Invertibility in nest algebras
HTML articles powered by AMS MathViewer
- by Avraham Feintuch and Alan Lambert PDF
- Proc. Amer. Math. Soc. 91 (1984), 573-576 Request permission
Abstract:
Let $\mathcal {F}$ denote a complete nest of subspaces of a complex Hilbert space $\mathcal {H}$, and let $\mathcal {C}$ denote the nest algebra defined by $\mathcal {F}$. Let $\mathcal {K}$ denote the ideal of compact operators on $\mathcal {H}$. If $\mathcal {F}$ has no infinite-dimensional gaps then $T \in \mathcal {C}$ is invertible in $\mathcal {C}$ if and only if it is invertible in $\mathcal {C} + \mathcal {K}$. An example is given of a nest with an infinite gap for which there exists an operator in $\mathcal {C}$ which is invertible in $\mathcal {C} + \mathcal {K}$ but not in $\mathcal {C}$.References
- J. A. Erdos, Operators of finite rank in nest algebras, J. London Math. Soc. 43 (1968), 391–397. MR 230156, DOI 10.1112/jlms/s1-43.1.391
- Thomas Fall, William Arveson, and Paul Muhly, Perturbations of nest algebras, J. Operator Theory 1 (1979), no. 1, 137–150. MR 526295
- A. Feintuch and R. Saeks, Extended spaces and the resolution topology, Internat. J. Control 33 (1981), no. 2, 347–354. MR 609087, DOI 10.1080/00207178108922927 —, System theory—a Hilbert space approach, Academic Press, New York, 1982.
- I. C. Gohberg and M. G. Kreĭn, Theory and applications of Volterra operators in Hilbert space, Translations of Mathematical Monographs, Vol. 24, American Mathematical Society, Providence, R.I., 1970. Translated from the Russian by A. Feinstein. MR 0264447
- M. S. Livshitz, On a certain class of linear operators in Hilbert space, Rec. Math. [Mat. Sbornik] N.S. 19(61) (1946), 239–262 (Russian, with English summary). MR 0020719
- J. R. Ringrose, On some algebras of operators, Proc. London Math. Soc. (3) 15 (1965), 61–83. MR 171174, DOI 10.1112/plms/s3-15.1.61 A. Feintuch, Invertibility in nest algebras, preprint.
Additional Information
- © Copyright 1984 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 91 (1984), 573-576
- MSC: Primary 47C05; Secondary 47A05
- DOI: https://doi.org/10.1090/S0002-9939-1984-0746092-2
- MathSciNet review: 746092