Polynomial control on weighted stability bounds and inversion norms of localized matrices on simple graphs

Abstract

The (un)weighted stability for some matrices is one of essential hypotheses in timefrequency analysis and applied harmonic analysis. In the first part of this paper, we show that for a localized matrix in a Beurling algebra, its weighted stabilities for different exponents and Muckenhoupt weights are equivalent to each other, and reciprocal of its optimal lower stability bound for one exponent and weight is controlled by a polynomial of reciprocal of its optimal lower stability bound for another exponent and weight. Inverse-closed Banach subalgebras of matrices with certain off-diagonal decay can be informally interpreted as localization preservation under inversion, which is of great importance in many mathematical and engineering fields. Let B(ℓ p w ) be the Banach algebra of bounded operators on the weighted sequence space ℓ p w on a simple graph. In the second part of this paper, we prove that Beurling algebras of localized matrices on a simple graph are inverse-closed in B(ℓ p w ) for all 1 ≤ p < ∞ and Muckenhoupt Ap-weights w, and the Beurling norm of the inversion of a matrix A is bounded by a bivariate polynomial of the Beurling norm of the matrix A and the operator norm of its inverse A −1 in B(ℓ p w ).

Differential subalgebras and norm-controlled inversion

“…The readers may refer to the survey papers [18,27,37], the recent papers [14,34,36] and references therein for historical remarks and recent advances. Given an element A in a Banach algebra A with the identity I, we define its spectral set σ A (A) and spectral radius ρ A (A) by…”

“…For the applications to some mathematical and engineering fields, the widely-used algebras B of infinite matrices and integral operators are the operator algebras B(ℓ p ) and B(L p ), 1 ≤ p ≤ ∞, which are symmetric only when p = 2. In [1,15,36,38,42,48], inverse-closedness of localized matrices and integral operators in B(ℓ p ) and B(L p ), 1 ≤ p ≤ ∞, are discussed, and in [14], Beurling algebras B p,α with 1 ≤ p < ∞ and α > d(1 − 1/p) are shown to admit polynomial norm-controlled inversion in nonsymmetric algebras B(ℓ p ), 1 ≤ p < ∞. It is still widely open to discuss Wiener's lemma and norm-controlled inversion when B and A are not * -algebras and B is not a symmetric algebra.…”