The spectral theorem for well-bounded operators

Doust, Ian; Bozhou, Qiu

doi:10.1017/s1446788700031827

J Aust Math Soc A

1993

DOI: 10.1017/s1446788700031827

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The spectral theorem for well-bounded operators

Ian Doust

Qiu Bozhou

Abstract: Well-bounded operators are those which possess a bounded functional calculus for the absolutely continuous functions on some compact interval. Depending on the weak compactness of this functional calculus, one obtains one of two types of spectral theorem for these operators. A method is given which enables one to obtain both spectral theorems by simply changing the topology used. Even for the case of well-bounded operators of type (B), the proof given is more elementary than that previously in the literature.

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Cited by 4 publications

References 10 publications

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Compact well-bounded operators

Qingping¹,

Doust²

2001

Glasgow Math. J.

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Abstract. Every compact well-bounded operator has a representation as a linear combination of disjoint projections reminiscent of the representation of compact self-adjoint operators. In this note we show that the converse of this result holds, thus characterizing compact well-bounded operators. We also apply this result to study compact well-bounded operators on some special classes of Banach spaces such as hereditarily indecomposable spaces and certain spaces constructed by G. Pisier.1991 Mathematics Subject Classification. Primary 47B40. Secondary 34L10, 47A60, 47B07.1. Introduction. Well-bounded operators are defined as those which possess a functional calculus for the absolutely continuous functions on some compact interval ½a; b of the real line. Well-bounded operators were introduced by Smart [17] and Ringrose [14] in order to provide a theory for Banach space operators that was similar to the successful theory of self-adjoint operators on Hilbert space, but which included operators whose spectral expansions may only converge conditionally.On a general Banach space the integral representation theorems which one obtains for these operators are much less satisfactory than those obtained for selfadjoint operators. Nonetheless, even on an arbitrary Banach space, every compact well-bounded operator can be written in the form

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Compact well-bounded operators

Qingping¹,

Doust²

2001

Glasgow Math. J.

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Functional Calculus, Integral Representations, and Banach Space Geometry

Doust

deLaubenfels²

1994

Quaestiones Mathematicae

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Well-bounded operators on nonreflexive Banach spaces

Qingping¹,

Doust

1996

Proc. Amer. Math. Soc.

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Abstract. Every well-bounded operator on a reflexive Banach space is of type (B), and hence has a nice integral representation with respect to a spectral family of projections. A longstanding open question in the theory of wellbounded operators is whether there are any nonreflexive Banach spaces with this property. In this paper we extend the known results to show that on a very large class of nonreflexive spaces, one can always find a well-bounded operator which is not of type (B). We also prove that on any Banach space, compact well-bounded operators have a simple representation as a combination of disjoint projections.

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The spectral theorem for well-bounded operators

Cited by 4 publications

References 10 publications

Compact well-bounded operators

Compact well-bounded operators

Functional Calculus, Integral Representations, and Banach Space Geometry

Well-bounded operators on nonreflexive Banach spaces

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