On the Abstract Subordinated Exit Equation

and Applied Analysis 3 As application, we consider the holomorphic case and we prove the following result: Theorem 1.3. We suppose that P is aC0-contraction holomorphic semigroup on B and β be a Bochner subordinator satisfying ∫1 0 1 s βt ds < ∞, t > 0. 1.9 Then each zero-integrable P-exit law ψ is subordinated to a unique P-exit law φ. Moreover, φ is given by φt ( q a ) PtVq ( ψ ) − bAPtVq(ψ) ∫∞ 0 ( PsVq ( ψ ) − Ps tVq(ψ))ν ds , t > 0, 1.10 where a, b, and ν are the parameters of β. The condition 1.9 is fulfilled for the fractional power subordinator and the Dirac subordinator. 2. C0-Contraction Semigroup For the following notions and properties about C0-contraction semigroups, we will refer essentially to 14, 15 cf. also 16, 17 . Let B, ‖ · ‖ be a real Banach space and let I be the identity operator on B. For a linear operator T : B → B, we denote also by ‖T‖ : sup‖f‖≤1‖Tf‖ the norm of T . If ‖T‖ < ∞, T is said to be bounded. We consider 0,∞ endowed with its Borel field A and a measure μ on 0,∞ ,A . We say that a property holds μ.a.e. if the set for which this property fails is μ-negligible. A B-valued function X : 0,∞ → B is said simple if there exists a disjoint sequence {Ai ∈ A : μ Ai < ∞}1≤i≤n and X1, . . . , Xn ∈ B such that X t ∑n i 1 Xi 1Ai t for all t > 0. A B-valued function X : 0,∞ → B is also denoted by X : Xt t>0. In this paper, we consider the integral in Bochner sense for functions X : a, b ⊂ 0,∞ → Bwhich are μ-strongly measurable i.e., there exists a sequence of simple functions Xn : a, b → B satisfying limn→∞‖Xn −X‖ 0, μ.a.e. . For such functions X, it is known that X is μ-Bochner integrable if and only if ∫b a ‖X s ‖ μ ds < ∞ cf. 15, page 133 . For such functions X, it is also known that for each bounded linear operator T : B → B, we have