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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let denote an infinite dimensional Banach space over the complex field, the bounded linear operators on and a closed subspace of . An element of is called a conservative operator. Some sufficient conditions for to be in the boundary, , of the maximal group, , of invertible elements are determined. For example, if , is such that (i) is the topological direct sum of and , (ii) is an automorphism on , then . Also, the complement of the closure of is discussed. This is an extension of another paper by the same authors [**6**].

**[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)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
46L20

Retrieve articles in all journals with MSC: 46L20

Additional Information

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