Stafney's lemma holds for several ``classical'' interpolation methods (Q2880648)

Let \(B=(B_0,B_1)\) be a Banach pair and let \(X=(X_0,X_1)\) be a pair of pseudolattices. The space \(J(X,B)\) consists of all \(B_0\cap B_1\)-valued sequences \(\{b_n\}\), such that the sequence \(\{e^{jn}b_n\}\) is in the space \(X_j(B_j)\), \(j=0,1\), with the natural norm. Let \(s\) be a complex number, \(1<|s|<e\). The space \(B_{X,s}\) consists of all elements of the form \(b=\sum s^n b_n,\) where \(\{b_n\}\in J(X,B),\) with the norm NEWLINE\[NEWLINE \|b\|_{B_{X,s}}=\inf\{\|\{b_n\}\|_{J(X,B)}: b=\sum s^n b_n\}. \eqno(1) NEWLINE\]NEWLINE The main result of the paper states that if \(b\in B_0\cap B_1\) then the infimum in \((1)\) can be taken with respect to all \(B_0\cap B_1-\)valued sequences \(\{b_n\}\) with finite support under the condition that this is a dense subset in \(J(X,B)\).