Cholesky Decomposition - Generalization

Generalization

The Cholesky factorization can be generalized to (not necessarily finite) matrices with operator entries. Let be a sequence of Hilbert spaces. Consider the operator matrix

\mathbf{A} = \begin{bmatrix} \mathbf{A}_{11} & \mathbf{A}_{12} & \mathbf{A}_{13} & \; \\ \mathbf{A}_{12}^* & \mathbf{A}_{22} & \mathbf{A}_{23} & \; \\ \mathbf{A} _{13}^* & \mathbf{A}_{23}^* & \mathbf{A}_{33} & \; \\ \; & \; & \; & \ddots \end{bmatrix}

acting on the direct sum

where each

is a bounded operator. If A is positive (semidefinite) in the sense that for all finite k and for any

we have, then there exists a lower triangular operator matrix L such that A = LL*. One can also take the diagonal entries of L to be positive.

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