Convergence estimates for the Magnus expansion I. Banach algebras

We review and provide simplified proofs related to the Magnus expansion, and improve convergence estimates. Observations and improvements concerning the Baker--Campbell--Hausdorff expansion are also made. In this Part I, we consider the general Banach algebraic setting. We show that the (cumulative) convergence radius of the Magnus expansion is $2$; and of the Baker--Campbell--Hausdorff series is $\mathrm C_2=2.89847930\ldots$.Comment: Part I of original submission arXiv:1709.01791v1, rewritten and expande

Spectral calculations on locally convex vector spaces I

We develop a holomorphic functional calculus for (multivalued linear) operators on locally convex vector spaces. This includes the case of fractional powers along Lipschitz curves.Comment: 18 page

Factorization of Laurent series over commutative rings

We generalize the Wiener-Hopf factorization of Laurent series to more general commutative coefficient rings, and we give explicit formulas for the decomposition. We emphasize the algebraic nature of this factorization.Comment: 8 page

Convergence estimates for the Magnus expansion II. C∗C^*-algebras

We review and provide simplified proofs related to the Magnus expansion, and improve convergence estimates. Observations and improvements concerning the Baker--Campbell--Hausdorff expansion are also made. In this Part II, we prove growth estimates regarding the Magnus expansion in the setting Hilbert space operators near the critical cumulative norm $\pi$. We also conduct a detailed geometrical study in the case of $2\times2$ real matrices, leading to some special normal forms via the Magnus expansion.Comment: Part II of original submission arXiv:1709.01791v1, basically unrewritte