An example in the theory of $AC$-operators

Doust, Ian; Gillespie, T. A.

doi:10.1090/s0002-9939-00-05733-6

Proc. Amer. Math. Soc.

2000

DOI: 10.1090/s0002-9939-00-05733-6

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An example in the theory of $AC$-operators

Ian Doust

T. A. Gillespie

Abstract: Abstract. AC-operators are a generalization in the context of well-boundedness of normal operators on Hilbert space. It was shown by Doust and Walden that compact AC-operators have a representation as a conditionally convergent sum reminiscent of the spectral representations for compact normal operators. In this representation, the eigenvalues must be taken in a particular order to ensure convergence of the sum. Here we show that one cannot replace the ordering given by Doust and Walden by the more natural one… Show more

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Compact AC(σ) Operators

Ashton¹,

Doust²

2009

Integr. equ. oper. theory

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All compact AC(σ) operators have a representation analogous to that for compact normal operators. As a partial converse we obtain conditions which allow one to construct a large number of such operators. Using the results in the paper, we answer a number of questions about the decomposition of a compact AC(σ) operator into real and imaginary parts. (2000). 47B40. Mathematics Subject Classification

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Compact AC(σ) Operators

Ashton¹,

Doust²

2009

Integr. equ. oper. theory

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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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An example in the theory of $AC$-operators

Cited by 2 publications

References 6 publications

Compact AC(σ) Operators

Compact AC(σ) Operators

Compact well-bounded operators

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