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Quadratic forms in unitary operators. (English) Zbl 0889.47007

Let \(u_1, \dots, u_n\) be unitary operators on a Hilbert space \(H\). Then the operator \(\sum u_i\otimes\bar u_i\) acts on the Hilbertian tensor product \(H\otimes_2\bar H\), where \(\bar H\) is the complex conjugate of \(H\). Let \(B(H)\) denote the space of bounded operators on \(H\) equipped with the usual norm.
The author shows: For any \(n\)-tuple \(u_1, \dots ,u_n\) of unitary operators on \(B(H)\) we have the sharp inequality \(|\sum_{k=1}^nu_k\otimes\bar u_k|\geq 2\sqrt {n-1}\). As an application he proves an inequality for the quasiregular representation of \(U(N)\).
Reviewer: E.Ellers (Toronto)

MSC:

47A30 Norms (inequalities, more than one norm, etc.) of linear operators
47A63 Linear operator inequalities
15A63 Quadratic and bilinear forms, inner products
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References:

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