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.