Invariant subspaces for polynomially bounded operators

Abstract

Let T be a polynomially bounded operator on a Banach space X whose spectrum contains the unit circle. Then T Ã has a nontrivial invariant subspace. In particular, if X is reflexive, then T itself has a nontrivial invariant subspace. This generalizes the well-known result of Brown, Chevreau, and Pearcy for Hilbert space contractions. r

Invariant Subspaces for Banach Space Operators with a Multiply Connected Spectrum

“…We will define intervals I λ,Ω analogous to ones defined in [1]. Let us fix a positive number η such that the circles centered at λ j with radius r j + η and the circle centered at 0 with radius 1 − η are pairwise disjoint.…”

“…Ambrozie and Müller generalized this result to Banach space operators by proving that the adjoint of a polynomially bounded operator whose spectrum contains the unit circle has an invariant subspace [1]. Recall that a bounded linear operator T defined on a complex Banach space X, is said to be polynomially bounded if there exists a constant K > 0 such that p(T ) ≤ K sup{|p(λ)| : |λ| ≤ 1} for all polynomials p, and the constant K is said to be the polynomial bound of T .…”

“…In order to adapt Ambrozie and Müller's result to the case of an annulus, besides the main tools used in [1], we basically needed some estimates concerning Poisson kernels in an annulus, which act as representing measures, and an inner function whose derivative does not vanish on the boundary. We can construct such a function using Blaschke products on the annulus as defined in [11] (See [12]).…”

Positivity in the theory of supercyclic operators

“…They proved (see [10], [11]) that if the spectrum of a Hilbert space contraction T is rich (namely, the spectrum of T contains the unit circle), then T has a non-trivial non-cyclic vector. In this direction, some recent advances have been made by C. Ambrozie and V. Müller (see [2]) who improved the result of Brown-Chevreau and Pearcy to polynomially bounded operators.…”

“…By (2) and (3), F x,x is a non-empty subsemigroup of the unit circle T. Let us suppose that F x,x = T. By (2) and (3) F x,y = T for all y ∈ X, thus y ∈ M x for all y ∈ X, and therefore the set {Sx : S ∈ M is weakly dense in X.…”