Complex powers of nondensely defined operators

Marko Kostic
2011 Publications de l'Institut Mathématique (Beograd)  
The power (−A) b , b ∈ C is defined for a closed linear operator A whose resolvent is polynomially bounded on the region which is, in general, strictly contained in an acute angle. It is proved that all structural properties of complex powers of densely defined operators with polynomially bounded resolvent remain true in the newly arisen situation. The fractional powers are considered as generators of analytic semigroups of growth order r > 0 and applied in the study of corresponding incomplete
more » ... sponding incomplete abstract Cauchy problems. In the last section, the constructed powers are incorporated in the analysis of the existence and growth of mild solutions of operators generating fractionally integrated semigroups and cosine functions. KOSTIĆ The central theme of this paper is the construction of the power ( By the usual series argument, we have that, under our standing hypotheses (♦) and (♦♦), there exist d ∈ (0, 1], C ∈ (0, 1), ε ∈ (0, 1] and M > 0 such that: is oriented so that Im(λ) increases along Γ 2 (α, ε, C) and that Im(λ) decreases along Γ 1 (α, ε, C) and Γ 3 (α, ε, C). Since there is no risk for confusion, we also write Γ for Γ(α, ε, C). The method established by Straub in [32] , the idea of Martínez and Sanz [22] in their construction of complex powers of nonnegative operators and the notion of stationary dense operators introduced by Kunstmann in [17] are essentially utilized in our analysis. We remove density assumptions from the definition of an (analytic) semigroup of growth order r > 0 and consider the negatives of constructed powers as the integral generators [18] of such semigroups. We also refer the reader to the constructions of powers obtained by deLaubenfels, Yao, Wang [7] and deLaubenfels, Pastor [8] in the framework of the theory of C-regularized semigroups. Suppose, for the time being, that A is densely defined and α 0. Then we introduce the complex powers of the operator −A as follows [32, 15] . Using the arguments given in the proof of [15, Proposition 3.1], we have that, for every ) dλ exists and defines a bounded linear operator. Then, for every b ∈ C, we define the operator J b by D(J b ) := D(A Re(b)+α +2 ) and J b x := I(b)x, −(α + 2) Re(b) < −(α + 1), I(b − Re(b) + α − 2)(−A) Re(b)+α +2 x, otherwise. Arguing as in [15, Proposition 3.2], we have (1.1) J b x =
doi:10.2298/pim1104047k fatcat:pdqgophdhjbwzmucclbtmmnvny