Let 𝒜 and ℬ be nest algebras of
operators on a Hilbert space with finite-rank nest projections 𝒩𝒜= {P(n)} and
𝒩ℬ= {Q(n)}, n ∈N, respectively. Let Pn= P(n+1)−P(n) and Qn= Q(n+1)−Q(n)
be the block diagonal projections for the two nests, 𝒜 and ℬ are thus the upper
triangular matrices with respect to the decompositions determined by {Pn}n∈N
and {Qn}n∈N respectively. It is easy to see that 𝒜 is isomorphic to ℬ if
and only if rank Pn=rank Qn for all n. J. Plastiras has shown that the
quasitriangular algebra 𝒜 + K(H), that is 𝒜 plus the compact operators, is
isomorphic to ℬ + K(H) if and only if there exist integers n0 and m0 so that
rank P(n0+n)=rank Q(m0+n) for all n. Using different techniques this paper shows
that the image of 𝒜 in the Calkin algebra 𝒜 is isomorphic to ℬ if and only if
there exist integers n0 and m0 so that rank Pn0+n=rank Qm0+n for all
n.
