Computation of functions of certain operator matrices (Q1318213)

The following result is obtained: Theorem 1. Let \(B_ 1\), \(C_ 1\in {\mathcal L}(H,K_ 1)\) and \(B_ 2\), \(C_ 2\in {\mathcal L}(H,K_ 2)\). Suppose \(f\) is a function which is holomorphic in an open set \(D\) containing the spectrum \((C^*_ 1 B_ 1+ C^*_ 2 B_ 2)\) and 0. Then \[ f\left(\left[{B_ 1 C^*_ 1\atop B_ 2 C^*_ 1} {B_ 1 C^*_ 2\atop B_ 2 C^*_ 2}\right]\right)= \left[ {f(0)I+ B_ 1 RC^*_ 1\atop B_ 2 RC^*_ 1} {B_ 1 R^*_ 2\atop f(0)I+ B_ 2 RC^*_ 2}\right], \] where \(R= g(C^*_ 1 B_ 1+ C^*_ 2 B_ 2)\) and \(g\) is the holomorphic extension to \(D\) of \(g(z)= [f(z)- f(0)]/z\). Here \(H\), \(K_ 1\), \(K_ 2\) are Hilbert spaces and \({\mathcal L}(H,K)\) denotes the space of bounded linear operators from \(H\) to \(K\). Some illustrations are given.