On a Certain Class of *-Algebras of Unbounded Operators

Araki, Huzihiro; Jurzak, J. P.

doi:10.2977/prims/1195183293

Cited by 33 publications

(10 citation statements)

References 8 publications

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“…Now (8) Remark. On JT, the associated bornological topology coincides with the order topology, or equivalently, the ^-topology considered in [6]. It was shown in [32] that this topology is different from the uniform topology, in general.…”

Section: R°s)°t=t^q=r°(s°t) We Show That Rmentioning

confidence: 88%

“…Then D is reflexive [9,29]. Furthermore, there exists a sequence (A n ) in L + (D) such that the following conditions are satisfied (see, e.g., [6,22] In particular, the space X and the space of continuous sesquilinear forms, considered in [6], are isomorphic as linear spaces.…”

Section: £(D D + ) Contains Both the Algebra L + (D} And The Algebramentioning

confidence: 99%

“…However, it is closely related to the product of operators on partial inner product spaces which was defined in [3]. Linear spaces with a partially defined multiplication were previously considered also in [4,5,6,11].…”

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confidence: 99%

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The Completion of the Maximal Op*-Algebra on a Frechet Domain

Kürsten¹

1986

Publ. Res. Inst. Math. Sci.

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This paper investigates the completion of the maximal Op*-algebra L + (D) of (possibly) unbounded operators on a dense domain D in a Hilbert space. It is assumed that D is a Frechet space with respect to the graph topology. Let D + denote the strong dual of D, equipped with the complex conjugate linear structure. It is shown that the completion of L + (D} (endowed with the uniform topology) is the space of continuous linear operators X (D, D + ) . This space is studied as an ordered locally convex space with an involution and a partially defined multiplication. A characterization of bounded subsets of D in terms of self-adjoint operators is given. The existence of special factorizations for several kinds of operators is proved. It is shown that the bounded operators are uniformly dense in § 1. Introduction Non-normable topological *-algebras satisfying various completeness conditions have been studied in several papers (see, e.g., [7,8,10,12,22,23,31,34]). However, these conditions are not fulfilled for the maximal *-algebra L + (D] of (possibly unbounded) operators on a dense linear subspace D of some Hilbert space H (for precise definitions, see Section 2). On the other hand, L + (D) is one of the most important unbounded operator algebras because it contains all *-algebras of operators on a fixed domain D.It is the aim of this paper to study the completion of L + (D) with respect to the uniform topology. We assume throughout that D is a Frechet space in the topology defined by the graph norms of operators belonging to L + (D}. However, some of the results can be obtained for more general domains D by the same proofs (see Remark 3 after Proposition 3.8 and the remarks after Proposition 5.1 and Corollary 5.6).Among [24,25]. However, it is closely related to the product of operators on partial inner product spaces which was defined in [3]. Linear spaces with a partially defined multiplication were previously considered also in [4,5,6,11].The pattern of the paper is as follows. In Section 2, we recall some definitions, notations, and some known or easy results. The study of the space JC(D, D + ] will be continued in [21]. In particular, we show there that X(D 9 D + ] is the second strong dual of its subspace of completely continuous operators. In [20, 33], the methods of the present paper are applied to the investigation of closed ideals in L + (D\ Acknowledgements

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Section: R°s)°t=t^q=r°(s°t) We Show That Rmentioning

confidence: 88%

Section: £(D D + ) Contains Both the Algebra L + (D} And The Algebramentioning

confidence: 99%

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The Completion of the Maximal Op*-Algebra on a Frechet Domain

Kürsten¹

1986

Publ. Res. Inst. Math. Sci.

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“…Let Bi and B2 be symmetric linear operators and let A be a selfadjoint operator acting on the same Hilbert space M such that D(Bi) Ç P(A) and DÍB2) Ç DÍA). We assume that there exists a constant A such that (1) ||5,£>|| < A||(A-M'M| for/= 1,2 and for all ^> G/? (Bi).…”

Section: Conditionsmentioning

confidence: 99%

Strongly commuting selfadjoint operators and commutants of unbounded operator algebras

Schmüdgen¹

1988

Proc. Amer. Math. Soc.

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ABSTRACT. Let A\ and A2 be (unbounded) selfadjoint operators on a Hubert space M which commute on a dense linear subspace of U. To conclude that A\ and Ai strongly commute, additional assumptions are necessary. Two propositions which contain such additional conditions are proved in §1. In §2 we define different commutants of unbounded operator algebras (form commutant, weak unbounded commutant, strong unbounded commutant) and we discuss the relations "between them and their bounded parts. In §3 we construct a selfadjoint »-representation of the polynomial algebra in two variables for which the form commutant is different from the weak unbounded commutant.

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“…Interesting properties about topologies [19], states [20], commutants [21] have been established in that situation.…”

Section: Introductionmentioning

confidence: 99%