Unitary invariance and spectral variation.

*(English)*Zbl 0657.15019An 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.

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 |

##### Keywords:

weakly unitary invariant norm; matrix norm; numerical radius; pinching inequality; spectral variation; normal operator
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\textit{R. Bhatia} and \textit{J. A. R. Holbrook}, Linear Algebra Appl. 95, 43--68 (1987; Zbl 0657.15019)

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##### References:

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