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Hyperbolic Composition Operators and the Invariant Subspace Problem


Valentin Matache


About 20 years ago, three mathematicians proved that the Invariant Subspace Problem, which is still open for Hilbert space operators, can be reduced to the study of the invariant subspace lattice of a single operator: a hyperbolic composition operator acting on the Hilbert Hardy space H2 of all functions analytic in the open unit disc, having square sumable Maclaurin coefficients. Shortly after, the adviser for this project proved in [3] that this amounts to choosing H2 – functions u and deciding if the closed, doubly invariant, cyclic subspace generated by u under the action of such an operator can be atoms of the invariant subspace lattice of that operator if u is not an eigenfunction.
The student working on this project is required to complete successfully the Complex Variables course offered at UNO, then read selected topics (as instructed by the adviser) in:

[1] Rudin, Walter. Real and complex analysis. Third edition. McGraw-Hill Book Co., New York, 1987.
[2] Shapiro, Joel H. Composition operators and classical function theory. Universitext: Tracts in Mathematics. Springer-Verlag, New York, 1993.

in order to be able to read and understand the papers:

[3] Matache, Valentin On the minimal invariant subspaces of the hyperbolic composition operator. Proc. Amer. Math. Soc. 119 (1993), no. 3, 837–841.
[4] Matache, Valentin The eigenfunctions of a certain composition operator. Studies on composition operators (Laramie, WY, 1996), 121–136, Contemp. Math., 213, Amer. Math. Soc., Providence, RI, 1998.
[5] Shapiro, Joel H. The invariant subspace problem via composition operators—redux. Topics in operator theory. Volume 1. Operators, matrices and analytic functions, 519–534, Oper. Theory Adv. Appl., 202, Birkhäuser Verlag, Basel, 2010.

In the second phase of this project , the student taking it, is supposed to produce, under guidance, refinements and further developments of the results in papers [3]—[5].