Induced norms of the Schur multiplier operator
Consider on \(M_ n\), the space of complex \(n\times n\)-matrices, the spectral norm \(\| A\|_{\infty}=\sup_{x\neq 0}\| Ax\| /\| x\|\) and the numerical-radius-norm \(w(A)=\sup_{x\neq 0}| <Ax,x>| /\| x\|^ 2\), where \(<, >\) denotes the standard inner product and \(\| x\| =<x,x>^{1/2}\) the Euclidean norm. For fixed \(A\in M_ n\) define the Schur multiplier operator \(S_ A: M_ n\to M_ n\) by \(S_ A: X\mapsto A\circ X\), the Schur (or Hadamard) product of A and X. Denoting with \(\| S_ A\|_{\infty}\), resp. \(\| S_ A\|_ w\) the induced norm of \(S_ A\) with respect to \(\| \cdot \|_{\infty}\), resp. w(\(\cdot)\), Haagerup [cf. \textit{V. I. Paulsen}, Completely bounded maps and dilations, Pitman Res. Notes Math. Ser., 146 (1986; Zbl 0614.47006)] obtained necessary and sufficient conditions in order that \(\| S_ A\|_{\infty}\leq 1.\) Here the authors prove the following theorem: Let \(A=(a_{ij})\in M_ n\). Equivalent are: (i) \(\| S_ A\|_ w\leq 1\); (ii) \(A=B^*WB\) with \(B^*B\circ I\leq I\), \(W^*W\leq I\); (iii) \(a_{ij}=<Wx_ j,x_ i>\) for \(x_ 1,...,x_ n\in {\mathbb{C}}^ n\), \(W\in M_ n\) where \(\| x_ i\| \leq 1\), \(W^*W\leq I\); (iv) \(\exists 0\leq R\in M_ n:\left( \begin{matrix} R\\ A^*\end{matrix} \begin{matrix} A\\ R\end{matrix} \right)\geq 0\), \(R\circ I\leq I\). This result is the w-counterparty of the Haagerup theorem. The proof, which is well-presented, involves an extension theorem for unital maps and an extension theorem for completely positive maps from \(C^*\)-algebras into \({\mathcal B}({\mathcal H})\). Next the result is used to obtain a new proof of the Haagerup theorem, and some consequences are listed, e.g. that \(\| S_ A\|_{\infty}=\| S_ A\|_ w\) for Hermitian A and that \(\| S_ A\|_{\infty}=\| S_ A\|_ w=1\) if A is unitary.
scientific article; zbMATH DE number 4189933