# Abstract/Details

## Invariant subspaces for Banach space operators with a multiply connected spectrum

2006 2006

### Abstract (summary)

Ambrozie and Müller proved that the adjoint of a polynomially bounded operator whose spectrum contains the unit circle has a nontrivial invariant subspace. This result generalizes the well known result of Brown, Chevreau, and Pearcy which applies to Hilbert space contractions. We extend Ambrozie and Müller's result by proving an analogous result where the unit circle is replaced by the boundary of a multiply connected set. Indeed, we consider a multiply connected complex domain Ω = [special characters omitted] *B¯*(*λ _{j}*,

*r*) where [special characters omitted] denotes the unit disk and

_{j}*B¯*(

*λ*,

_{ j}*r*) ⊂ [special characters omitted] denotes the closed disk centered at

_{j}*λ*with radius

_{j}*r*for

_{j}*j*= 1,...,

*n*. We show that if

*T*is a bounded linear operator on a Banach space

*X*whose spectrum contains ∂Ω and does not contain the points λ

_{1}, λ

_{2},...,

*λ*, such that for some constant

_{n}*k*> 0 the inequalities ||

*p*(

*T*)|| ≤

*k*sup{|

*p*(λ)| : |λ| ≤ 1} and ||

*p*(

*r*(

_{j}*T*-

*λ*)

_{ j}I^{-1})|| ≤

*k*sup{|

*p*(λ)| : |λ| ≤ 1} are satisfied for all polynomials

*p*and

*j*= 1,...,

*n*, then there exists a nontrivial common invariant subspace for

*T** and (

*T*-

*λ*)*

_{j}I^{-1}.