Discrete Triebel-Lizorkin spaces and expansive matrices

We provide a characterization of two expansive dilation matrices yielding equal discrete anisotropic Triebel-Lizorkin spaces. For two such matrices A and B, it is shown that ḟαp,q(A) = ḟαp,q(B) for all α ∈ R and p, q ∈ (0, ∞] if and only if the set ʲ B⁻ʲ : j ∈ Z\ is finite, or in the trivial case when $p = q$ and |(A)|α + 1/2 - 1/p = |(B)|α + 1/2 - 1/p. This provides an extension of a result by Triebel for diagonal dilations to arbitrary expansive matrices. The obtained classification of dilations is different from corresponding results for anisotropic Triebel-Lizorkin function spaces.