On the Resolvent of a Dilation for Polynomially Bounded Operators

Let H denote a separable, complex, Hilbert space and let L(H) denote the algebra of all bounded linear operators on H. Determining the structure of an arbitrary operator in L(H) has been one of the most studied topics in operator theory. In particular, the problem whether every operator T in L(H) has a nontrivial invariant subspace is still open. The most recent partial result in this direction is the famous theorem of Brown, Chevreau, and Pearcy [6] (see also [1]) which states that every contraction T in L(H) whose spectrum contains the unit circle, possesses a nontrivial invariant subspace. The ideas and techniques employed were introduced by Scott Brown in [5] and they were further developed in the work of Bercovici, Brown, Chevreau, Foias, and Pearcy, to mention just a few. An extensive bibliography on this subject (which is often denoted as “the theory of dual algebras”) can be found in [2], or, more updated in [10]. The success of the aforementioned techniques in the case of contraction operators generated a renewed interest in the class of polynomially bounded operators. Recall that an operator T in L(H) is said to be polynomially bounded (notation: T ∈ (PB)(H) or T ∈(PB)) if there exists an M ≥ 1 such that ‖p(T )‖ ≤M sup{|p(ζ)| : |ζ| = 1} ∀ polynomial p. (1)