Extremal positive and self-adjoint extensions of suboperators

Abstract

F. ollowing Halmos [1] a suboperator is a map from a subspace of a (complex) Hilbert space into the whole space which is a restriction of some bounded linear transformation, an operator on the space. A suboperator is said to be subpositive, subself-adjoint (and so on) if it is a restriction of a positive, self-adjoint (and so on) operator of the space. The characterization problem of subself-adjoint suboperators is solved by Krein, see [2]. Krein shows that a bounded and symmetric linear map from a subspace of a Hilbert space has a symmetric extension to the whole space (hence a self-adjoint extension) with the same bound. Krein's method of proof yields the normpreserving smallest, resp. largest self-adjoint extensions, in the usual ordering of self-adjoint operators on Hilbert space, too. The extension theorem for subpositive suboperators, due to one of the authors [3], led naturally to the reduction of Krein's theorem to the just mentioned positive extendibility problem. Another characterization of subpositive suboperators is given by Halmos [1]. The purpose of this note is first to show that the extension of a subpositive suboperator constructed in [3] is of minimal norm and smallest in the ordering of self-adjoint operators, between all the positive extensions. The existence of a largest positive extension (with the same norm) is then a simple consequence of the existence of the smallest positive extension. The extremal norm-preserving extension problem with respect to subself-adjoint suboperators is thus reduced to the corresponding, just mentioned question for subpositive suboperators.