Eigenspaces of families of Toeplitz operators with rational matrix symbols (Q2266428)

Let us consider a family \(T(z)\), \(z\in {\mathbb{C}}\), of Toeplitz operators acting on the space of all square summable \({\mathbb{C}}^ n\)-valued sequences and assume that the multiplier of the operator \(T(z)\) is given by an \(n\times n\)-matrix function \(C(\cdot,z)\) having the form \[ C(\lambda,z)=\sum^{m}_{j=-m}\lambda^ jC_ j(z),\quad |\lambda| =1, \] where \(C_ j\) \((j=-m,...,m)\) are matrix polynomials. Furthermore, assume that for some \(z_ 0\in {\mathbb{C}}\), the function \(C(\cdot,z_ 0)\) is invertible. Let \(\phi (T)=\{z\in {\mathbb{C}}:\) \(\det C(\lambda,z)\neq 0\) for every \(\lambda\) from the unit circle\}. It is known that the set \({\mathbb{C}}\setminus \phi(T)\) consists of a finite number of bounded smooth arcs, and thus \(\phi(T)\) is the union \(\Delta_ 1\cup...\cup \Delta_ p\) of a finite number of open connected components. Moreover, the function \({\mathbb{C}}\ni z\mapsto \alpha(z)= \dim(Ker T(z))\) takes some constant value \(\mu_ j\) for all but finitely many points in each set \(\Delta_ j\). Put \(\mu(z)=\min \{\mu_ j: j\in P(z)\},\) \(z\in {\mathbb{C}}\), where \(P(z)=\{j\in \{1,...,p\}: z\in Cl \Delta_ j\}.\) The main theorem of the paper says that the inequality \(\alpha(z)\leq \mu(z)\) holds for all but finitely many points of the complex plane. This theorem has an analogue for Toeplitz operators with multipliers C(\(\cdot,z)\) of the form \(C(\lambda,z)=R(\lambda)-zId,\) where R is a rational matrix function with the poles off the unit circle. The following corollary is a simple consequence of this result. The eigenvalues of a selfadjoint Toeplitz operator, with rational matrix multiplier whose entries have no poles on the unit circle, form a finite set in the complex plane.