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

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 − t z d μ ( t ) for all ℜ z ⩾ 0 , C Ψ ( 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 − t A X e − t D d μ ( t ) is well defined for any X ∈ B ( H ) as is the newly introduced Taylor remainder transformer R n ( f ; D , A ) X : = f ( A ) X − ∑ k = 0 n 1 k ! ∑ i = 0 k ( − 1 ) i k i A k − i X D i f ( k ) ( D ) . If A , D * are also ( n + 1 ) -accretive, ∑ k = 0 n + 1 ( − 1 ) k n + 1 k A n + 1 − k X D k ∈ C Ψ ( H ) and | | · | | Ψ is Q* norm, then | | · | | Ψ norm estimates for ∑ k = 0 n + 1 n + 1 k A k A n + 1 − k 1 2 R n ( f ; D , A ) X ∑ k = 0 n + 1 n + 1 k D n + 1 − k D * k 1 2 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.