The spectral properties of certain linear operators and their extensions
HTML articles powered by AMS MathViewer
- by Bruce A. Barnes PDF
- Proc. Amer. Math. Soc. 128 (2000), 1371-1375 Request permission
Abstract:
Let $H$ be a Hilbert space with inner-product $(x,y)$, and let $R$ be a bounded positive operator on $H$ which determines an inner-product, $\langle x,y\rangle =(Rx,y), x, y\in H$. Denote by $H^-$ the completion of $H$ with respect to the norm $\|x\|=\langle x,x\rangle ^{1/2}$. In this paper, operators having certain relationships with $R$ are studied. In particular, if $T=SR^{1/2}$ where $S\in B(H)$, then $T$ has an extension $T^-\in B(H^-)$, and $T$ and $T^-$ have essentially the same spectral and Fredholm properties.References
- Bruce A. Barnes, The spectral and Fredholm theory of extensions of bounded linear operators, Proc. Amer. Math. Soc. 105 (1989), no. 4, 941–949. MR 955454, DOI 10.1090/S0002-9939-1989-0955454-4
- Bruce A. Barnes, Essential spectra in a Banach algebra applied to linear operators, Proc. Roy. Irish Acad. Sect. A 90 (1990), no. 1, 73–82. MR 1092709
- Bruce A. Barnes, Common operator properties of the linear operators $RS$ and $SR$, Proc. Amer. Math. Soc. 126 (1998), no. 4, 1055–1061. MR 1443814, DOI 10.1090/S0002-9939-98-04218-X
- R. G. Douglas, On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc. 17 (1966), 413–415. MR 203464, DOI 10.1090/S0002-9939-1966-0203464-1
- Vasile I. Istrăţescu, Introduction to linear operator theory, Monographs and Textbooks in Pure and Applied Mathematics, vol. 65, Marcel Dekker, Inc., New York, 1981. MR 608969
- Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
- Joseph I. Nieto, On the essential spectrum of symmetrizable operators, Math. Ann. 178 (1968), 145–153. MR 233221, DOI 10.1007/BF01350656
- A. C. Zaanen, Linear Operators, North-Holland Pub. Co., Amsterdam, 1953.
Additional Information
- Bruce A. Barnes
- Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403
- Email: barnes@math.uoregon.edu
- Received by editor(s): June 23, 1998
- Published electronically: August 5, 1999
- Communicated by: David R. Larson
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 1371-1375
- MSC (1991): Primary 47A10, 47A30
- DOI: https://doi.org/10.1090/S0002-9939-99-05326-5
- MathSciNet review: 1664321