Abstract
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 $$\displaystyle{\|[g(D),y]\| \leq A_{0}\|y\| + A_{1}\|[D,y]\| + A_{2}\|[D,[D,y]]\| +\ldots +A_{n}\|[D,[D,\ldots [D,y]\ldots ]]\|.}$$ 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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