The complex inversion formula in UMD spaces for families of bounded operators. (English) Zbl 1252.34065
Following a method developed by M. Haase [J. Aust. Math. Soc. 84, No. 1, 73–83 (2008; Zbl 1149.47031)], the authors establish sufficient conditions to guarantee the validity of the complex inversion formula in UMD Banach spaces for the wide class of \((a, k)\)- regularized families. Some preliminaries on vector-valued Fourier transform are given. The validity of the complex inversion formula for \((a, k)\)-regularized families, under certain conditions on the scalar kernels \(a(t)\) and \(k(t)\) is proved. The result of Haase for resolvent families follows as a special case for \(k(t) = 1\). In the special case \(a(t) = 1\), they provide a wide class of kernels \(k(t)\) such that the complex inversion formula holds.
Reviewer: S. L. Kalla (Ellisville)
MSC:
| 34G10 | Linear differential equations in abstract spaces |
| 47D06 | One-parameter semigroups and linear evolution equations |
| 44A10 | Laplace transform |
Citations:
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