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Bounded, conservative, linear operators and the maximal group. II


Authors: E. P. Kelly and D. A. Hogan
Journal: Proc. Amer. Math. Soc. 38 (1973), 298-302
MSC: Primary 46L20
DOI: https://doi.org/10.1090/S0002-9939-1973-0313832-9
MathSciNet review: 0313832
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Abstract: Let $ V$ denote an infinite dimensional Banach space over the complex field, $ B[V]$ the bounded linear operators on $ V$ and $ F$ a closed subspace of $ V$. An element of $ {\mathcal{T}_F} = \{ T\vert T \in B[V],T(F) \subseteq F\} $ is called a conservative operator. Some sufficient conditions for $ T \in {\mathcal{T}_F}$ to be in the boundary, $ \mathcal{B}$, of the maximal group, $ \mathcal{M}$, of invertible elements are determined. For example, if $ T \in {\mathcal{T}_F}$, is such that (i) $ V$ is the topological direct sum of $ \mathcal{R}(T)$ and $ N(T) \ne \{ \theta \} $, (ii) $ T$ is an automorphism on $ \mathcal{R}(T) \cap F$, then $ T \in \mathcal{B}$. Also, the complement of the closure of $ \mathcal{M}$ is discussed. This is an extension of another paper by the same authors [6].


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  • [1] I. D. Berg, A Banach algebra criterion for Tauberian theorems, Proc. Amer. Math. Soc. 15 (1964), 648-652. MR 29 #2574. MR 0165285 (29:2574)
  • [2] -, Open sets of conservative matrices, Proc. Amer. Math. Soc. 16 (1965), 719-724. MR 31 #3762. MR 0179514 (31:3762)
  • [3] J. Copping, Mercerian theorems and inverse transformations, Studia Math. 21 (1961/62), 177-194. MR 25 #3296. MR 0139869 (25:3296)
  • [4] N. Dunford and J. T. Schwartz, Linear operators. I: General theory, Pure and Appl. Math., vol. 7, Interscience, New York, 1958. MR 22 #8302. MR 0117523 (22:8302)
  • [5] J. Feldman and R. V. Kadison, The closure of the regular operators in a ring of operators, Proc. Amer. Math. Soc. 15 (1964), 909-916. MR 0068749 (16:935g)
  • [6] E. Kelly and D. Hogan, Bounded, conservative, linear operators and the maximal group, Proc. Amer. Math. Soc. 32 (1972), 195-200. MR 0290136 (44:7321)
  • [7] B. E. Rhoades, Triangular summability methods and the boundary of the maximal group, Math. Z. 105 (1968), 284-290. MR 37 #4461. MR 0228882 (37:4461)
  • [8] C. E. Rickart, General theory of Banach algebras, University Series in Higher Math., Van Nostrand, Princeton, N.J., 1960. MR 22 #5903. MR 0115101 (22:5903)
  • [9] R. Whitley, Conull and other matrices which sum a bounded divergent sequence, Amer. Math. Monthly 74 (1967), 798-801. MR 36 #3010. MR 0219940 (36:3010)
  • [10] A. Wilansky, Topological divisors of zero and Tauberian theorems, Trans. Amer. Math. Soc. 113 (1964), 240-251. MR 29 #6222. MR 0168967 (29:6222)
  • [11] B. Yood, Transformations between Banach spaces in the uniform topology, Ann. of Math. (2) 50 (1949), 486-503. MR 10, 611. MR 0029474 (10:611c)
  • [12] D. J. H. Garling and A. Wilansky, On a summability theorem of Berg, Crawford, and Whitley, Proc. Cambridge Philos. Soc. 71 (1972), 495-497. MR 0294946 (45:4014)

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DOI: https://doi.org/10.1090/S0002-9939-1973-0313832-9
Keywords: Banach algebra, maximal group, boundary of maximal group, bounded operator, conservative operator, projection, quasi-compact operator
Article copyright: © Copyright 1973 American Mathematical Society

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