Geodesic
Numerical ranges of powers of hermitian elements
Glasgow mathematical journal, Tome 28 (1986) no. 1, pp. 37-45

Voir la notice de l'article provenant de la source Cambridge University Press

An element k of a unital Banach algebra A is said to be Hermitian if its numerical rangeis contained in R; equivalently, ∥eitk∥ = 1(t ∈ R)—see Bonsall and Duncan [3] and [4]. Here we find the largest possible extent of V(kn), n ∈ N, given V(k) ⊆ [−1, 1], and so ∥k∥ ≤ 1: previous knowledge is in Bollobás [2] and Crabb, Duncan and McGregor [7]. The largest possible sets all occur in a single example. Surprisingly, they all have straight line segments in their boundaries. The example is in [2] and [7], but here we give A. Browder's construction from [5], partly published in [6]. We are grateful to him for a copy of [5], and for discussions which led to the present work. We are also grateful to J. Duncan for useful discussions. 
Crabb, M. J.; McGregor, C. M. Numerical ranges of powers of hermitian elements. Glasgow mathematical journal, Tome 28 (1986) no. 1, pp. 37-45. doi: 10.1017/S0017089500006315
@article{10_1017_S0017089500006315,
     author = {Crabb, M. J. and McGregor, C. M.},
     title = {Numerical ranges of powers of hermitian elements},
     journal = {Glasgow mathematical journal},
     pages = {37--45},
     year = {1986},
     volume = {28},
     number = {1},
     doi = {10.1017/S0017089500006315},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089500006315/}
}
TY  - JOUR
AU  - Crabb, M. J.
AU  - McGregor, C. M.
TI  - Numerical ranges of powers of hermitian elements
JO  - Glasgow mathematical journal
PY  - 1986
SP  - 37
EP  - 45
VL  - 28
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089500006315/
DO  - 10.1017/S0017089500006315
ID  - 10_1017_S0017089500006315
ER  - 
%0 Journal Article
%A Crabb, M. J.
%A McGregor, C. M.
%T Numerical ranges of powers of hermitian elements
%J Glasgow mathematical journal
%D 1986
%P 37-45
%V 28
%N 1
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089500006315/
%R 10.1017/S0017089500006315
%F 10_1017_S0017089500006315

[1] 1.Boas, R. P., Entire functions (Academic Press, 1954). Google Scholar

[2] 2.Bollobás, B., The numerical range in Banach algebras and complex functions of exponential type, Bull. London Math. Soc. 3 (1971), 27–33. Google Scholar | DOI

[3] 3.Bonsall, F. F. and Duncan, J., Numerical ranges of operators on normed spaces and elements of normed algebras. London Math. Soc. Lecture Notes 2 (Cambridge University Press, 1971). Google Scholar | DOI

[4] 4.Bonsall, F. F. and Duncan, J., Numerical ranges II, London Math. Soc. Lecture Notes 10 (Cambridge University Press, 1973). Google Scholar | DOI

[5] 5.Browder, A., States, Numerical ranges, etc., Proc. Brown Informal analysis Seminar, 1969. Google Scholar

[6] 6.Browder, A., On Bernstein's inequality and the norm of Hermitian operators, Amer. Math. Monthly 78 (1971), 871–873. Google Scholar | DOI

[7] 7.Crabb, M. J., Duncan, J. and McGregor, C. M., Some extremal problems in the theory of numerical ranges, Acta Math. 128 (1972), 123–142. Google Scholar | DOI

[8] 8.Halmos, P. R., A Hilbert Space Problem book, (Van Nostrand, 1967). Google Scholar

[9] 9.Hardy, G. H., A Course of Pure Mathematics, (Cambridge, ed. 5, 1928). Google Scholar

[10] 10.Luke, Y. L., Integrals of Bessel functions, (McGraw–Hill, 1962). Google Scholar

Cité par Sources :