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Assume T is a bounded linear operator on some Banach space Y, and that T has a bounded extension T on another space. In general almost nothing can be said concerning the relationship between the spectral and Fredholm properties of T and T . However, assuming the special condition that the range of T lies in K , it is shown that these properties are essentially the same for T and T .doi:10.1090/s0002-9939-1989-0955454-4 fatcat:4tkxzue74rf5jatyy4ky3b3rcy