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Journal of Operator Theory

Volume 65, Issue 1, Winter 2011  pp. 17-45.

Characteristic functions of liftings

Authors:  Santanu Dey (1) and Rolf Gohm (2)
Author institution: (1) Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India
(2) Institute of Mathematics and Physics, Aberystwyth University, Ceredigion SY23 3BZ, Wales, U.K.

Summary:  We introduce characteristic functions for certain contractive liftings of row contractions. These are multi-analytic operators which classify the liftings up to unitary equivalence and provide a kind of functional model. The most important cases are subisometric and coisometric liftings. We also identify the most general setting which we call reduced liftings. We derive properties of these new characteristic functions and discuss the relation to Popescu's definition of the characteristic function for completely non-coisometric row contractions. Finally we apply our theory to completely positive maps and prove a one-to-one correspondence between the fixed point sets of completely positive maps related to each other by a subisometric lifting.

Keywords:  Characteristic function, contractive lifting, row contraction, multi-analytic, subisometric dilation, coisometric, completely non-coisometric, completely positive map, fixed point

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