Operator- and Matrix-Valued Strip-Analytic Abu-Ghuwaleh Transforms: Wiener–Mellin Inversion, System Symbols, and Stable Recovery

We develop a unified operator- and matrix-valued strip-analytic extension of the Abu-Ghuwaleh transform program. The central object is a strongly measurable operator-valued orbit density whose boundary representation induces a continuous dilation-convolution operator acting on the Fourier transform of a weighted Hilbert-space-valued signal. In this setting the transform admits two complementary inversion mechanisms: Mellin contour inversion and contour-free Wiener--Mellin inversion on the logarithmic scale. We prove exact factorization formulas on named weighted signal spaces, derive branchwise Mellin diagonalization formulas with operator-valued system symbols, obtain inversion theorems under bounded invertibility assumptions, and formulate a log-scale Fourier multiplier representation suitable for FFT-based recovery. We then prove Young-type boundedness on the logarithmic side and stability estimates on frequency windows away from singularities of the multiplier. The finite-dimensional matrix case is obtained as a direct specialization of the Hilbert-space theory, and in that setting the Wiener inverse is derived from a standard matrix Wiener criterion. Finally, we isolate an explicit Gamma-type kernel family for which the system symbol is computable in closed form and yields concrete injectivity and stability constants. The paper is intended as the natural operator-theoretic successor to the scalar strip-analytic stage of the master-integral-transform program.