The orthogonal $u$-invariant of a quaternion algebra

In quadratic form theory over fields, a much studied field invariant is the $u$-invariant, defined as the supremum of the dimensions of anisotropic quadratic forms over the field. We investigate the corresponding notions of $u$-invariant for hermitian and for skew-hermitian forms over a division algebra with involution, with a special focus on skew-hermitian forms over a quaternion algebra with canonical involution. Under certain conditions on the center of the quaternion algebra, we obtain sharp bounds for this invariant.