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# Dynamics of Partial Differential Equations

## Volume 1 (2004)

### Number 3

### The Robbin-Salamon index theorem in Banach spaces with UMD

Pages: 303 – 337

DOI: http://dx.doi.org/10.4310/DPDE.2004.v1.n3.a2

#### Author

#### Abstract

Let $(A(t))_{t\in \mathbf{R}}$ be a continuous family ofunbounded operators on a UMD Banach space $X$, with $t$-independent domain$W$ dense and compactly embedded in $X$. Under these and other generaltechnical conditions, we prove that the operator$D_{A}=\frac{d}{dt}-A(\cdot )$ is a Fredholm operator between the spaces$W^{1,p}(\mathbf{R},X)\cap L^{p}(\mathbf{R},W)$ and$L^{p}(\mathbf{R},X)$ for every $p\in (1,\infty )$. We also characterizethe index of $D_{A}$ by the spectral flow of $A$. These results generalizethose obtained by Robbin and Salamon when $X$ and $W$ are Hilbert spaces,$p=2$ and the family $(A(t))_{t\in \mathbf{R}}$ is selfadjoint. Thehypotheses involved are satisfied by broad classes of second order ellipticdifferential operators with boundary conditions and by PDE systems with aHamiltonian-like structure. An application to the $L^{p}$ maximal regularityfor the nonautonomous Cauchy problem on the half-line is discussed in detail.

#### Keywords

Fredholm and semi-Fredholm operators, unbounded operator, Banach space with UMD, Rademacher boundedness, holomorphic semigroup, elliptic differentia operator, Hamiltonian system

#### 2010 Mathematics Subject Classification

Primary 46-xx. Secondary 35-xx, 47-xx.