Norm Estimates for Remainders of Noncommutative Taylor Approximations for Laplace Transformers Defined by Hyperaccretive Operators

Abstract

Let H be a separable complex Hilbert space, B(H) the algebra of bounded linear operators on H, μ a finite Borel measure on R+ with the finite (n+1)-th moment, f(z):=∫R+e−tzdμ(t) for all ℜz⩾0,CΨ(H), and ||·||Ψ the ideal of compact operators and the norm associated to a symmetrically norming function Ψ, respectively. If A,D∈B(H) are accretive, then the Laplace transformer on B(H),X↦∫R+e−tAXe−tDdμ(t) is well defined for any X∈B(H) as is the newly introduced Taylor remainder transformer Rn(f;D,A)X:=f(A)X−∑k=0n1k!∑i=0k(−1)ikiAk−iXDif(k)(D). If A,D* are also (n+1)-accretive, ∑k=0n+1(−1)kn+1kAn+1−kXDk∈CΨ(H) and ||·||Ψ is Q* norm, then ||·||Ψ norm estimates for ∑k=0n+1n+1kAkAn+1−k12Rn(f;D,A)X∑k=0n+1n+1kDn+1−kD*k12 are obtained as the spacial cases of the presented estimates for (also newly introduced) Taylor remainder transformers related to a pair of Laplace transformers, defined by a subclass of accretive operators.