Operator matrices in class \(C_{\rho }\)
A bounded linear operator \(T\) on the Hilbert space \(H\) is said to be a \(\rho\)-contraction \((\rho>0)\) if there exists a unitary operator \(U\) on a space \(K\) containing \(H\) such that \(T^n=\rho P_HU^n|H\) for all \(n\geq 1\), where \(P_H\) denotes the orthogonal projection from \(K\) onto \(H\). The class \({\mathcal C}_p\) of \(\rho\)-contractions was first introduced by Sz.-Nagy and Foias in the 1960s. The \(\rho\)-numerical radius of \(T\) is, by definition, \(w_\rho(T)=\inf\{r>0:T/r\in{\mathcal C}_\rho\}\). It is known that \(w_1(T)=\|T\|\), the operator norm of \(T\), and \(w(T)=w_2 (T)(=\sup\{\langle Tx,x\rangle:x\in H,\|x\|=1\})\), the numerical radius of \(T\). The main result of this paper, Theorem 1, gives necessary and sufficient conditions for \(T=\left[\begin{smallmatrix} T_1 & R\\ 0 & T_2 \end{smallmatrix} \right]\) to be in \({\mathcal C}_p\). For example, it is shown that this is the case if and only if \(T_1\) and \(T_2\) are in \({\mathcal C}_\rho\) and \(|\alpha|^2R^* L^\rho_\alpha(T_1)^{-1}R\leq L^\rho_{\overline \alpha}(T^*_2)\) for all \(\alpha\), \(|\alpha|\leq 1\), where \[ L^\rho_\alpha(A)=\rho I+(1-\rho)(\alpha A^*+\overline \alpha A)+ (\rho-2)|\alpha|^2AA^*. \] The latter is in turn equivalent to ``\(T_1\) and \(T_2\) are in \({\mathcal C}_\rho\) and for every \(\alpha\), \(|\alpha|\leq 1\), there exists an operator \(C_\alpha\) with \(\|C\|\leq 1\) such that \(|\alpha|R= L^\rho_\alpha(T_1)^{1/2}C_\alpha L^\rho_{\overline \alpha}(T^*_2)^{1/2}\)''. Such results recover the classical characterization of \(T=\left[\begin{smallmatrix} T_1 & R\\ 0 &T_2 \end{smallmatrix}\right]\) being a contraction. Among the various corollaries is the assertion that \(w\left(\left[ \begin{smallmatrix} T_1 & R\\ 0 &T_2 \end{smallmatrix}\right]\right)\leq 1\) if and only if \(w\left(\left[ \begin{smallmatrix} \text{Re}\,T_1 & R\\ 0 & \text{Re}\,T_2\end{smallmatrix} \right]\right)\leq 1\) and \(w\left(\left[\begin{smallmatrix} \text{Im}\,T_1 & R\\ 0 &\text{Im}\,T_2\end{smallmatrix} \right]\right)\leq 1\). In the final section, the authors illustrate these results by specializing them to the two-by-two matrix \(T=\left[\begin{smallmatrix} a & c\\ 0 & b \end{smallmatrix}\right]\).
scientific article; zbMATH DE number 5083607