The Toeplitz-Hausdorff Theorem Explained

Chandler Davis

doi:10.4153/CMB-1971-042-7

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Paul Halmos expressed [3, p. 110] the general dissatisfaction with the usual proofs of this famous and important theorem. They all make it seem like an accidental product of a computation. A more conceptual proof was devised by N. P. Dekker. In spite of the elegance of his proof, the one offered below may have some claim to be regarded as "the reason the theorem is true".

References

1. Davis, Ch., The shell of a Hilbert-space operator—II. Acta Sci. Math, (to appear).Google Scholar

2. Dekker, N. P., Joint numerical range and joint spectrum of Hilbert space operators, Amsterdam thesis, 1969.Google Scholar

3. Halmos, P. R., A Hilbert space problem book, Van Nostrand, Princeton, N.J., 1967.Google Scholar

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Kahan, W. 1973. Every 𝑛×𝑛 matrix 𝑍 with real spectrum satisfies ‖𝑍-𝑍*‖≤‖𝑍+𝑍*‖(log₂𝑛+0.038). Proceedings of the American Mathematical Society, Vol. 39, Issue. 2, p. 235.

Fox, David W. 1978. The convexity of the range of three Hermitian forms and of the numerical range of sesquilinear forms. Linear Algebra and its Applications, Vol. 22, Issue. , p. 191.

McIntosh, Alan 1978. The Toeplitz-Hausdorff Theorem and Ellipticity Conditions. The American Mathematical Monthly, Vol. 85, Issue. 6, p. 475.

Goldberg, Moshe 1979. On certain finite dimensional numerical ranges and numerical radii†. Linear and Multilinear Algebra, Vol. 7, Issue. 4, p. 329.

Marcus, Marvin 1979. SOME COMBINATORIAL ASPECTS OF NUMERICAL RANGE*. Annals of the New York Academy of Sciences, Vol. 319, Issue. 1, p. 368.

Das, K.C. and Embry-Wardrop, Mary 1980. Orthogonality and the numerical range. II. Linear Algebra and its Applications, Vol. 30, Issue. , p. 63.

Cybenko, George 1984. Restrictions of Normal Operators, Padé Approximation and Autoregressive Time Series. SIAM Journal on Mathematical Analysis, Vol. 15, Issue. 4, p. 753.

Bratteli, Ola Digernes, Trond and Elliott, George A. 1985. Operator Algebras and their Connections with Topology and Ergodic Theory. Vol. 1132, Issue. , p. 46.

Marcus, Marvin and Pesce, Claire 1987. Computer generated numerical ranges and some resulting theorems. Linear and Multilinear Algebra, Vol. 20, Issue. 2, p. 121.

Vershik, A. M. 1987. Quadratic forms positive on a cone and quadratic duality. Journal of Soviet Mathematics, Vol. 36, Issue. 1, p. 39.

Bebiano, Natália and Da Providěncia, Joáo 1988. On the boundary of theC-numerical range of a normal matrix. Linear and Multilinear Algebra, Vol. 23, Issue. 2, p. 145.

Choi, Man-Duen and Rosenthal, Peter 1994. A survey of Chandler Davis. Linear Algebra and its Applications, Vol. 208-209, Issue. , p. 3.

Holbrook, John A. 1995. Interpenetration of ellipsoids and the polynomial bound of a matrix. Linear Algebra and its Applications, Vol. 229, Issue. , p. 151.

Barvinok, A. I. 1995. Problems of distance geometry and convex properties of quadratic maps. Discrete & Computational Geometry, Vol. 13, Issue. 2, p. 189.

Feintuch, Avraham and Markus, Alexander 1999. The toeplitz-hausdorff theorem and robust stability theory. The Mathematical Intelligencer, Vol. 21, Issue. 3, p. 33.

Bourdon, Paul S. and Shapiro, Joel H. 2000. The Numerical Ranges of Automorphic Composition Operators. Journal of Mathematical Analysis and Applications, Vol. 251, Issue. 2, p. 839.

Bourdon, Paul S. and Shapiro, Joel H. 2002. When is zero in the numerical range of a composition operator?. Integral Equations and Operator Theory, Vol. 44, Issue. 4, p. 410.

Uhlig, Frank 2008. An inverse field of values problem. Inverse Problems, Vol. 24, Issue. 5, p. 055019.

Liu, Xuhua Tadesse, Dawit G. and Tam, Tin-Yau 2011. A range associated with skew symmetric matrix. Linear and Multilinear Algebra, Vol. 59, Issue. 10, p. 1105.

Dunkl, Charles F. Gawron, Piotr Holbrook, John A. Puchała, Zbigniew and Życzkowski, Karol 2011. Numerical shadows: Measures and densities on the numerical range. Linear Algebra and its Applications, Vol. 434, Issue. 9, p. 2042.

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