Norm‐controlled inversion in smooth Banach algebras, II

Abstract

We show that smoothness implies norm‐controlled inversion: the smoothness of an element a in a Banach algebra with a one‐parameter automorphism group is preserved under inversion, and the norm of the inverse a−1 is controlled by the smoothness of a and by spectral data. In our context smooth subalgebras are obtained with the classical constructions of approximation theory and resemble spaces of differentiable functions, Besov spaces or Bessel potential spaces. To treat ultra‐smoothness, we resort to Dales‐Davie algebras. Furthermore, based on Baskakov's work, we derive explicit norm control estimates for infinite matrices with polynomial off‐diagonal decay. This is a quantitative version of Jaffard's theorem.

Latent Agents in Networks: Estimation and Targeting

“…More precisely, the aforementioned result focuses on infinite matrices and establishes that Banach algebras of matrices with a polynomial off-diagonal decay property are inverse-closed Klotz 2010, 2014). A quantitative version of this result, with explicit bounds on decay parameters, is given in Gröchenig and Klotz (2014). Even though this result is for infinite matrices, viewing finite matrices as diagonal subblocks of infinite matrices immediately yields the following result, which we state without a proof:…”

Norm-controlled inversion in smooth Banach algebras, I

“…In this case the estimates for norm control can be improved significantly, even without the assumption that the ambient algebra is a C * -algebra. This will be the topic of Part II [18].…”

Differential subalgebras and norm-controlled inversion

“…for all A ∈ A being invertible in B [19,20,34,36]. The norm-controlled inversion is a strong version of Wiener's lemma.…”

“…The concept of differential subalgebras of order θ was introduced in [7,26,32] for θ = 1 and [12,20,36] for θ ∈ (0, 1). We also refer the reader to [3,15,19,20,21,25,33,34,41,42,43,45] for various differential subalgebras of infinite matrices, convolution operators, and integral operators with certain off-diagonal decay.…”

“…For m = 2, the estimate (4.3) to the inverse A −1 ∈ A is essentially established in [19,20] for θ = 1 and [36,40] for θ ∈ (0, 1), and a similar estimate is given in [34]. The reader may refer to [15,21,42,43,45] for norm estimation of the inverse of elements in Banach algebras of infinite matrices and integral operators with certain off-diagonal decay.…”