If is a upper triangular matrix on the Hilbert space , then -Weyl's theorem for and need not imply -Weyl's theorem for , even when . In this note we explore how -Weyl's theorem and -Browder's theorem survive for operator matrices on the Hilbert space.
Let and be finite groups and let be a hilbertian field. We show that if has a generic extension over and satisfies the arithmetic lifting property over , then the wreath product of and also satisfies the arithmetic lifting property over . Moreover, if the orders of and are relatively prime and is abelian, then any extension of by (which is necessarily a semidirect product) has the arithmetic lifting property.
Let be a free group of finite rank , let be the semigroup of endomorphisms of , and let be the group of automorphisms of .
Theorem. If is an automorphism of , then there is an such that for all .
- (a)
- For some is stationary.
- (b)
- For each there is a generic extension of in which does not exist and is non-stationary.