The squaring operation on A-generators of the Dickson algebra

We study the squaring operation Sq(0) on the dual of the minimal A-generators of the Dickson algebra. We show that this squaring operation is isomorphic on its image. We also give vanishing results for this operation in some cases. As a consequence, we prove that the Lannes-Zarati homomorphism vanishes (I) on every element in any finite Sq(0)-family in Ext(A)*(F-2, F-2) except possibly the family initial element, and (2) on almost all known elements in the Ext group. This verifies a part of the algebraic version of the classical conjecture on spherical classes.