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In this paper, we consider the sampled-data problem of interconnected systems, specifically, time- and space-invariant systems. Our main contribution is to provide sufficient conditions on well-posedness, stability, and contractiveness of sampled-data interconnected systems in the form of a group of linear operator inequalities (LOIs). And despite their infinite dimensionality, further reduce them to linear matrix inequalities (LMIs). The technique is also applicable when dynamics are spatially continuous, and measurement and actuation take place in spatially localized patches; namely, when a spatial, rather than temporal, sampled-data arrangement is present.