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

B.P Duggal

doi:10.1515/conop-2020-0120

Abstract

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,B m (P)= ( I - L A R B ) m ( P ) = ∑ j = 0 m ( - 1 ) j ( j m ) {\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. Unlike m-isometric and m-left invertible operators, commuting products and perturbations by commuting nilpotents of (m, I)-expansive operators do not result in expansive operators: using elementary algebraic properties of the left and right multiplication operators, a sufficient condition is proved. For Drazin invertible A and B ∈ B(ℋ), with Drazin inverses Ad and Bd, a sufficient condition proving (Ad, Bd) ^ (A, B) is (m − 1, P)-isometric (resp., (m − 1, P)-contractive) for m even (resp., m odd) is given, and a Banach space analogue of this result is proved.

Highlights

Let H denote an in nite dimensional complex Hilbert space, and let B(H) (resp., B(X)) denote the algebra of operators, i.e. bounded linear transformations, on H into itself
M and (A, B) ∈ (m, P)-expansive, we prove that (A, B)∧(A−, B− ) ∈ (n, P)-isometric for all n ≥ m− if m is even and (A, B)∧(A−, B− ) ∈ (m−, P)
We prove that a su cient condition for a successful transfer of these properties to (m, X)-expansive operators is the existence of “partial expansive sequences"

Summary

Introduction

Let H (resp., X) denote an in nite dimensional complex Hilbert (resp., Banach) space, and let B(H) (resp., B(X)) denote the algebra of operators, i.e. bounded linear transformations, on H (resp., X) into itself. An operator T ∈ B(H) is m-expansive for some positive integer m if m m T* Left invertible for all integers n ≥ m: (T*, T) ∈ (m, X)-expansive, T ∈ B(H), fails to imply (T*, T) ∈ (m + , X)-

Results

Conclusion