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Extensions of symmetric operators I: The inner characteristic function case

Given a symmetric linear transformation on a Hilbert space, a natural problem to consider is the characterization of its set of symmetric extensions. This problem is equivalent to the study of the partial isometric extensions of a fixed partial isometry. We provide a new function theoretic characterization of the set of all self-adjoint extensions of any symmetric linear transformation $B$ with equal indices and inner Livsic characteristic function $Θ_B$ by constructing a natural bijection between the set of self-adjoint extensions and the set of all contractive analytic functions $Φ$ which are greater or equal to $Θ_B$. In addition we characterize the set of all symmetric extensions $B'$ of $B$ which have equal indices in the case where $Θ_B$ is inner.