μ. It is required that the operator A has a dense domain and that the measure μ, as well as the value f(∞)), are real and non-negative. These three conditions are essential in the proof of the main results, but they are very restrictive, since important cases are excluded, as the fractional powers Aα of complex exponent α, or of base A non-densely defined. In this paper we present a reconstruction of the Hirsch theory, without using those hypothesis.

About fractional integrals in the space of locally integrable functions

Abstract In this paper we study the classical fractional integrals of Riemann-Liouville and Weyl considering them as operators defined in the largest functional space where these operators can be defined. We describe the fractional derivatives and integrals of Riemann-Liouville as the fractional powers of suitable non-negative operators, and consequently we get important properties directly. This is not the case for the fractional integral of Weyl; however, we can obtain some interesting properties for these operators.

n-th roots of a non-negative operator. Conditions for uniqueness

In the present paper we study which restrictions must be imposed on the n-th roots of certain non-negative closed operators A on a Banach space so that these roots are unique.Counterexamples are given to show that the two results on this subject in the previous literature are incorrect.Finally, we obtain an explicit formula relating the canonical root A1/n to another given non-negative n-th root B, and this allows us to establish the conditions for a given element φ to yield the same value by both roots. This point of view, which had not been considered up to now, provides simple conditions for global uniqueness that need only to be checked on the subspaces D∞(A)=∩n≥1D(An) and R∞(A)=∩n≥1R(An).