Semigroups Generated by Similarity Orbits of Semi-invertible Operators and Unitary Orbits of Isometries

Similarity invariant semigroups generated by non-Fredholm operators

“…Every operator B € # ( # ) witi infinite nullity and co-nullity is a product of 12 operators similar to A. [7, and N x onto M. Since r a n y C N- 1 , we have r a n W y C M and YWYW = 0. A product of two operators similar to al © Y is…”

“…Since A is not in the set C + K(H), by [7,Proposition 1] there exists a product Z of two operators similar to A that has the form all © B for some a > 0 and some bounded operator B.…”

“…Then [7,Lemma 1] states that every invertible operator is a product of 6 (or more) operators similar to bilateral shift of infinite multiplicity. In particular, P' 1 and Q~\ can both be expressed as products of operators similar to aU.…”

“…They proved that if operator A & 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A. The author extended their results to semi-invertible operators [7] and later to semi-Fredholm operators in a so far unpublished article.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 ind A.…”