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A bounded linear operator is essentially normal if its self-commutator is compact. By essential equivalence of two operators, we mean unitary equivalence modulo compact operators. The celebrated BDF Theorem classies all essentially normal operators up to essential equivalence by

means of the essential spectrum and index data. The original proof of BDF theorem relies on the idea of associating to every compact Hausdor space X the group Ext(X) of *-monomorphisms from the continuous functions on X into the Calkin algebra. It turns out that for planar sets, this group is isomorphic to the group of homomorphisms from the rst cohomotopy group into the set of integers.

We will discuss two special cases of this theorem. The rst of which, usually known as Weyl-von Neumann theorem, says that two self-adjoint operators are essentially equivalent if and only they have same essential spectrum. In terms of the extension group, this result says that Ext(X) must be trivial

for any compact subset X of the real line. The second interesting case identies the extension group of the unit circle with the group of integers.Discipline/Coordinating Entity: Mathematics