Commutators and analytic families of operators

This work connects the theory of commutators with analytic families of oper ators in abstract interpolation theory. Our main result asserts that if {L- xi}(0 less than or equal to Re xi less than or equal to 1) is an analytic f amily of operators satisfying some conditions. then [L-theta, Omega] + (L-x i)' (theta) : (A) over bar(theta) --> (B) over bar(theta) is bounded. From this, we can deduce the boundedness of the commutator [L-theta, Omega] : (A ) over bar(delta)'((theta)) --> (B) over bar(delta'(theta)).