Implementation of derivations and invariant subspaces

• Published in: Israel journal of mathematics 2003-01, Vol.134 (1), p.1-28
• Main Authors: Kissin, E., Lomonosov, V. I., Shulman, V. S.
• Format: Article
• Language: English
• Subjects: Algebra; Hilbert space; Invariants; Linear operators; Mathematical analysis; Mathematics; Operators (mathematics); Subspaces
• ISSN: 0021-2172; 1565-8511
• DOI: 10.1007/BF02787401

The paper studies operator implementations of derivations of algebras. Letπ andϱ be irreducible representations of an algebraA on Banach spacesX andY. A linear mapδ:A →B(Y, X) is a (π, ρ)-derivations ifδ(ab)=π(a)δ(b)+δ(a)ρ(b). It is bimodule-closable ifπ(an) → 0,ρ(an) → 0 andδ(an) →B implyB=0. A closed operatorF fromY intoX implementsδ ifF ρ(a)−π(a)F ⊆δ(a), fora ∈A. It is shown that ifX, Y are reflexive and either the closure of the algebra {π(a)+ρ(a):a ∈A} or both algebrasπ (A),ρ (A) contain compact operators, then the set Imp(δ) of all implementations is not empty for any bimoduleclosable (π, ρ)-derivationδ, and either contains aminimal operator, or amaximal operator, or two families of operatorsRλ ⊆Gλ,λ ∈ ℂ, such thatRλ ⊆T ⊆Gλ for eachT ∈ Imp(δ) and someλ.These results are applied to the study of norm-closed operator algebrasB on Banach spacesX with only one invariant subspaceL. It is proved that, ifL contains compact operators,X is reflexive andL has approximation property, thenB contains all compact “corner” operators:BX ⊆L andBL=0. IfL has a closed complement, the same is true if the closure of the block-diagonal part ofB contains compact operators. IfX is non-reflexive,B may have no “corner” operators. If, however,B consists of compact operators then its weak closure contains all “corner” operators. A description is given of algebras of compact operators on Hilbert spaces with only one invariant subspace.