Geodesic
On the maximal dual pairs of invariant subspaces of $J$-self-adjoint operators
Matematičeskie zametki, Tome 7 (1970) no. 4, pp. 443-447.

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In the $J$-spaces $\mathfrak{H}=\mathfrak{H}_1\oplus\mathfrak{H}_2$, with the infinite-dimensional components $\mathfrak{H}_k=P_k\mathfrak{H}$ ($k=1,2$), we can always find an operator $A$, for which there are at least two distinct invariant maximal dual pairs, such that if $[x,x]=0$ and $[Ax,x]=0$, then $x=0$. 
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     author = {H. Langer},
     title = {On the maximal dual pairs of invariant subspaces of $J$-self-adjoint operators},
     journal = {Matemati\v{c}eskie zametki},
     pages = {443--447},
     publisher = {mathdoc},
     volume = {7},
     number = {4},
     year = {1970},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1970_7_4_a8/}
}
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H. Langer. On the maximal dual pairs of invariant subspaces of $J$-self-adjoint operators. Matematičeskie zametki, Tome 7 (1970) no. 4, pp. 443-447. http://geodesic.mathdoc.fr/item/MZM_1970_7_4_a8/