Restricted Linear Constrained Minimization of quadratic functionals

In this work a linearly constrained minimization of a positive semidefinite quadratic functional is examined. Our results are concerning infinite dimensional real Hilbert spaces, with a singular positive operator related to the functional, and considering as constraint a singular operator. The difference between the proposed minimization and previous work on this problem, is that it is considered for all vectors perpendicular to the kernel of the related operator or matrix.