In the first part of the paper, it is proved that for 1 < p < ∞ the couple (K_θ ^p, K_θ ^∞) of coinvariant subspaces of the shift operator on the unit circle is K-closed in the couple (L_p(T),L^∞ (T)). This property underlies basically all problems of real interpolation for the first couple. Also, a weighted analog of the above statement is established. In the second part it is shown that, given two closed ideals I and J in a uniform algebra such that the complex conjugate of I ∩ J is not included in some of them, the sum I + J̅ is not closed. Though the methods of study in the two parts are quite different, the topics are related by the fact that the question treated in the second part emerged during the work on the first.