Polynomial Control on Weighted Stability Bounds and Inversion Norms of Localized Matrices on Simple Graphs

Abstract

The (un)weighted stability for some matrices on a graph is one of essential hypotheses in time-frequency 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. Banach algebras of matrices with certain off-diagonal decay is of great importance in many mathematical and engineering fields, and its inverse-closed property can be informally interpreted as localization preservation. Let $${{\mathcal {B}}}(\ell ^p_w)$$ be the Banach algebra of bounded linear operators on the weighted sequence space $$\ell ^p_w$$ on a graph. In the second part of this paper, we prove that Beurling algebras of localized matrices on a connected simple graph are inverse-closed in $${{\mathcal {B}}}(\ell ^p_w)$$ for all $$1\le p<\infty $$ and Muckenhoupt $$A_p$$ -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 $${{\mathcal {B}}}(\ell ^p_w)$$ .