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We study the stability of an abstract thermoelastic system, which consists of an abstract elastic equation with or without inertial term and a heat equation of Cattaneo type. A complete analysis is carried out on how the decay rates of the system depend on the parameter region and the so-called 'wave speeds'. Specifically, the parameter region for exponential stability of such a system is characterized. Furthermore, optimality of the rate of polynomial decay is obtained under certain conditions. The proofs rely on interpolation inequalities, frequency domain condition for stability, and detailed spectral analysis. [ABSTRACT FROM AUTHOR]