Steinberg symbols modulo the trace class, holonomy, and limit theorems for Toeplitz determinants

Suppose that $\phi=\psi z^\gamma$ where $\gamma\in Z_+$ and $\psi \in \text{\rm Lip}_\beta,\,{1\over 2}<\beta<1$, and the Toeplitz operator $T_\psi$ is invertible. Let $D_n(T_\phi)$ be the determinant of the Toeplitz matrix $((\hat\phi _{i,j}))=((\hat\phi _{i-j})),\quad 0\leq i,j\leq n ,$ where $\hat \phi_k={1\over 2\pi}\int_0^{2\pi} \phi(\theta)e^{-ik\theta}\, d\theta$. Let $P_n$ be the orthogonal projection onto $\ker {S^*}^{n+1}=\bigvee\{1,e^{i\theta}, e^{2i\theta},\ldots, e^{in\theta}\},$where $S=T_z$; set $Q_n=1-P_n$, let $H_\omega$ denote the Hankel operator associated to $\omega$, and set $\tilde\omega(t)=\omega({1\over t})$ for $t\in \mathbb{T}$. For the Wiener-Hopf factorization $\psi=f\bar g$ where $f, g$ and ${1\over f },{1\over g}\in \text{\rm Lip}_\beta\cap H^\infty(\mathbb{T} ), {1\over 2}<\beta<1$, put $E(\psi)=\exp\sum_{k=1}^\infty k(\log f)_k(\log \bar g)_{-k}$, $G(\psi)=\exp(\log\psi)_0.$

Theorem A.     $D_n(T_\phi)=(-1)^{(n+1)\gamma} G(\psi)^{n+1}E(\psi) G({\bar g\over f})^\gamma$

$\cdot \det\bigg((T_{{f\over \bar g}z^{n+1}}\cdot [1-H_{\bar g\over f} Q_{n-\gam... ...^{\alpha-1},z^{\tau-1})\bigg)_{\gamma \times \gamma} \cdot [1+O(n^{1-2\beta})].$

Let $H^2(\mathbb{T} )= {\mathcal X}\dotplus {\mathcal Y}$ be a decomposition into $T_\phi T_{\phi^{-1}}$invariant subspaces, ${\mathcal X}= \bigcap_{n=1}^\infty\operatorname{ran} (T_\phi T_{\phi^{-1}})^n$and ${\mathcal Y}=\bigcup _{n=1}^\infty\ker (T_\phi T_{\phi^{-1}})^n$, so that $T_\phi T_{\phi^{-1}}$ restricted to ${\mathcal X}$ is invertible, ${\mathcal Y}$ is finite dimensional, and $T_\phi T_{\phi^{-1}}$ restricted to ${\mathcal Y}$ is nilpotent. Let $\{w_\alpha\}_1^\gamma$ be the basis $\{T_f z^\alpha\}_0^{\gamma-1}$ for the null space of $T_\phi T_{\phi^{-1}}$, and let $u_\alpha$ be the top vector in a Jordan root vector chain of length $m_\alpha+1$ lying over $(-1)^{m_\alpha}w_\alpha$, i.e., $(T_\phi T_{\phi^{-1}})^{m_\alpha}u_\alpha =(-1)^{m_\alpha}w_\alpha$where $m_\alpha=\max\{m\in Z_+:\exists x\,\text{\rm so that} (T_\phi T_{\phi^{-1}})^mx=w_\alpha\}^{-1}$.

Theorem B.     $E( \psi) G({\bar g\over f})^\gamma=$ ${\prod_{\lambda\in\sigma(T_{\phi} T_{\phi^{-1}})\setminus \{0\}}\,\lambda}\over \det( u_\alpha,T_{1\over g}z^{\tau-1})$ $=\left (\bar g\cup f\times {\bar g\over f}\cup z^\gamma\right )(\mathbb{T} )$, the holonomy of a Deligne bundle with connection defined by the factorization $\phi= f\bar gz^\gamma$.

Note that the generalizations of the Szegö limit theorem for $D_n(T_\phi)$which have appeared in the literature with $1$ instead of $[1-H_{\bar g\over f} Q_{n-\gamma} H_{({f\over \bar g})^{\tilde{}}}]^{-1}$ have the defect that the limit of ${D_n(T_\phi)\over (-1)^{(n+1)\gamma} G(\psi)^{n+1} \det(T_{{f\over \bar g}z^{n+1}}z^{\alpha-1},z^{\tau-1})}$ does not exist in general. An example is given with $D_n(T_\phi)\neq 0$yet $D_{\gamma-1}(T_{{f\over \bar g}z^{n+1}})=0$ for infinitely many $n$.