Norm Estimates for a Semigroup Generated by the Sum of Two Operators with an Unbounded Commutator

  • Michael Gil’Email author


Let A be the generator of an analytic semigroup \((e^{At})_{t\ge 0}\) on a Banach space \({\mathcal {X}}\), B be a bounded operator in \({\mathcal {X}}\) and \(K=AB-BA\) be the commutator. Assuming that there is a linear operator S having a bounded left-inverse operator \(S_l^{-1}\), such that \(\int _{0}^{\infty } \Vert Se^{At}\Vert \Vert e^{Bt}\Vert dt<\infty \), and the operator \(KS_l^{-1}\) is bounded and has a sufficiently small norm, we show that \(\int _{0}^{\infty } \Vert e^{(A+B)t}\Vert dt<\infty \), where \((e^{(A+B)t})_{t\ge 0}\) is the semigroup generated by \(A+B\). In addition, estimates for the supremum- and \(L^1\)-norms of the difference \(e^{(A+B)t}-e^{At}e^{Bt}\) are derived.


Banach space semigroups perturbations commutator 

Mathematics Subject Classification

47D06 47D60 



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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsBen Gurion University of the NegevBeer-ShevaIsrael

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