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The Robbin–Salamon index theorem in Banach spaces with UMD. (English) Zbl 1090.47006

The Robbin–Salamon index theorem is well-known for a continuous family of unbounded linear operators \((A(t)) _{t\in \mathbb R}\) with \(t\)-independent dense domain \(W\) which is compactly embedded in a Hilbert space \(H\) and with \(A(t)\) selfadjoint. This theorem says that the operator \(\frac{d} {dt}-A(\;)\) is a Fredholm operator between the spaces \(W^{1,2}(\mathbb R,H)\cap L^{2}(\mathbb R,W)\) and \(L^{2}(\mathbb R,H)\) [J. Robbin and D. Salamon, Bull.Lond.Math.Soc.27, No. 1, 1–33 (1995; Zbl 0859.58025)].
In the paper under review, the author proves that the Robbin–Salamon index theorem holds in a general Banach space \(X\) with UMD (unconditionality of martingale difference). More precisely, the author shows if \(A(t)\) has \(t\)-independent domain \(W\) which is compactly embedded in \(X\), then the operator \(D_{A}=\frac{d}{dt}-A(\;)\) is Fredholm between the spaces \(W^{1,p}(\mathbb R,X)\cap L^{p}(\mathbb R,W)\) and \(L^{p}(\mathbb R,X\)), for \(p>1\). The index of the operator \(D_{A}\) is also characterized by the spectral flow of \(A\). As application, the author proposes to study the \(L^{p}\) maximal regularity for some Cauchy problem.

MSC:

47A53 (Semi-) Fredholm operators; index theories
47D06 One-parameter semigroups and linear evolution equations
35K20 Initial-boundary value problems for second-order parabolic equations
37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
47N20 Applications of operator theory to differential and integral equations
58J30 Spectral flows

Citations:

Zbl 0859.58025
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