The asymptotic behaviour of the reduced minimum modulus of a Fredholm operator

Abstract

Let γ ( S ) \gamma (S) denote the reduced minimum modulus of a linear operator S S acting in a complex Banach space X X , and let I I denote the identity on X X . In this paper it is shown that for a (not necessarily bounded) Fredholm operator T T acting in X X , the limit lim n → ∞ γ ( T n ) 1 / n {\lim _{n \to \infty }}\gamma {({T^n})^{1/n}} exists and is equal to the supremum of all positive numbers δ \delta such that the dimension of the null space and the codimension of the range of T − λ I T - \lambda I are constant on 0 > | λ | > δ 0 > |\lambda | > \delta .

Generalized spectrum and commuting compact perturbations

“…Conversely, suppose that k(A) = p is finite. Then, by [ [8] we can choose functionals in (2) such that the vectors in (2) and (1) are biorthogonal; i.e., x£ rk -j+1 (x m ,,) = 1, if k = m and j = i, x£ rk -j+1 (x mJ ) = 0 in the other cases. Let X 1 be the subspace in X spanned by vectors in (1)…”

“…To prove the other inequality suppose that e>0, and let p denote the total multiplicity of the jumps having absolute value less than V(A) -E. AS in the proof of [18, Theorem 1.1 (II)] (using Theorem 2.1 instead of Kato's decomposition theorem [9, Theorem 4]) we conclude that the space X decomposes into the direct sum of two closed subspaces Z and Y which are /4-invariant, dim Z = p and Z is the direct sum of the finite dimensional summands at the jumping points k v (A),...,X P (A) (where each jump appears consecutively according to its multiplicity). Let P 2 Acknowledgements. I am grateful to Professor Laura Burlando for the examples in the Remark 4.4, and to Professor Jean-Philippe Labrousse for comments and verification of the proof of Theorem 2.1.…”

A Toeplitz-like Operator with Rational Symbol Having Poles on the Unit Circle II: The Spectrum

“…That the index is constant on connected components of the complement of the essential spectrum in fact holds for any unbounded densely defined operator (see [15,Theorem VII.5.2]; see also [4,Proposition XI.4.9] for the bounded case; for a much more refined analysis of this point see [6]).…”

Generalized inverses and the maximal radius of regularity of a Fredholm operator

“…♦ We recall now the Kato decomposition for Fredholm operators (cf. for instance [4]). Namely, X decomposes into two T -invariant closed subspaces X 0 and X 1 and, if T i is the restriction of T to X i , i = 0, 1, then :…”