# Journal of Operator Theory

Volume 56, Issue 1, Summer 2006 pp. 89-110.

On multi-commutators and sums of squares of generators of one parameter groups**Authors**: Davide Di Giorgio (1) and A.F.M. ter Elst (2)

**Author institution:**(1) Dipartimento di Matematica, Universit\`a di Pisa, Via Buonarroti 2, 56127 Pisa, Italy

(2) Department of Mathematics, University of Auckland, P.B. 92019, Auckland, New Zealand

**Summary:**Let $\cx$ be a Banach space and let $A_1,\dots ,A_d$ be the generators of strongly continuous groups. We prove that under suitable assumptions (roughly speaking if all the multi-commutators of $A_1,\dots ,A_d$ of order $r+1$ are zero) every linear combination of the multi-commutators is closable on its natural domain and the closure generates a strongly continuous group. Moreover the sum of the squares of $A_1,\dots ,A_d$ is closable and the closure of $-\sum\limits_{k=1}^{d} A_k^2$ generates a holomorphic semigroup. Finally, as an application of our theorem we obtain the Kolmogorov operator and the Grushin operator.

**Keywords:**Semigroups, commutators, integrability of Lie algebras

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