Compact Operators on Matrix Domains via the Hausdorff Measure of Non-Compactness

This paper gives a quantitative compactness framework for operators on matrix domains of triangular matrices. Let XA={x:Ax∈X} be equipped with the natural norm induced by an invertible triangle A. Because UAx=Ax is an isometric isomorphism, the Hausdorff measure of non-compactness of an operator defined on XA can be evaluated after transferring the operator to the model space X. For a matrix B:XA→Y, the associated operator is represented by B̂=BA-1; for endomorphisms of XA the relevant conjugate is ABA-1. Exact formulas are obtained for operators from ℓp(A) into c₀, and two-sided estimates are obtained for maps into c. For targets ℓr, 1≤r<∞, compactness is characterized by vanishing tail operator norms. Specializations to ℓ1(A) give an explicit column-tail criterion, while a Hilbert-Schmidt condition yields a practical sufficient condition on ℓ2(A). Examples based on first-order weighted differences show how the abstract transfer becomes a computable condition on matrix coefficients. The results extend the standard use of the Hausdorff measure of non-compactness on BK spaces to a general matrix-domain setting and clarify the role of the inverse triangle in compactness tests.

Keywords: Hausdorff measure of non-compactness; compact operator; matrix domain; BK space; infinite matrix; difference sequence space.