Numerical range of operators acting on Banach spaces

Jahedi, Khadijeh ; Yousefi, Bahmann

Czechoslovak Mathematical Journal, Tome 62 (2012) no. 2, pp. 495-503

Abstract (VO)

The aim of the paper is to propose a definition of numerical range of an operator on reflexive Banach spaces. Under this definition the numerical range will possess the basic properties of a canonical numerical range. We will determine necessary and sufficient conditions under which the numerical range of a composition operator on a weighted Hardy space is closed. We will also give some necessary conditions to show that when the closure of the numerical range of a composition operator on a small weighted Hardy space has zero.

Détail

DOI : 10.1007/s10587-012-0024-7

Classification : 30H10, 47A12, 47B33, 47B37
Keywords: numerical range; weighted Hardy space; compact operator; composition operator

Bibliographie

[1] Bauer, F. L.: On the field of values subordinate to a norm. Numer. Math. 4 (1962), 103-113. | DOI | MR | Zbl

[2] Bonsall, F. F., Duncan, J.: Numerical Ranges II. London Mathematical Society Lecture Note Series. 10. London: Cambridge University Press. VII (1973). | MR | Zbl

[3] Cowen, C. C., MacCluer, B. D.: Composition Operators on Spaces of Analytic Functions. Studies in Advanced Mathematics. Boca Raton, FL: CRC Press. xii (1995). | MR | Zbl

[4] Gustafson, K. E., Rao, D. K. M.: Numerical Range. The Field of Values of Linear Operators and Matrices. Universitext. New York, NY: Springer. xiv (1996). | MR | Zbl

[5] Halmos, P. R.: A Hilbert Space Problem Book. Graduate Texts in Mathematics, 19. New York-Heidelberg-Berlin: Springer-Verlag. XVII (1982). | MR | Zbl

[6] Hausdorff, F.: Der Wertvorrat einer Bilinearform. Math. Zeitschr. 3 (1919), 314-316 German. | DOI | MR

[7] Jahedi, K., Yousefi, B.: On the weighted Hardy spaces. J. Math. Extension 1 (2006), 45-55. | MR

[8] Lumer, G.: Semi-inner-product spaces. Trans. Am. Math. Soc. 100 (1961), 29-43. | DOI | MR | Zbl

[9] Seddighi, K., Yousefi, B.: On the reflexivity of operators on function spaces. Proc. Am. Math. Soc. 116 (1992), 45-52. | DOI | MR | Zbl

[10] Shapiro, J. H.: Composition Operators and Classical Function Theory. Universitext: Tracts in Mathematics. New York: Springer-Verlag. xiii (1993). | MR | Zbl

[11] Shapiro, J.: Compact composition operators on spaces of boundary-regular holomorphic functions. Proc. Am. Math. Soc. 100 (1987), 49-57. | DOI | MR | Zbl

[12] Sheilds, A. L.: Weighted shift operators and analytic function theory. Topic Operator Theory, math. Survays 13 (1974), 49-128. | DOI

[13] Toeplitz, O.: Das algebraische Analogen zu einem Satze von Fejér. Math. Zs. 2 (1918), 187-197. | DOI | MR

[14] Yang, K. B., Zhou, Z. H.: Essentional norm of difference of composition operators on Bloch space. Czech. Math. J. 60 (2010), 1139-1152. | DOI | MR

[15] Yousefi, B.: On the space $l^p(\beta)$. Rend. Circ. Mat. Palermo, II. Ser. 49 (2000), 115-120. | MR

[16] Yousefi, B., Jahedi, K.: Application of Rosenthal-Dor Theorem on Banach space of formal power series. Islamic Azad University Journal of Sciences. Fall (2001), 3147-3168. | MR

[17] Yousefi, B.: Unicellularity of the multiplication operator on Banach spaces of formal power series. Stud. Math. 147 (2001), 201-209. | DOI | MR | Zbl

[18] Yousefi, B.: Bounded analytic structure of the Banach space of formal power series. Rend. Circ. Mat. Palermo (2) 51 (2002), 403-410. | DOI | MR | Zbl

[19] Yousefi, B., Jahedi, S.: Composition operators on Banach space of formal power series. Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (8) 6 (2003), 481-487. | MR

[20] Yousefi, B.: Strictly cyclic algebra of operators acting on Banach spaces $H^{p}(\beta)$. Czech. Math. J. 54 (2004), 261-266. | DOI | MR

[21] Yousefi, B., Soltani, R.: On the Hilbert space of formal power series. Honam Math. Jour. 26 (2004), 299-308. | MR

[22] Yousefi, B.: Composition operators on weighted Hardy spaces. Kyungpook Math. J. 44 (2004), 319-324. | MR | Zbl

[23] Yousefi, B., Dehghan, Y. N.: Reflexivity on weighted Hardy spaces. Southeast Asian Bull. Math. 28 (2004), 587-593. | MR | Zbl

[24] Yousefi, B.: On the eighteenth question of Allen Shields. Int. J. Math. 16 (2005), 37-42. | DOI | MR | Zbl

[25] Yousefi, B., Kashkuli, A. I.: Cyclicity and uncellularity of the differentiation operator on Banach spaces of formal power series. Math. Proc. R. Ir. Acad. 105A (2005), 1-7. | DOI | MR | Zbl

[26] Yousefi, B., Haghkhah, S.: Numerical range of composition operators on weighted Hardy spaces. Int. J. Contemp. Math. Sci. 2 (2007), 1341-1346. | DOI | MR | Zbl