Extensions of symmetric operators
In functional analysis, one is interested in extensions of symmetric operators acting on a Hilbert space.Of particular importance is the existence, and sometimes explicit constructions, of self-adjoint extensions.This problem arises, for example, when one needs to specify domains of self-adjointness for formal expressions of observables in quantum mechanics.The unifying theme is that each problem has an operator-theoretic characterization which gives a corresponding parametrization of solutions.More specifically, finding self-adjoint extensions, with various requirements, of symmetric operators is equivalent to finding unitary extensions of suitable partial isometries.In general, a symmetric operator is self-adjoint if the domain of its adjoint,In the present context, it is a convenient fact that every densely defined, symmetric operatorThis can be shown by invoking the symmetric assumption and Riesz representation theorem.In the next section, a symmetric operator will be assumed to be densely defined and closed.Another way of looking at the issue is provided by the Cayley transform of a self-adjoint operator and the deficiency indices.to have a self-adjoint extension, as follows: Theorem — A necessary and sufficient condition forDefine the deficiency subspaces of A by In this language, the description of the self-adjoint extension problem given by the theorem can be restated as follows: a symmetric operatorare defined as the dimension of the orthogonal complements of the domain and range: Theorem — A partial isometryA symmetric operator has a unique self-adjoint extension if and only if both its deficiency indices are zero.This notion leads to the von Neumann formulae:[5] Theorem — SupposeOn the subspace of absolutely continuous function that vanish on the boundary, define the operatorWe will see that extending A amounts to modifying the boundary conditions, thereby enlargingEvery partial isometry can be extended, on a possibly larger space, to a unitary operator.Consequently, every symmetric operator has a self-adjoint extension, on a possibly larger space.While the extension problem for general symmetric operators is essentially that of extending partial isometries to unitaries, for positive symmetric operators the question becomes one of extending contractions: by "filling out" certain unknown entries of a 2 × 2 self-adjoint contraction, we obtain the positive self-adjoint extensions of a positive symmetric operator.Before stating the relevant result, we first fix some terminology.Using this machinery, one can explicitly describe the structure of general matrix contractions.The Cayley transform for general symmetric operators can be adapted to this special case.a contraction defined by which have matrix representation[clarification needed] It is easily verified that theand its projection onto its domain is self-adjoint, then it is clear that its inverse Cayley transform defined onThe symmetric property follows from its projection onto its own domain being self-adjoint and positivity follows from contractivity., its Cayley transform is a contraction satisfying the stated "partial" self-adjoint property.The unitarity criterion of the Cayley transform is replaced by self-adjointness for positive operators.Therefore, finding self-adjoint extension for a positive symmetric operator becomes a "matrix completion problem".This can always be done and the structure of such contractions gives a parametrization of all possible extensions.