Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-23T05:19:13.009Z Has data issue: false hasContentIssue false
  • English
  • Français

Strongly analytic spaces in spectral decomposition

Published online by Cambridge University Press:  18 May 2009

Ridgley Lange  and
Shengwang Wang
Ridgley Lange
Affiliation:
Department of Mathematics, Central Michigan University, Mount Pleasant, Michigan 48859, U.S.A.
Shengwang Wang
Affiliation:
Department of Mathematics, Central Michigan University, Mount Pleasant, Michigan 48859, U.S.A.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It is now well-known that decomposable operators have a rich structure theory; in particular, an operator is decomposable iff its adjoint is [3]. There are many other criteria for decomposability [8], [9]. In Theorem 2.2 of this paper (see below) we give several new ones. Some of these (e.g. (ii), (iii)) are “relaxations” of conditions given in [7] and [8]. Assertion (vi) is a version of a result in [10]. Characterizations (iv)and (v) are novel in two respects. For instance, (v) states that an operator Tcan be “patched” together into a decomposable operator if it has an invariant subspace Y such that T | Y and the coinduced operator T | Y are both decomposable. Secondly, in this way the strongly analytic subspace appears in the theory of spectral decomposition.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 30 , Issue 3 , September 1988 , pp. 249 - 257
Copyright
Copyright © Glasgow Mathematical Journal Trust 1988

References

1.Albrecht, E., On two questions of I. Colojoara and C. Foias, Manuscripta Math., 25 (1978), 1–15.CrossRefGoogle Scholar
2.Bishop, E., A duality theory for an arbitrary operator, Pacific J. Math., 9 (1959), 379–397.CrossRefGoogle Scholar
3.Erdelyi, I. and Wang, S., A local spectral theory for closed operators, London Math. Soc., Lecture Note Series No 105 (Cambridge University Press, 1985).CrossRefGoogle Scholar
4.Foias, C., On the maximal spectral spaces of a decomposable operator, Rev. Roumaine Math. PuresAppl., 15 (1970), 1599–1606.Google Scholar
5.Kato, T., Perturbation theory for linear operators (Springer-Verlag, 1980).Google Scholar
6.Lange, R., Strongly analytic subspaces, in Operator Theory and Functional Analysis, Research Notes in Math. 38, Pitman Advanced Publishing Program, (San Francisco, 1979), 16–30.Google Scholar
7.Lange, R., On generalization of decomposability, Glasgow Math. J., 22 (1981), 77–81.CrossRefGoogle Scholar
8.Lange, R., Duality and asymptotic spectral decompositions, Pacific J. Math., 121 (1986), 93–108.CrossRefGoogle Scholar
9.Lange, R. and Wang, S., New criteria of decomposable operators, Illinois J. Math., to appear.Google Scholar
10.Radjabalipour, M., On decomposition of operators, Michigan Math. J., 21 (1974), 265–275.Google Scholar
11.Radjabalipour, M., Equivalence of decomposable and 2-decomposable operators, Pacific J. Math., 77 (1978), 243–247.CrossRefGoogle Scholar
12.Shulberg, G., Decomposable restrictions and extensions, J. Math. Anal. Appl., 83 (1981), 144–158.CrossRefGoogle Scholar
13.Snader, J. C., Bishop's condition (β), J. Math. Anal. Appl., in print.Google Scholar
14.Wang, S., A characterization of strongly decomposable operators and its duality theorem, Ada Math. Sinica, 29 (1986), 145–155.Google Scholar