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arXiv · 1110.5036

Numerical radius and distance from unitary operators

Catalin Badea·Michel Crouzeix
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Abstract

Denote by w(A) the numerical radius of a bounded linear operator A acting on Hilbert space. Suppose that A is invertible and that the numerical radius of A and of its inverse are no greater than 1+e for some non-negative e. It is shown that the distance of A from unitary operators is less or equal than a constant times $e^{1/4}$. This generalizes a result due to J.G. Stampfli, which is obtained for e = 0. An example is given showing that the exponent 1/4 is optimal. The more general case of the operator $\rho$-radius is discussed for $\rho$ between 1 and 2.

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Catalin Badea, Michel Crouzeix. 2011-10-23. Numerical radius and distance from unitary operators. https://arxiv.org/abs/1110.5036

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