?. Ker,

, If ind(?) is defined then P U P is Fredholm on Ran P and ind(?) equals minus its Fredholm index : ind(?) = dim ker QU * Q ? dim ker QU Q

, dim ker (? 2 ? I) ? Ran P ? ? dim ker (? 2 ? I) ? Ran P

, ? t ? U (t) is norm continuous and unitary and for ?(t) = U * (t)P U (t): 1 / ? ? ess (?(t) 2 ) then Z ? ind

, For unitaries U 0 , U 1 such that U 1 ? U 0 is a compact operator

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