On (m, P)-expansive operators: products, perturbation by nilpotents, Drazin invertibility

Abstract

A generalisation of m-expansive Hilbert space operators T ∈ B(ℋ) [18, 20] to Banach space operators T ∈ B(𝒳) is obtained by defining that a pair of operators A, B ∈ B(𝒳) is (m, P)-expansive for some operator P ∈ B(𝒳) if Δ A,Bm(P)= (I-LARB)m(P)=∑j=0m(-1)j(jm){\left( {I - {L_A}{R_B}} \right)^m}\left( P \right) = \sum\nolimits_{j = 0}^m {{{\left( { - 1} \right)}^j}\left( {_j^m} \right)}AjPBj≤0; LA(X) = AX and RB(X)=XB.

Keywords

• hilbert/banach space
• left/right multiplication operator
• (m, p)-expansive operators
• m-left invertible/m-isometric operators
• product of operators
• perturbation by nilpotents
• drazin invertible operators
• 47a05, 47a55
• secondary 47a11, 47b47