Solvability and Spectrum of Infinite Matrix Operators on XA

Let be a Banach sequence space, let be an invertible triangle, and equip the matrix domain with the norm This paper studies solvability and spectral properties of infinite matrix operators acting on . The central observation is that the coordinate map , , is an isometric isomorphism, so every bounded operator on is similar to on . The similarity yields exact transfer principles for the resolvent, spectrum, point spectrum, approximate point spectrum, continuous and residual spectra, Fredholmness, index and essential spectrum. A corresponding solvability formula converts on into on . When and are given by infinite matrices, explicit formulas for the associated conjugate matrix are obtained under standard row-convergence hypotheses. Diagonal and shift models illustrate how nontrivial operators on difference-type matrix domains can have spectra computed without direct coordinate recursion. Perturbation, pseudospectral, compact-perturbation and Neumann-series criteria are also recorded. The framework provides a general mechanism for transporting spectral questions from a matrix domain to a familiar model space.