 In fact "TP" = "rP" and ("T"/"r")"n" → "P" as "n" → ∞ so this projection "P" (which is called the Perron projection) approximates ("T"/"r")"n" as "n" increases, and each of its columns is an eigenvector of "T".More generally if "T" is a compact operator then all non-zero points in σ("T") are isolated and so any finite subset of them can be used to decompose "T".The associated spectral projection always has finite rank.