C-regularized solutions of ill-posed problems defined by strong strip-type operators

Cauchy problems of the form d u d t = q (A) u + h (t) {\frac{du}{dt}=q(A)u+h(t)}, 0 < t < T {0<t<T}, u (0) = φ {u(0)=\varphi}, are studied in a Banach space X where A is a strong strip-type operator and q (A) {q(A)} is a complex polynomial in A. In this case, the spectrum of A lies within a horizontal strip of height θ, and so potentially neither A nor - A {-A} generates a strongly continuous semigroup on X. Therefore, depending on the definition of q (A) {q(A)}, the original problem may be severely ill-posed. We utilize a functional calculus for strip-type operators in order to define an approximate operator f β (A) {f{\beta}(A)} such that f β (A) {f{\beta}(A)} is bounded for each β > 0 {\beta>0} and f β (A) χ → q (A) χ {f{\beta}(A)\chi\rightarrow q(A)\chi} as β → 0 {\beta\rightarrow 0} for χ in a suitable domain. We show that this approximation gives rise to regularization for the original problem with respect to a graph norm related to C-regularized semigroups. We also fit the theory of the paper into a special case where iA generates a bounded, strongly continuous group on X. Under this assumption, which implies that A is a strong strip-type operator of height 0, results follow for a wide variety of ill-posed PDEs in L p {L{p}} spaces.