COMPLETING THE FUNCTIONAL CALCULUS 5
In Section 6 the cl[ int{ f(5(E)) } ] is characterized. Section 7 presents the
characterization of int{ cl[ f(5(E)) ] } .
From general topology we know that it is not possible to continue the
iteration of the operations int and cl indefinitely and obtain distinct sets. In
Section 8 the remaining possible distinct iterations are discussed. Indeed,
there are only two more : int{ cl[ int{ f(5(E)) } ] } and cl[ int{ cl[ f(5(E)) ] } ] .
In the first case, we give the desired characterization. In the second we do
not complete the characterization. Indeed, an initial attempt to write
necessary conditions for an operator to belong to cl[ int{ cl[ f(5(E)) ] }
resulted in such a long list that we became convinced that the point of
diminishing returns had been reached. We do, however, characterize the
biquasitriangular and compact operators that belong to cl[ int{ cl[ f(5(E)) ] } ]
as well as the set cl[ int{ cl 5(E) } ] .
In a forthcoming paper [23], the second author studies questions in
the Calkin algebra similar to those studied here.
The authors would like to thank Raul E Curto and Man-duen Choi for
their helpful comments and discussions concerning the contents of this
paper.
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