Category theorems for stable operators on Hilbert spaces

Abstract

We discuss the two closely related, but different concepts of weak and almost weak stability for the powers of a contraction on a separable Hilbert space. Extending Halmos' and Rohlin's theorems in ergodic theory as a model, we show that the set of all weakly stable contractions is of first category while the set of all almost weakly stable contractions is of second category and is residual. Analogous statements for unitary and isometric operators are also proved.

9 pages. To appear in Acta Scientiarum Mathematicarum (Szeged)

Cited by in corpus (7)

• Linear dynamical systems on Hilbert spaces: typical properties and explicit examples
• Rigidity of contractions on Hilbert spaces
• The space of contractive $C_0$-semigroups is a Baire space
• Weakly and almost weakly stable C_0-semigroups
• Typical operators admit common cyclic vectors
• On typical properties of Hilbert space operators
• Category theorems for stable semigroups