Let $A$ denote a unital algebra of bounded linear operators on an infinite
dimensional real or complex Banach space $X$.
$A$ is said to have the closure property if every densely defined
linear operator on $X$ which commutes with
the closure of A in the weak operator topology
(WOT) is closeable.
$T \in L(X)$ is said to have the closure property if $A_T$ does, where
$A_T$ denotes the algebra of polynomials in $T$.
This new concept is motivated by deep classical work of W.B.~Arveson.
I shall discuss how some of his results may be formulated in terms of the
closure property, give some structural results we have obtained, and present
some examples of operators with the closure property.
The following conjecture, if true, would yield a very broad generalization
of Arveson's main result.

Conjecture.
Let $A$ have the closure property.
Then either $A$ has a non-trivial closed invariant linear subspace, or the WOT closure of $A$ coincides with L(X)$.