Abstract.
In complex, separable, infinite-dimensional Hilbert space there exist 5 proper dense operator ranges with the property that every operator leaving each of them invariant is a scalar multiple of the identity. The algebra of operators leaving a pair of proper dense operator ranges invariant can have an infinite nest of invariant subspaces. A slight extension of Foiaş' Theorem shows that it can also have a non-trivial reducing subspace.
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Submitted: July 13, 2001¶ Revised: December 6, 2001.
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Longstaff, W. Small Transitive Families of Dense Operator Ranges. Integr. equ. oper. theory 45, 343–350 (2003). https://doi.org/10.1007/s000200300009
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DOI: https://doi.org/10.1007/s000200300009