ISSN:
1420-8989
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract LetT be an operator on an infinite dimensional Hilbert space $$\mathcal{H}$$ with eigenvectorsv i , ‖v i ‖=1,i=1, 2, ..., andsp{v i ⋎i∈n} dense in $$\mathcal{H}$$ . Suppose that {v i } is a Schauder basis for $$\mathcal{H}$$ . We denote byA T the ultraweakly closed algebra generated byT andI, the identity operator on $$\mathcal{H}$$ . For any nonnegative sequence of scalars $$\left\{ {\alpha ,with = \sum\nolimits_1^\infty {\alpha _1 } = 1} \right\},$$ , we associate an ultraweakly (normal) continuous linear functional $$\phi _\alpha = \sum\nolimits_1^\infty {\alpha _j } \omega _v$$ where $$\phi _\alpha \left( A \right) = \lim _n \sum\nolimits_1^n {\alpha _j } \omega _v ,$$ , and $$\omega _v ,\left( A \right) =〈 Av_1 ,v_1 〉$$ for allA∈A T . We denote the set of all such linear functionals onA TbyF(T). The question that we investigate in this paper is whether each linear functional φα inF(T) is a vector state, i.e. does φα=ωx for some unit vectorx in $$\mathcal{H}$$ ?
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01272121
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