Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-25T14:34:52.101Z Has data issue: false hasContentIssue false
  • English
  • Français

Orthonormal systems in Banach spaces and their applications

Published online by Cambridge University Press:  24 October 2008

N. J. Kalton  and
G. V. Wood
N. J. Kalton
Affiliation:
University College, Swansea
G. V. Wood
Affiliation:
University College, Swansea
Get access Rights & Permissions [Opens in a new window]

Extract

By an orthonormal system in a general complex Banach space, we mean a collection {eα: α ∈ } it vectors such that, for each α, there is an hermitian (in the numerical range sense, see (4)) projection Pα whose range is lin (eα) and such that PαPβ = 0, if α ≠ β. This paper is devoted to the study of orthonormal systems in general Banach spaces, and their applications to problems of characterizing isometries and hermitian operators.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 79 , Issue 3 , May 1976 , pp. 493 - 510
Copyright
Copyright © Cambridge Philosophical Society 1976

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Berkson, E.A characterization of complex Hilbert spaces. Bull. London Math. Soc. 2 (1970), 313–315.CrossRefGoogle Scholar
(2)Berkson, E.Hermitian projections and orthogonality in Banach spaces. Proc. London Math. Soc. (1) 24 (1972), 101–118.Google Scholar
(3)Bessaga, C. and Peczynski, A.On bases and unconditional convergence of series in Banach space. Studia Math. 17 (1958), 151–164.Google Scholar
(4)Bonsall, F. F. and Duncan, J.Numerical ranges of operators on normed spaces and of elements of normed algebras, vol. I (Cambridge University Press, 1971).Google Scholar
(5)Bonsall, F. F. and Duncan, J.Numerical ranges of operators on nonmed spaces and of elements of normed algebras, vol. II (Cambridge University Press, 1974).Google Scholar
(6)Davis, W. J.Separable Banach spaces with only trivial isometries. Rev. Roumaine Math. Pures Appl. 16 (1971), 1051–1054.Google Scholar
(7)Fleming, R. J. and Jamison, J. E.Hermitian and adjoint abelian operators on certain Banach spaces. Pacific J. Math. 52 (1974), 67–85.Google Scholar
(8)Fleming, R. J. and Jamison, J. E.Isometries on certain Barash spaces. J. London Math. Soc. (2) 9 (1974), 121–127.Google Scholar
(9)Dunford, N. and Scawartz, J. T.Linear Operators, vol. I (Interscience, New York, 1958).Google Scholar
(10)De Groot, J. and Wille, R. J.Rigid continua and topological group-pictures, Arch. Math. (Basel) 9 (1958), 441–446.Google Scholar
(11)Jordan, P. and von Neumann, J.On inner-products in linear metric spaces. Ann. of Math. (2) 36 (1935), 719–723.Google Scholar
(12)Lumer, G.Semi-inner-product spaces. Trans. Amer. Math. Soc. 100 (1961), 29–43.Google Scholar
(13)Lumer, G.On isometries of reflexive Orlicz spaces. Ann. Inst. Fourier (Grenoble) 13 (1963), 99–109.CrossRefGoogle Scholar
(14)Luxemburg, W. A. J. Banach function spaces, Thesis Delft Techn. Univ. 1955.Google Scholar
(15)Luxemburg, W. A. J. and Zaanen, A. C.Some remarks on Banach function spaces. Indagationes Math. 18 (1956), 110–119.Google Scholar
(16)Palmer, T. W.Unbounded normal operators on Banach spaces. Trans. Amer. Math. Soc. 133 (1968), 385–414.Google Scholar
(17)Rolewicz, S.Metric Linear Spaces (PWN Warsaw, 1972).Google Scholar
(18)Schneider, H. and Turner, R. E. L.Matrices Hermitian for an absolute norm. Linear and Multilinear Algebra 1 (1973), 9–31.CrossRefGoogle Scholar
(19)Singer, I.Bases in Banach Spaces I (Springer, Berlin, 1971).Google Scholar
(20)Sobczyk, A.Projection of (m) onto its subspace (c 0). Bull. Amer. Math. Soc. 47 (1941), 938–947.Google Scholar
(21)Tam, K. W.Isometries of certain function spaces. Pacific J. Math. 31 (1969), 233–246.Google Scholar
(22)Tong, A. E.Diagonal submatrices of matrix maps. Pacific J. Math. 32 (1970), 551–559.Google Scholar
(23)Torrance, E. Adjoints of operators on Banach spaces. Ph.D. Thesis, Illinois, 1968.Google Scholar