Similarity Invariant Semigroups Generated by Non-fredholm Operators

Let A be a bounded operator on a separable, infinite dimensional Hilbert space H. Moreover, assume that A is not in the set C + K(H) of operators expressible as a sum of a scalar multiple of identity and a compact operator. What is the smallest similarity invariant semigroup containing operator A? Equivalently, which operators can be expressed as products of operators, similar to A? A partial answer to the above question was obtained in 2003 by Fong and Sourour [6]. They proved that if operator A 6∈ C +K(H) is invertible, every invertible operator is a product of operators, similar to A. The author extended their results to semi-invertible operators [7] and later to semi-Fredholm operators in a so far unpublished article. In these cases we must account for the Fredholm index, which makes precisely specifying the semigroup very difficult; we only manage to prove that it contains all operators with index that is a large enough multiple of indA. In this article we consider the operators that are not semi-Fredholm, termed nonFredholm operators. We shall see that, although harder to prove, the results are more conclusive than in the case of semi-Fredholm operators. Throughout this article we assume that H is a separable, infinite dimensional Hilbert space. All operators appearing in the article are bounded. We shall denote the null-space of the operator A by kerA and its range by ranA. The nullity of an operator is defined as