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Unitary invariance and spectral variation. (English) Zbl 0657.15019
An operator norm is called weakly unitary invariant (wui) if it is invariant under similarity transformation with a unitary matrix. The numerical radius is an example of such norm. In the first part of the paper, the authors present other examples of wui norms, the ways how to generate them, and some representation theorems. They also prove the so called pinching inequality for wui norms, which was proved earlier by I. C. Gohberg and M. G. Krein [Introduction to the theory of linear nonselfadjoint operators (1969; Zbl 0181.135; Russian original 1965; Zbl 0138.078)] for more special norms.
In the second part, an inequality has been proved that bounds the spectral variation when a normal operator A is replaced by another normal B in terms of the arclength of any normal path from A to B, computed using wui norm. This result fills in the lacuna in the proof of related result of the first author [Trans. Am. Math. Soc. 272, 323-331 (1982; Zbl 0488.15010)]. Other results are obtained by application of those of P. R. Halmos [A Hilbert space problem book, 2nd. ed. (1982; Zbl 0496.47001)] and R. Bouldin [Proc. Am. Math. Soc. 80, 277-282 (1980; Zbl 0459.47016)]. The paper is completed by some results on the behaviour of the spectral variation and normal pahts in the neighbourhood of a given normal operator.
Reviewer: Z.Dostal

MSC:
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
15A09 Theory of matrix inversion and generalized inverses
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[1] Bhatia, R., Analysis of spectral variation and some inequalities, Trans. amer. math. soc., 272, 323-331, (1982) · Zbl 0488.15010
[2] R. Bhatia, C. Davis, and P. Koosis, An extremal problem in Fourier analysis with applications to operator theory, to appear. · Zbl 0674.42002
[3] Bhatia, R.; Davis, C.; Mcintosh, A., Perturbation of spectral subspaces and solution of linear operator equations, Linear algebra appl., 52-53, 45-67, (1983) · Zbl 0518.47013
[4] Bhatia, R.; Friedland, S., Variation of grassman powers and spectra, Linear algebra appl., 40, 1-18, (1981) · Zbl 0469.15004
[5] Bhatia, R.; Holbrook, J.A.R., Short normal paths and spectral variation, (), 377-382 · Zbl 0568.15012
[6] Bouldin, R., Best approximation of a normal operator in the Schatten p-norm, (), 277-282 · Zbl 0459.47016
[7] Davis, C., Various averaging operations onto subalgebras, Illinois J. math., 3, 538-553, (1959) · Zbl 0087.11404
[8] Fong, C.-K.; Holbrook, J.A.R., Unitarily invariant operator norms, Canad. J. math., 35, 274-299, (1983) · Zbl 0477.47005
[9] C.-K. Fong, H. Radjavi, and P. Rosenthal, Norms for matrices and operators, to appear. · Zbl 0661.47010
[10] Gohberg, J.C.; Krein, M.G., Introduction to the theory of linear non-selfadjoint operators, (1969), Amer. Math. Soc · Zbl 0181.13504
[11] Halmos, P.R., ()
[12] Halmos, P.R., Spectral approximants of normal operators, (), 51-58 · Zbl 0274.47016
[13] Hewitt, E.; Ross, K.A., ()
[14] Lidskii, V.B., On the eigenvalues of sums and products of symmetric matrices (in Russian), Dokl. akad. nauk SSSR, 75, 769-772, (1950)
[15] Marshall, A.W.; Olkin, I., Inequalities, (1979), Academic · Zbl 0131.24903
[16] Schatten, R., Norm ideals of completely continuous operators, (1960), Springer · Zbl 0090.09402
[17] Von Neumann, J.; Von Neumann, J., Some matrix inequalities and metrization of matric space, (), 1, 205-214, (1937), also in · JFM 63.0037.03
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