Suppose we have two Banach spaces X and Y each of them having the Schur property (weakly convergent sequences are norm convergent). Does it follows that X+Y has the Schur property  ? Note that this is trivialy true when the sum is direct.

Any proof (or disproof), or references will be apreciated.

Suppose we have two Banach spaces X and Y each of them having the Schur property (weakly convergent sequences are norm convergent). Does it follows that X+Y has the Schur property? Note that this is trivially true when the sum is direct.

Any proof (or disproof), or references will be appreciated.