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We give a basis for the vector space generated by rational Hodge classes oftype (2,2) on the Hilbert square of a projective K3 surface general in itsrank, which is a subspace of the singular cohomology ring with rationalcoefficients: we use Nakajima operators and an algebraic model developed byLehn and Sorger as main tools. We then obtain a basis of the lattice generatedby integral Hodge classes of type (2,2) on the Hilbert square of a projectiveK3 surface general in its rank: we exploit lattice theory, a theorem by Qin andWang and a result by Ellingsrud, G\"ottsche and Lehn.

Hodge classes of type (2, 2) on Hilbert squares of projective K3 surfaces