Abstract
A classical conjecture on nontrivial invariant subspaces for Hilbert-space contractions reads as follows. “a C 1.-contraction has a nontrivial invariant subspace”. This turns out to be equivalent to a second conjecture, namely, “if a contraction is a quasiaffine transform of a unitary operator, then it has a nontrivial invariant subspace”. Although these are still unsolved problems, it can be proved that if a C 1. -contraction has no nontrivial invariant subspace then it is a quasiaffine transform of its own unitary extension which is reductive and has an invariant dense and totally cyclic linear manifold. This paper presents a brief review, based on [7] and [9], on the equivalence between the above conjectures.
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Kubrusly, C.S. (2000). Invariant Subspaces and Quasiaffine Transforms of Unitary Operators. In: Balakrishnan, A.V. (eds) Semigroups of Operators: Theory and Applications. Progress in Nonlinear Differential Equations and Their Applications, vol 42. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8417-4_16
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DOI: https://doi.org/10.1007/978-3-0348-8417-4_16
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