Cloning of Symmetric Operators

Abstract

Given a closed densely defined symmetric operator S in a separable Hilbert space, by means of a non-real and non-imaginary complex number z and the corresponding deficiency subspace $${{\mathfrak {N}}}_z$$ of S we define a new closed symmetric operator S(z) with dense domain. We prove that the operator S(z) preserves various properties of S. When the deficiency indices of S are equal a bijection of the set of all selfadjoint extensions of S onto the set of all selfadjoint extensions of S(z) is established. We consider in detail the case when a symmetric operator S is nonnegative.