Irreducible subspaces of separable Hilbert space

A question about definition: Let $\mathcal{H}$ be a separable Hilbert space over $\mathbb{C}$, with $B(\mathcal{H})$ the bounded linear operators on it. What does it mean to have an irreducible subspace for a subalgebra $A \in B(\mathcal{H})$. Is it any subspace that does not contain a proper invariant subspace under $A$, or a closed subspace that does not contain a proper invariant subspace under $A$?

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Small addedndum to Gerald Edgar's remarks: there has been some work on "operator ranges" (also called "paraclosed subspaces"), namely the linear manifolds inside $H$ that are of the form $T(H)$ for some continuous linear operator $T:H\to H$. –  Yemon Choi Jun 13 '12 at 22:51