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Local and global solutions of well-posed integrated Cauchy problems

Pedro J. Miana — 2008

Studia Mathematica

We study the local well-posed integrated Cauchy problem v ' ( t ) = A v ( t ) + ( t α ) / Γ ( α + 1 ) x , v(0) = 0, t ∈ [0,κ), with κ > 0, α ≥ 0, and x ∈ X, where X is a Banach space and A a closed operator on X. We extend solutions increasing the regularity in α. The global case (κ = ∞) is also treated in detail. Growth of solutions is given in both cases.

Spectral mapping inclusions for the Phillips functional calculus in Banach spaces and algebras

Eva FašangováPedro J. Miana — 2005

Studia Mathematica

We investigate the weak spectral mapping property (WSMP) μ ̂ ( σ ( A ) ) ¯ = σ ( μ ̂ ( A ) ) , where A is the generator of a ₀-semigroup in a Banach space X, μ is a measure, and μ̂(A) is defined by the Phillips functional calculus. We consider the special case when X is a Banach algebra and the operators e A t , t ≥ 0, are multipliers.

Integrated version of the Post-Widder inversion formula for Laplace transforms

José E. GaléMaría M. MartínezPedro J. Miana — 2011

Studia Mathematica

We establish an inversion formula of Post-Widder type for λ α -multiplied vector-valued Laplace transforms (α > 0). This result implies an inversion theorem for resolvents of generators of α-times integrated families (semigroups and cosine functions) which, in particular, provides a unified proof of previously known inversion formulae for α-times integrated semigroups.

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