 Since "T" is compact and α is non-zero, we have formula_31 is a finite-dimensional (and therefore closed) non-zero formula_25-invariant sub-space (because the operators all commute with "T", we have for formula_33 and formula_34, that formula_35).In particular we definitely have formula_36 Thus we could in principle argue by induction over dimension, yielding that formula_37 has a common eigenvector for formula_25.Theorem 1.