For a self-adjoint unbounded operator D on a Hilbert space H, a bounded operator y on H and some Borel functions g(t) we establish inequalities of the type [g(D),y]≤ Aoy+A1[D,y]+A2[D,[D,y]]+ +AN[D,[D,...[D,y]...]]The proofs take place in a space of infinite matrices with operator entries, and in this setting it is possible to approximate the matrix associated to [g(D),y] by the Schur product of a matrix approximating [D, y] and a scalar matrix. A classical inequality on the norm of Schur products may then be applied to obtain the results.
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