Numerical radius and distance from unitary operators

Details Numerical radius and distance from unitary operators Numerical radius and distance from unitary operators

2011-10-23

arxiv.org/abs/1110.5036v2

Mathematics

Functional Analysis

Final version : new title and several other changes

Scientific paper

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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