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Vector-valued Laplace transforms and Cauchy problems. (English) Zbl 0978.34001
Monographs in Mathematics. 96. Basel: Birkhäuser. xi, 523 p. (2001).
The present monograph (by 4 authors!) is apparently an important and very interesting presentation of the abstract Cauchy problem treated especially by means of the (vector-valued) Laplace and Laplace-Stieltjes transforms. It appears as a worthy continuation of the classical books by Hille [Am. Math. Soc. (1948; Zbl 0033.06501)] and Hille and Phillips [Am. Math. Soc. (1957; Zbl 0078.10004)] about functional analysis and semigroups.
The basic idea reads: If \(A\) is a closed linear operator on a Banach space \(X\), one considers the Cauchy problem (“initial value problem”) \[ u'(t)= Au(t),\quad t\geq 0,\quad u(0)= x, \] where \(x\in X\) is given. If \(u(\cdot)\) is an exponentially bounded continuous function, which is also a (mild) solution, that is: \[ \int^t_0 u(s) ds\in D(A)\quad\text{and}\quad u(t)= x+ A\int^t_0 u(s) ds,\quad t\geq 0, \] and if one considers the Laplace transform: \[ \widehat u(\lambda)= \int^\infty_0 e^{-\lambda t}u(t) dt, \] which converges for large \(\lambda\), then \((\lambda- A)\widehat u(\lambda)= x\) (\(\lambda\) large), and conversely. Thus, if \(\lambda\in \rho(A)\) – the resolvent set of \(A\) – then \(\widehat u(\lambda)= (\lambda- A)^{-1}x\).
This fundamental relationship indicates that the Laplace transform is the link between solutions and resolvents, between Cauchy problems and spectral properties of operators.
A further study concerns criteria to decide whether a given function is a Laplace transform. Such results – in the vector-valued case – when applied to the resolvent of an operator, would give information on the solvability of the Cauchy problem.
Finally, let us note that our aim here is not to provide a summary – or appreciation – of the wealth of concepts contained in this 500 pages book. We invite all interested mathematicians to study this monograph, or any part of it. The reward should be considerable, as always is when reading great mathematics.

34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
44-02 Research exposition (monographs, survey articles) pertaining to integral transforms
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
46F12 Integral transforms in distribution spaces
44A10 Laplace transform
34G10 Linear differential equations in abstract spaces
47A10 Spectrum, resolvent
47D06 One-parameter semigroups and linear evolution equations
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.