On polynomially Riesz operators

A bounded linear operator A on a Banach space X is said to be "polynomially Riesz", if there exists a nonzero complex polynomial p such that the image p(A) is Riesz. In this paper we give some characterizations of these operators. Let C denote the set of all complex numbers and let X and Y be infinite dimensional Banach spaces. We denote by B(X) the set of all linear bounded operators on X and by K(X) the set of all compact operators on X. We write (A) = { ∈ C : A − is not invertible} for the spectrum of A ∈ B(X). For A ∈ B(X) let N(A) denote the null-space and R(A) the range of A. We set (A) = dimN(A) and (A) = dimX=R(A) = codimR(A). Sets of upper and lower semi-Fredholm operators, respectively, are defined as Φ+(X) = {A ∈ B(X) : (A) < ∞ and R(A) is closed}; and Φ−(X) = {A ∈ B(X) : (A) < ∞}: For upper and lower semi-Fredholm operators the index is defined by i(A) = (A) − (A). If A ∈ Φ+(X)\Φ−(X), then i(A) = −∞, and if A ∈ Φ−(X)\Φ+(X), then i(A) = +∞. The set of Fredholm operators is defined as Φ(X) = Φ+(X) ∩ Φ−(X): An operator A ∈ B(X) is relatively regular (or 1-invertible) if there exists B ∈ B(X) such that ABA = A. It is well-known that A is relatively regular if and only if R(A) and N(A) are closed and complemented subspaces of X. We say that an operator A ∈ B(X) is left Fredholm, and write A ∈ Φl(X), if A is a relatively regular upper semi-Fredholm operator, while we say that A is right Fredholm, and write A ∈ Φr(X), if A is a relatively regular lower semi-Fredholm operator. In other words, A is left Fredholm if R(A) is a closed and complemented subspace of X and (A)< ∞, while A is right Fredholm if N(A) is a complemented subspace of X and (A)< ∞. An operator A ∈ B(X) is called a Weyl operator if it is Fredholm of index zero. We shall say that A ∈ B(X) is upper semi-Weyl if it is upper semi-Fredholm and i(A) ≤ 0, while A is lower semi-Weyl if it is lower semi-Fredholm and i(A) ≥ 0. An operator A ∈ B(X) is left (right) Weyl if A is left (right) Fredholm and i(A) ≤ 0 (i(A) ≥ 0). Denote by asc(A) (dsc(A)) the ascent (the descent) of A ∈ B(X), i.e. the smallest non-negative integer n such that N(A n ) = N(A n+1 ) (R(A n ) = R(A n+1 )). If such n does not exist, then asc(A) = ∞ (dsc(A) = ∞).