Approximation of the quadratic numerical range of block operator matrices (Q1946544)

Let \({\mathcal H}={\mathcal H}_1\oplus {\mathcal H}_2\) be a decomposition of a complex Hilbert space \({\mathcal H}\) into the orthogonal sum of two subspaces and let \({\mathcal A}=[A_{ij}]_{i,j=1}^{2}\) be the block operator matrix representation of a (not necessarily bounded) linear operator \({\mathcal A}\) on \({\mathcal H}\). The quadratic numerical range \(W^2({\mathcal A})\) is the set of all eigenvalues of \(2\times 2\) matrices \({\mathcal A}_{x_1,x_2}=[ \langle A_{ij}x_j,x_i\rangle/\| x_i\| \|x_j\|]_{i,j=1}^{2}\), where \(x_1\in {\mathcal D}(A_{11})\cap {\mathcal D}(A_{21})\) and \(x_2\in {\mathcal D}(A_{12})\cap {\mathcal D}(A_{22})\) are non-zero vectors and \({\mathcal D}(\cdot)\) denotes the domain of an operator. In the paper, the computation of \(W^2({\mathcal A})\) by projection methods is considered. These methods yield a subset of the quadratic numerical range under very weak hypotheses. Only when one wants to generate the whole set \(W^2({\mathcal A})\), some extra assumptions are necessary. Some analytical results about the large-\(\lambda\) shape of the quadratic numerical range for certain classes of block operator matrices are proved as well and applied to a Hain-Lüst problem. The paper illustrates the practical difficulties associated with the computation of quadratic numerical ranges.