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Minimization of Constrained Quadratic forms in Hilbert Spaces

A common optimization problem is the minimization of a symmetric positive definite quadratic form $< x,Tx >$ under linear constrains. The solution to this problem may be given using the Moore-Penrose inverse matrix. In this work we extend this result to infinite dimensional complex Hilbert spaces, making use of the generalized inverse of an operator. A generalization is given for positive diagonizable and arbitrary positive operators, not necessarily invertible, considering as constraint a singular operator. In particular, when $T$ is positive semidefinite, the minimization is considered for all vectors belonging to $\mathcal{N}(T)^\perp$.