Stability of localized operators

摘要

Let l~p, l≤p ≤∞, be the space of all p-summable sequences and C~a be the convolution operator associated with a summable sequence a. It is known that the l~p-stability of the convolution operator C_a for different 1≤ p≤∞ are equivalent to each other, i.e., if C_a has l~p-stability for some 1≤ p≤∞ then C_a has l~q-stability for all 1≤q≤∞. In the study of spline approximation, wavelet analysis, time-frequency analysis, and sampling, there are many localized operators of non-convolution type whose stability is one of the basic assumptions. In this paper, we consider the stability of those localized operators including infinite matrices in the Sj_strand class, synthesis operators with generating functions enveloped by shifts of a function in the Wiener amalgam space, and integral operators with kernels having certain regularity and decay at infinity. We show that the l~p-stability (or L~p-stability) of those three classes of localized operators are equivalent to each other, and we also prove that the left inverse of those localized operators are well localized.