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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. Other applications of solutions to this problem can be seen in various moment problems. This article discusses a few related problems of this type. 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. == Symmetric operators == Let ''H'' be a Hilbert space. A linear operator ''A'' acting on ''H'' with dense domain Dom(''A'') is symmetric if :<''Ax'', ''y''> = <''x'', ''Ay''>, for all ''x'', ''y'' in Dom(''A''). If Dom(''A'') = ''H'', the Hellinger-Toeplitz theorem says that ''A'' is a bounded operator, in which case ''A'' is self-adjoint and the extension problem is trivial. In general, a symmetric operator is self-adjoint if the domain of its adjoint, Dom(''A *''), lies in Dom(''A''). When dealing with unbounded operators, it is often desirable to be able to assume that the operator in question is closed. In the present context, it is a convenient fact that every symmetric operator ''A'' is closable. That is, ''A'' has a smallest closed extension, called the ''closure'' of ''A''. This can be shown by invoking the symmetric assumption and Riesz representation theorem. Since ''A'' and its closure have the same closed extensions, it can always be assumed that the symmetric operator of interest is closed. In the sequel, a symmetric operator will be assumed to be densely defined and closed. Problem ''Given a densely defined closed symmetric operator A, find its self-adjoint extensions.'' This question can be translated to an operator-theoretic one. As a heuristic motivation, notice that the Cayley transform on the complex plane, defined by : maps the real line to the unit circle. This suggests one define, for a symmetric operator ''A'', : on ''Ran''(''A'' + ''i''), the range of ''A'' + ''i''. The operator ''UA'' is in fact an isometry between closed subspaces that takes (''A'' + ''i'')''x'' to (''A'' - ''i'')''x'' for ''x'' in Dom(''A''). The map : is also called the Cayley transform of the symmetric operator ''A''. Given ''UA'', ''A'' can be recovered by : defined on ''Dom''(''A'') = ''Ran''(''U'' - 1). Now if : is an isometric extension of ''UA'', the operator : acting on : is a symmetric extension of ''A''. Theorem The symmetric extensions of a closed symmetric operator ''A'' is in one-to-one correspondence with the isometric extensions of its Cayley transform ''UA''. Of more interest is the existence of ''self-adjoint'' extensions. The following is true. Theorem A closed symmetric operator ''A'' is self-adjoint if and only if Ran (''A'' ± ''i'') = ''H'', i.e. when its Cayley transform ''UA'' is a unitary operator on ''H''. Corollary The self-adjoint extensions of a closed symmetric operator ''A'' is in one-to-one correspondence with the unitary extensions of its Cayley transform ''UA''. Define the deficiency subspaces of ''A'' by : and : In this language, the description of the self-adjoint extension problem given by the corollary can be restated as follows: a symmetric operator ''A'' has self-adjoint extensions if and only if its Cayley transform ''UA'' has unitary extensions to ''H'', i.e. the deficiency subspaces ''K''+ and ''K''- have the same dimension. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「extensions of symmetric operators」の詳細全文を読む スポンサード リンク
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