Invariant subspaces for $H^2$ spaces of $\sigma$-finite algebras

We show that a Beurling type theory of invariant subspaces of noncommutative $H^2$ spaces holds true in the setting of subdiagonal subalgebras of $\sigma$-finite von Neumann algebras. This extends earlier work of Blecher and Labuschagne for finite algebras, and complements more recent contributions in this regard by Bekjan, and Chen, Hadwin and Shen in the finite setting, and Sager in the semifinite setting. We then also introduce the notion of an analytically conditioned algebra, and go on to show that in the class of analytically conditioned algebras this Beurling type theory is part of a list of properties which all turn out to be equivalent to the maximal subdiagonality of the given algebra.