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 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 7→ R 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 − nΣ k=0 1 k! kΣ i=0 (−1)i(ki )Ak−iXDi f (k)(D). If A, D∗ are also (n + 1)-accretive, Σn+1 k=0 (−1)k(n+1 k )An+1−kXDk ∈ CΨ(H) and || · ||Ψ is Q∗ norm, then || · ||Ψ norm estimates for 􀀀 Σn+1 k=0 (n+1 k )AkAn+1−k 1 2Rn( f ; D, A)X 􀀀 Σn+1 k=0 (n+1 k )Dn+1−kD∗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.