Unitary invariance and spectral variation

Bhatia, Rajendra ; Holbrook, John A. R. (1987) Unitary invariance and spectral variation Linear Algebra and its Applications, 95 . pp. 43-68. ISSN 0024-3795

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Official URL: http://linkinghub.elsevier.com/retrieve/pii/002437...

Related URL: http://dx.doi.org/10.1016/0024-3795(87)90026-7


We call a norm on operators or matrices weakly unitarily invariant if its value at operator A is not changed by replacing A by U AU, provided only that U is unitary. This class includes such norms as the numerical radius. We extend to all such norms an inequality 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 the norm in question. Related results treat the local metric geometry of the "manifold" of normal operators. We introduce a representation for weakly unitarily invariant matrix norms in terms of function norms over the unit ball, and identify this correspondence explicitly in certain cases.

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