Keywords: complex powers, non-negative operators, almost non-negative operators, G-regularized resolvent.

The theory of fractional powers of non-negative operators, [1,4], has been extended to the operators with polynomially bounded resolvent, [3,6,7]. All of them are almost non-negative operators, which are considered in [5], where we construct their complex powers and we prove their properties. However, the maximal operators of a G-regularized type, considered by other authors, see [2], do not belong to the aforementioned classes and they are a particular case of a wide class of operators of which we call the slightly non-negative operators, and we denote by . In this paper, we study the general properties of these operators and we construct their fractional powers and we extend the former theory in the case of injective base operators.

We said that a lineal operator A defined in a complex Banach space X, is slightly non-negative, if its resolvent set contains the non-negative real axis and there exists two non negative integers m, n and some , such that, for all

Then, we said that A belongs to the class , and stands for the union of all . Obviously, the class is contained in and , and the class of almost non-negative ones is contained in some .

The first main result is the location of the spectrum of our operators. This fact allows us to construct slightly non-negative operators, which are not almost non-negative. Hence, the class strictly contains all of the above mentioned classes. Next we prove that if A belongs to , then for all complex exponent , with real part lower than -m-1, the Komatsu operator , [4], is well-defined on the range of certain bounded operators. In addition, for all integer numbers r, h, such that , the operator is bounded. When the base operator A is injective, this result is fundamental, since for the exponents with real part lower than -m-1, the power may be defined by diagonalization, in the form , by choosing s, t large enough. For the rest of exponents, we may define one, by . Notice that is a bounded operator, because the real part of always is lower than -m-1. Obviously, both definitions do not depend on the choice of s, t, n and m.

Our complex powers, verify the main properties of a satisfactory theory of complex powers. In fact, we prove the coherence with the integer exponents (i.e., and coincide with the base operator A and the identity operator, respectively), and the additivity, which asserts that the operator is always an extension of , and both operators coincide, when the domain of contains the domain of . The injectivity of base operator A, implies that its complex power is injective. Further, the operator belongs to the class , as well. We prove that the inverse of coincides with the operator , which coincides with .

The last part of the paper is devoted to extending the theory to the class of operators with G-regularized resolvent verifying the estimate (34). In this case, we see that the operator is bounded, under the same assumptions for , r and s. Thus, if the operator A is injective, we can reason in the similar way. For with real part lower than -m-1, we define , and for the remaining cases, . As before, both definitions are independent of the choice of s, t, n and m. All the processes run in the same form. Likewise, similar results are obtained.

1

A.V. Balakrishnan, "Fractional powers of closed operators and the semigroups generated by them", Pacific J. Math., 10, 419-437, 1960.

2

R. deLaubenfels, J. Pastor, "Fractional powers and logarithms via regularized semigroups", Proceedings volume of the Semigroups of operators: Theory and Applications 2001 Conference (Rio de Janeiro). To appear.

3

R. deLaubenfels, F. Yao, S. Wang, "Fractional powers of operators of regularized type", Journal of Mathematical Analysis and Applications, 199, 910-993, 1996. doi:10.1006/jmaa.1996.0182

4

C. Martínez, M. Sanz, "The Theory of Fractional Powers of Operators", North-Holland Mathematics Studies, Elsevier Science Publishers B.V., 2001.

5

C. Martínez, M. Sanz, A. Redondo, "Fractional powers of almost non-negative operators", Frac. Calc. & Applied Analysis, 8(2), 201-230, 2005.

6

F. Periago, B. Straub, "A functional calculus for almost sectorial operators and applications to abstract evolution equations", J. evol. equ., 2, 41-68, 2002. doi:10.1007/s00028-002-8079-9

7

B. Straub, "Fractional powers of operators with polynomially bounded resolvent and the semigroups generated by them", Hiroshima Math. J., 24, 529-548, 1994.

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