Structured invariant spaces of vector valued rational functions, Hermitian matrices, and a generalization of the Iohvidov laws
The theory of finite dimensional reproducing kernel spaces with an indefinite inner product is applied to study the relations between inertia of principal submatrices of Hermitian solutions to certain matrix equations. Let be given matrix equations \[ (1)\quad P-(Z_ w)^*PZ_{w^*}=V^*JV \] \[ (2)\quad PZ_{w^*}- (Z_{w^*})^*P=(2\pi i)^{-1}V^*JV \] where \(Z_ w\) is the \(n\times n\) upper triangular Jordan block with the eigenvalue \(w\in {\mathbb{C}}\), J is a given \(2\times 2\) matrix which is indefinite Hermitian and unitary, and the \(2\times n\) matrix V is given as well. Hermitian Toeplitz and Hankel matrices are, in particular, solutions of equations of these types. One of the main results (in a simplified form): Let \(P=P^*\) be a singular solution to (1), where \(| w| \neq 1\) and the first column in V is nonzero. Let \(P_{[j]}\) be the upper left \(j\times j\) block of P, and let r be the maximal index j such that \(P_{[j]}\) is invertible \((r=0\) if there is no such j). Denote by \(k\times k\) the size of the largest all zero block in the upper left hand corner of P [the Schur complement of \(P_{[r]}]\) if \(r=0\) \([r>0]\). Then \[ \mu_{\pm}(P_{[r+j]})=\mu_{\pm}(P_{[r]}),\quad i=1,...,k \] \[ \mu_{\pm}(P_{[r+j]})=\mu_{\pm}(P_{[r]})+j-k,\quad j=k+1,...,n-r. \] Here by \(\mu_+(A)\) (resp. \(\mu_ -(A))\) we denote the number of positive (resp. negative) eigenvalues of a Hermitian matrix A, counted with multiplicities. Analogous results are proven for \(| w| =1\) and for Hermitian solutions P of equation (2). The proofs are based on the identification of P as the Gram matrix of an appropriately chosen basis in some finite dimensional vector space of rational column functions. This vector space is invariant under generalized shifts, carries an indefinite (generally degenerate) inner product, and satisfies a suitable identity. There are stated (with proofs appearing elsewhere) some results in the same spirit concerning the relations between ranks of principal submatrices of solutions P to equations \[ P-(Z_{w^*})^*PZ_{w^*}=U^*JV \] \[ PZ_{w^*}- (Z_{w^*})^*P=(2\pi i)^{-1}U^*JV, \] where \(Z_ w\) and J are as above.
finite dimensional reproducing kernel spaces
scientific article; zbMATH DE number 4158862
scientific article