Steinberg symbols modulo the trace class, holonomy, and limit theorems for Toeplitz determinants
Authors: Carey Richard , Pincus Joel 
Published: October 2005

Suppose thatϕ=ψzγ\phi =\psi z^\gammawhereγ∈Z+\gamma \in Z_+andψ∈Lipβ,12>β>1\psi \in \operatorname {Lip}_\beta ,\,{1\over 2}>\beta >1, and the Toeplitz operatorTψT_\psiis invertible. LetDn(Tϕ)D_n(T_\phi )be the determinant of the Toeplitz matrix((ϕ^i,j))=((ϕ^i−j)),0≤i,j≤n,((\hat \phi _{i,j}))=((\hat \phi _{i-j})),\quad 0\leq i,j\leq n ,whereϕ^k=12π∫02πϕ(θ)e−ikθdθ\hat \phi _k={1\over 2\pi }\int _0^{2\pi } \phi (\theta )e^{-ik\theta }\, d\theta. LetPnP_nbe the orthogonal projection ontoker⁡S∗n+1=⋁{1,eiθ,e2iθ,…,einθ},\ker {S^*}^{n+1}=\bigvee \{1,e^{i\theta }, e^{2i\theta },\ldots , e^{in\theta }\},whereS=TzS=T_z; setQn=1−PnQ_n=1-P_n, letHωH_\omegadenote the Hankel operator associated toω\omega, and setω~(t)=ω(1t)\tilde \omega (t)=\omega ({1\over t})fort∈Tt\in \mathbb {T}. For the Wiener-Hopf factorizationψ=fg¯\psi =f\bar gwheref,gf, gand1f,1g∈Lipβ∩H∞(T),12>β>1{1\over f },{1\over g}\in \operatorname {Lip}_\beta \cap H^\infty (\mathbb {T}), {1\over 2}>\beta >1, putE(ψ)=exp⁡∑k=1∞k(log⁡f)k(log⁡g¯)−kE(\psi )=\exp \sum _{k=1}^\infty k(\log f)_k(\log \bar g)_{-k},G(ψ)=exp⁡(log⁡ψ)0.G(\psi )=\exp (\log \psi )_0.

Theorem A.Dn(Tϕ)=(−1)(n+1)γG(ψ)n+1E(ψ)G(g¯f)γ⋅det((Tfg¯zn+1⋅[1−Hg¯fQn−γH(fg¯)~]−1zα−1,zτ−1))γ×γ⋅[1+O(n1−2β)].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 })].

LetH2(T)=X∔YH^2(\mathbb {T})= {\mathcal X}\dotplus {\mathcal Y}be a decomposition intoTϕTϕ−1T_\phi T_{\phi ^{-1}}invariant subspaces,X=⋂n=1∞ran⁡(TϕTϕ−1)n{\mathcal X}= \bigcap _{n=1}^\infty \operatorname {ran} (T_\phi T_{\phi ^{-1}})^nandY=⋃n=1∞ker⁡(TϕTϕ−1)n{\mathcal Y}=\bigcup _{n=1}^\infty \ker (T_\phi T_{\phi ^{-1}})^n, so thatTϕTϕ−1T_\phi T_{\phi ^{-1}}restricted toX{\mathcal X}is invertible,Y{\mathcal Y}is finite dimensional, andTϕTϕ−1T_\phi T_{\phi ^{-1}}restricted toY{\mathcal Y}is nilpotent. Let{wα}1γ\{w_\alpha \}_1^\gammabe the basis{Tfzα}0γ−1\{T_f z^\alpha \}_0^{\gamma -1}for the null space ofTϕTϕ−1T_\phi T_{\phi ^{-1}}, and letuαu_\alphabe the top vector in a Jordan root vector chain of lengthmα+1m_\alpha +1lying over(−1)mαwα(-1)^{m_\alpha }w_\alpha, i.e.,(TϕTϕ−1)mαuα=(−1)mαwα(T_\phi T_{\phi ^{-1}})^{m_\alpha }u_\alpha =(-1)^{m_\alpha }w_\alphawheremα=max{m∈Z+:∃xso that(TϕTϕ−1)mx=wα}−1m_\alpha =\max \{m\in Z_+:\exists x\,\text {so that} (T_\phi T_{\phi ^{-1}})^mx=w_\alpha \}^{-1}.

Theorem B.E(ψ)G(g¯f)γ=E( \psi ) G({\bar g\over f})^\gamma =∏λ∈σ(TϕTϕ−1)∖{0}λdet(uα,T1gzτ−1){\prod _{\lambda \in \sigma (T_{\phi } T_{\phi ^{-1}})\setminus \{0\}}\,\lambda }\over \det ( u_\alpha ,T_{1\over g}z^{\tau -1})=(g¯∪f×g¯f∪zγ)(T)=\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 factorizationϕ=fg¯zγ\phi = f\bar gz^\gamma.

Note that the generalizations of the Szegö limit theorem forDn(Tϕ)D_n(T_\phi )which have appeared in the literature with11instead of[1−Hg¯fQn−γH(fg¯)~]−1[1-H_{\bar g\over f} Q_{n-\gamma } H_{({f\over \bar g})^{\tilde {}}}]^{-1}have the defect that the limit ofDn(Tϕ)(−1)(n+1)γG(ψ)n+1det(Tfg¯zn+1zα−1,zτ−1){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 withDn(Tϕ)≠0D_n(T_\phi )\neq 0yetDγ−1(Tfg¯zn+1)=0D_{\gamma -1}(T_{{f\over \bar g}z^{n+1}})=0for infinitely manynn.