Triangularizability of Polynomially Compact Operators

An operator on a complex Banach space is polynomially compact if a non-zero polynomial of the operator is compact, and power compact if a power of the operator is compact. Theorems on triangularizability of algebras (resp. semigroups) of compact operators are shown to be valid also for algebras (resp. semigroups) of polynomially (resp. power) compact operators, provided that pairs of operators have compact commutators. 1 Basic definitions and properties An operator T on an infinite-dimensional complex Banach space X is called polynomially compact if there exists a non-zero complex polynomial p such that the operator p(T ) is compact. If T k is compact for some k we say that T is a power compact operator. Trivial examples of polynomially compact operators are compact and algebraic operators; the sum of a compact and an algebraic operator is also a polynomially compact operator: if p(A) = 0 and K is compact, then p(A + K) is obviously compact. A polynomially compact operator on a Hilbert space is a compact perturbation of an algebraic operator, see Section 4 and [3, Theorem 2.4]. The monic polynomial p of the smallest degree for which the operator p(T ) is compact is called the minimal polynomial of the polynomially compact operator T . Note that if A is algebraic with minimal polynomial p then p is in general only divisible by (and not necessarily equal to) the minimal polynomial of A as a polynomially compact operator; for example, a non-zero finite-rank projection is algebraic with a minimal polynomial of degree 2 and polynomially compact with a minimal polynomial of degree 1. Obviously, the minimal polynomial of a polynomially compact operator T is the minimal polynomial of the algebraic element π(T ) of the Calkin algebra B(X )/K(X ), where B(X ) denotes the algebra of all bounded linear operators on X , and K(X ) the ideal of compact operators. A semigroup S on X is a subset of B(X ) which is closed under multiplication of operators, and I ⊆ S is an ideal if the implication A ∈ I, B, C ∈ S ∪ {I} =⇒ BAC ∈ I holds. An algebra A on X is a linear subset of B(X ) which is also a semigroup. A closed subspace M of X is said to be invariant (respectively, hyperinvariant) for a family of operators F if A(M) ⊆ M for every A ∈ F (respectively, B(M) ⊆ M for every B which commutes with all A ∈ F). Propositions 1.1 and 1.3 show that polynomially compact operators have many of the wellknown spectral properties of compact operators. We include the proofs for the sake of completeness.