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

Abstract:

Suppose that $\phi =\psi z^\gamma$ where $\gamma \in Z_+$ and $\psi \in \operatorname {Lip}_\beta ,\,{1\over 2}<\beta <1$, 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 ${1\over f },{1\over g}\in \operatorname {Lip}_\beta \cap H^\infty (\mathbb {T}), {1\over 2}<\beta <1$, put $E(\psi )=\exp \sum _{k=1}^\infty k(\log f)_k(\log \bar g)_{-k}$, $G(\psi )=\exp (\log \psi )_0.$

Theorem A. $D_n(T_\phi )=(-1)^{(n+1)\gamma } G(\psi )^{n+1}E(\psi ) G({\bar g\over f})^\gamma \cdot \det \bigg ((T_{{f\over \bar g}z^{n+1}}\cdot [1-H_{\bar g\over f} Q_{n-\gamma } H_{({\frac {f}{\bar g}})^{\tilde {}}} ]^{-1}z^{\alpha -1},z^{\tau -1})\bigg )_{\gamma \times \gamma } \cdot [1+O(n^{1-2\beta })].$

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 $m_\alpha =\max \{m\in Z_+:\exists x\,\text {so that} (T_\phi T_{\phi ^{-1}})^mx=w_\alpha \}^{-1}$.

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 $n$.