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Let 
$H^{2}$ be the Hardy space over the bidisk. It is known that Hilbert–Schmidt invariant subspaces of 
$H^{2}$ have nice properties. An invariant subspace which is unitarily equivalent to some invariant subspace whose continuous spectrum does not coincide with 
$\overline{\mathbb{D}}$ is Hilbert–Schmidt. We shall introduce the concept of splittingness for invariant subspaces and prove that they are Hilbert–Schmidt.
