Re: strong convergence


ok, i found an example myself. take E to be the kernel of the map T ->
tr(TS) for S a trace class operator, which is not of finite rank. this
space is ultrastrongly closed, but not strongly closed, since the
linear functional is not strongly continuous (but ultrastrongly).
also: ultrastrongly closed spaces have the property that their
intersection with bounded closed sets is strongly closed.

i would like to rephrase my question:

does anybody know an example of a linear subspace E of B(H), the
algebra of bounded linear operators on the hilbert space H, which has
the following properties:

- E is closed under the adjoint operation,
- every bounded and ultrastrongly convergent net of elements from E
has its
limit in E,
- E is not ultrastrongly closed.

best, andreas thom
.

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