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Commutators and normal operators

Published online by Cambridge University Press:  18 May 2009

M. J. Crabb  and
P. G. Spain
M. J. Crabb
Affiliation:
University of Glasgow, Glasgow G12 8QQ
P. G. Spain
Affiliation:
University of Glasgow, Glasgow G12 8QQ
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Let X be a Banach space and L(X) the Banach algebra of bounded linear operators on X. An operator T in L(X) is hermitian if ∥eitT∥ = 1 (t ∈ R), and is normal if T = R + iJ where R and J are commuting normal operators; R and J are then determined uniquely by T, and we may write T* = R–iJ. These definitions extend those for operators on Hilbert spaces. More details may be found in [1].

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 18 , Issue 2 , July 1977 , pp. 197 - 198
Copyright
Copyright © Glasgow Mathematical Journal Trust 1977

References

REFERENCES

1.Bonsall, F. F. and Duncan, J., Complete normed algebras (Springer-Verlag, 1973).CrossRefGoogle Scholar
2.Dowson, H. R., Gillespie, T. A. and Spain, P. G., A commutativity theorem for hermitian operators, Math. Ann. 220 (1976), 215–217.CrossRefGoogle Scholar
3.Palmer, T. W., Unbounded normal operators on Banach spaces, Trans. Amer. Math. Soc. 133 (1968), 385–414.CrossRefGoogle Scholar
4.Putnam, C. R., On normal operators in Hilbert space, Amer. J. Math. 73 (1951), 357–362.CrossRefGoogle Scholar