Direct and inverse abstract Cauchy problems with fractional powers of almost sectorial operators

We derive the explicit solution operator of an abstract Cauchy problem involving a time-variable coefficient and a fractional power of an almost sectorial operator. The time-variable coefficient is recovered by solving the inverse abstract Cauchy problem using the solution operator representation. As a complement, we also study similar problems by considering almost sectorial operators that depend on a time-variable.

💡 Research Summary

The paper investigates both direct and inverse abstract Cauchy problems that involve a time‑dependent coefficient and a fractional power of an almost sectorial operator. An almost sectorial operator A on a complex Banach space X is defined by two conditions: (a) its spectrum lies in a sector S_ω, and (b) its resolvent satisfies a power‑type bound ‖(z−A)^{-1}‖ ≤ C_μ |z|^{γ} for any z outside a larger sector S_μ, where −1 < γ < 0 and 0 ≤ ω < π. Because γ is negative, A does not generate a C₀‑semigroup, but for any α ∈ (0, π/(2ω)) the fractional power A^α can be defined via a functional calculus, and the family
 T_α(t) = (2πi)^{-1} ∫_{Γ_θ} e^{-t z^α} (z−A)^{-1} dz
forms an analytic semigroup of growth order κ = 1 + γ/α. This semigroup is singular at t = 0 (it is not strongly continuous there) but enjoys the usual semigroup properties for t > 0, including the commutation relation A^β T_α(t) = T_α(t) A^β for any β with Re β > 0.

Direct problem.
The authors consider the initial‑value problem
 u′(t) = φ(t) A^α u(t), u(0) = u₀ ∈ D(A^α),
where φ :