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CivilComp Proceedings
ISSN 17593433 CCP: 84
PROCEEDINGS OF THE FIFTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 14
Complex Powers for Slightly NonNegative Operators C. Martínez^{1}, M. Sanz^{1} and A. Redondo^{2}
^{1}Department of Applied Mathematics, University of Valéncia, Burjassot, Spain
, "Complex Powers for Slightly NonNegative Operators", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Fifth International Conference on Engineering Computational Technology", CivilComp Press, Stirlingshire, UK, Paper 14, 2006. doi:10.4203/ccp.84.14
Keywords: complex powers, nonnegative operators, almost nonnegative operators, Gregularized resolvent.
Summary
The theory of fractional powers of nonnegative operators, [1,4], has been
extended to the operators with polynomially bounded resolvent, [3,6,7]. All of
them are almost nonnegative operators, which are considered in [5], where we
construct their complex powers and we prove their properties. However, the maximal
operators of a Gregularized 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 nonnegative 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 nonnegative, if its resolvent set contains the nonnegative 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 nonnegative 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 nonnegative operators, which are not almost nonnegative. 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 m1, the Komatsu operator , [4], is welldefined 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 m1, 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 m1. 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 Gregularized 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 m1, 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. References
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