Bounded, conservative, linear operators and the maximal group. II
HTML articles powered by AMS MathViewer
- by E. P. Kelly and D. A. Hogan PDF
- Proc. Amer. Math. Soc. 38 (1973), 298-302 Request permission
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 ${\mathcal {T}_F} = \{ T|T \in B[V],T(F) \subseteq F\}$ 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].References
- I. David Berg, A Banach algebra criterion for Tauberian theorems, Proc. Amer. Math. Soc. 15 (1964), 648–652. MR 165285, DOI 10.1090/S0002-9939-1964-0165285-6
- I. David Berg, Open sets of conservative matrices, Proc. Amer. Math. Soc. 16 (1965), 719–724. MR 179514, DOI 10.1090/S0002-9939-1965-0179514-7
- J. Copping, Mercerian theorems and inverse transformations, Studia Math. 21 (1961/62), 177–194. MR 139869, DOI 10.4064/sm-21-2-177-194
- Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
- Jacob Feldman and Richard V. Kadison, The closure of the regular operators in a ring of operators, Proc. Amer. Math. Soc. 5 (1954), 909–916. MR 68749, DOI 10.1090/S0002-9939-1954-0068749-2
- E. P. Kelly Jr. and D. A. Hogan, Bounded, conservative, linear operators and the maximal group, Proc. Amer. Math. Soc. 32 (1972), 195–200. MR 290136, DOI 10.1090/S0002-9939-1972-0290136-3
- B. E. Rhoades, Triangular summability methods and the boundary of the maximal group, Math. Z. 105 (1968), 284–290. MR 228882, DOI 10.1007/BF01125969
- Charles E. Rickart, General theory of Banach algebras, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0115101
- Robert Whitley, Conull and other matrices which sum a bounded divergent sequence, Amer. Math. Monthly 74 (1967), 798–801. MR 219940, DOI 10.2307/2315795
- Albert Wilansky, Topological divisors of zero and Tauberian theorems, Trans. Amer. Math. Soc. 113 (1964), 240–251. MR 168967, DOI 10.1090/S0002-9947-1964-0168967-X
- Bertram Yood, Transformations between Banach spaces in the uniform topology, Ann. of Math. (2) 50 (1949), 486–503. MR 29474, DOI 10.2307/1969464
- 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 294946, DOI 10.1017/s0305004100050775
Additional Information
- © Copyright 1973 American Mathematical Society
- 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