Stability of localized operators

Let ℓp, 1⩽p⩽∞, be the space of all p-summable sequences and Ca be the convolution operator associated with a summable sequence a. It is known that the ℓp-stability of the convolution operator Ca for different 1⩽p⩽∞ are equivalent to each other, i.e., if Ca has ℓp-stability for some 1⩽p⩽∞ then Ca has ℓq-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 Sjöstrand 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 ℓp-stability (or Lp-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.