Relations Between Generalized Growth Conditions and Several Classes of Convexoid Operators

Furuta, Takayuki

Canadian journal of mathematics, Tome 29 (1977) no. 5, pp. 1010-1030

Résumé

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.

Détail

DOI : 10.4153/CJM-1977-099-0

Bibliographie

[1] 1. Berberian, S. K., Approximate proper vectors, Proc. Amer. Math. Soc. 13 (1962), 111–114. Google Scholar

[2] 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] 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] 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] 5. Fujii, M. and Tamaki, K., On normal approximate spectrum, III, Proc. Japan Acad. 48 (1972), 389–393. Google Scholar

[6] 6. Furuta, T., A note on two inequalities correlated to unitary p-dilatations, Proc. Japan Acad. 45 (1969), 561–564. Google Scholar

[7] 7. Furuta, T. Some theorems on unitary p-dilations of Sz.-Nagy and Foias, Acta Sci. Math. 33 (1972), 119–122. Google Scholar

[8] 8. Furuta, T. Some characterizations of convexoid operators, Rev. Roum. Math. Pures et Appl. 18 (1973), 893–900. Google Scholar

[9] 9. Furuta, T. and Nakamoto, R., On tensor products of operators, Proc. Japan Acad. 45 (1969), 680–685. Google Scholar

[10] 10. Furuta, T. and Nakamoto, R. On the numerical range of an operator, Proc. Japan Acad. 41 (1971), 279–284. Google Scholar

[11] 11. Halmos, P. R., A Hilbert space problem book (Van Nostrand, Princeton, 1967). Google Scholar

[12] 12. Holbrook, J., On the power-bounded operators of Sz.-Nagy and Foias, C., Acta Sci. Math. 21 (1968), 299–310. Google Scholar

[13] 13. Lebow, A., On von Neumann s theory of spectral set, J. Math. Anal. Appl. 7 (1963), 64–90. Google Scholar

[14] 14. Lin, C.-S., On a family of generalized numerical ranges, Can. j. Math. 26 (1974), 678–685. Google Scholar

[15] 15. Luecke, G. R., A class of operators on Hilbert space, Pacific J. Math. 1 (1971), 153–156. Google Scholar

[16] 16. Luecke, G. R. Topological properties of paranormal operators, Trans. Amer. Math. Soc. 172 (1972), 35–43. Google Scholar

[17] 17. Orland, G. H., On a class of operators, Proc. Amer. Math. Soc. 15 (1964), 75–79. Google Scholar

[18] 18. Patel, S. M., A note on a class of operators, to appear. Google Scholar

[19] 19. Putnam, C. R., On the spectra of semi-normal operators, Trans. Amer. Math. Soc. 110 (1965), 509–523. Google Scholar

[20] 20. Putnam, C. R. An inequality for the area of hyponormal spectra, Math. Z. 116 (1970), 323–330. Google Scholar

[21] 21. Saito, T., Numerical ranges of tensor products of operators, Tohoku Math. J. 11 (1967), 98–100. Google Scholar

[22] 22. Saito, T. A theorem on boundary spectra, Acta Sci. Math. Szeged. 33 (1972), 101–104. Google Scholar

[23] 23. Schreiber, M., On the spectrum of a contraction, Proc. Amer. Math. Soc. 12 (1961), 709–713. Google Scholar

[24] 24. Schreiber, M. Numerical range and spectral sets, Michigan Math. J. 10 (1963), 283–288. Google Scholar

[25] 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] 26. Sz.-Nagy, B. and Foias, C. Harmonic analysis of operators on Hilbert space (North-Holland, 1970). Google Scholar

[27] 27. Williams, J. P., Minimal spectral sets of compact operators, Acta Sci. Math. Szeged. 28 (1967), 93–106. Google Scholar