Journal of Operator Theory

Volume 58, Issue 2, Fall 2007 pp. 387-414.

Dichotomy and Fredholm properties of evolution equations

Authors: Yuri Latushkin (1), Alin Pogan (2), and Roland Schnaubelt (3)
Author institution: (1) Department of Mathematics, University of Missouri-Columbia, Columbia, MO 65211, USA
(2) Department of Mathematics, University of Missouri- Columbia, Columbia, MO 65211, USA
(3) Department of Mathematics, Institute of Analysis, University of Karlsruhe, 76128 Karlsruhe, Germany

Summary: Under minimal assumptions, we characterize the Fredholm property and compute the Fredholm index of abstract differential operators $-\frac{\mathrm d}{\mathrm dt}$ $+A(\cdot)$ act ing on spaces of functions $f:\mathbb{R}\to X$. Here $A(t)$ are (in general) unbounded operators on the Banach space $X$ and our results are formulated in terms of exponential dichotomies on two halflines for the propagator solving the evolution equation $\dot{u}(t)=A(t)u(t)$ in a mild sense.

Keywords: Fredholm operator and index, exponential dichotomy, node operator, evolution family, evolution equation, weighted shift operator, input-output method.

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