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Factorisation of orthogonal projectors

We study the problem of a special factorisation of an orthogonal projector P acting in the Hilbert space L2(R) with dimkerP < ∞. In particular, we prove that the orthogonal projector P admits a special factorisation in the form P = V V ∗, where V is an isometric upper-triangular operator in the Banach algebra of all linear continuous operators in L2(R). Moreover, we give an explicit formula for the operator V .