Hostname: page-component-848d4c4894-p2v8j Total loading time: 0.001 Render date: 2024-06-01T22:53:25.072Z Has data issue: false hasContentIssue false
  • English
  • Français

Relations Between Generalized Growth Conditions and Several Classes of Convexoid Operators

Published online by Cambridge University Press:  20 November 2018

Takayuki Furuta
Takayuki Furuta*
Affiliation:
Hirosaki University, Bunkyo-Cho 3, Hirosaki, Aomori-Ken 036, Japan
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.

In this paper we shall discuss some classes of bounded linear operators on a complex Hilbert space. If T is a bounded linear operator T acting on the complex Hilbert space H, then the following two inequalities always hold:

where σ(T) indicates the spectrum of T, W(T) denotes the numerical range of T defined by W(T) = {(Tx, x) : ||x|| = 1 and x ∊ H} and means the closure of W(T) respectively.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 29 , Issue 5 , 01 October 1977 , pp. 1010 - 1030
Copyright
Copyright © Canadian Mathematical Society 1977

References

1. Berberian, S. K., Approximate proper vectors, Proc. Amer. Math. Soc. 13 (1962), 111–114.Google Scholar
2. Berberian, S. K. An extension of WeyVs theorem to a class of not necessarily normal operators, Michigan Math. J. 19 (1969), 273–279.Google Scholar
3. Berberian, S. K. Conditions on an operator implying Re <r(T) = <r(Re T), Trans. Amer. Math. Soc. 154 (1971), 267–272.Google Scholar
4. Fujii, M., On some examples of non-normal operators, I, II and III, Proc. Japan Acad. Jf.7 (1971), 458-463; 49 (1973), 118-123 and 49 (1973), 124–129.Google Scholar
5. Fujii, M. and Tamaki, K., On normal approximate spectrum, III, Proc. Japan Acad. 48 (1972), 389–393.Google Scholar
6. Furuta, T., A note on two inequalities correlated to unitary p-dilatations, Proc. Japan Acad. 45 (1969), 561–564.Google Scholar
7. Furuta, T. Some theorems on unitary p-dilations of Sz.-Nagy and Foias, Acta Sci. Math. 33 (1972), 119–122.Google Scholar
8. Furuta, T. Some characterizations of convexoid operators, Rev. Roum. Math. Pures et Appl. 18 (1973), 893–900.Google Scholar
9. Furuta, T. and Nakamoto, R., On tensor products of operators, Proc. Japan Acad. 45 (1969), 680–685.Google Scholar
10. Furuta, T. and Nakamoto, R. On the numerical range of an operator, Proc. Japan Acad. 41 (1971), 279–284.Google Scholar
11. Halmos, P. R., A Hilbert space problem book (Van Nostrand, Princeton, 1967).Google Scholar
12. Holbrook, J., On the power-bounded operators of Sz.-Nagy and Foias, C., Acta Sci. Math. 21 (1968), 299–310.Google Scholar
13. Lebow, A., On von Neumann s theory of spectral set, J. Math. Anal. Appl. 7 (1963), 64–90.Google Scholar
14. Lin, C.-S., On a family of generalized numerical ranges, Can. j. Math. 26 (1974), 678–685.Google Scholar
15. Luecke, G. R., A class of operators on Hilbert space, Pacific J. Math. 1 (1971), 153–156.Google Scholar
16. Luecke, G. R. Topological properties of paranormal operators, Trans. Amer. Math. Soc. 172 (1972), 35–43.Google Scholar
17. Orland, G. H., On a class of operators, Proc. Amer. Math. Soc. 15 (1964), 75–79.Google Scholar
18. Patel, S. M., A note on a class of operators, to appear.Google Scholar
19. Putnam, C. R., On the spectra of semi-normal operators, Trans. Amer. Math. Soc. 110 (1965), 509–523.Google Scholar
20. Putnam, C. R. An inequality for the area of hyponormal spectra, Math. Z. 116 (1970), 323–330.Google Scholar
21. Saito, T., Numerical ranges of tensor products of operators, Tohoku Math. J. 11 (1967), 98–100.Google Scholar
22. Saito, T. A theorem on boundary spectra, Acta Sci. Math. Szeged. 33 (1972), 101–104.Google Scholar
23. Schreiber, M., On the spectrum of a contraction, Proc. Amer. Math. Soc. 12 (1961), 709–713.Google Scholar
24. Schreiber, M. Numerical range and spectral sets, Michigan Math. J. 10 (1963), 283–288.Google Scholar
25. Sz.-Nagy, B. and Foias, C., On certain classes of power bounded operators in Hilbert space, Acta Sci. Math. 27 (1966), 17–25.Google Scholar
26. Sz.-Nagy, B. and Foias, C. Harmonic analysis of operators on Hilbert space (North-Holland, 1970).Google Scholar
27. Williams, J. P., Minimal spectral sets of compact operators, Acta Sci. Math. Szeged. 28 (1967), 93–106.Google Scholar