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<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis left-parenthesis ModifyingAbove phi With caret Subscript i comma j Baseline right-parenthesis right-parenthesis equals left-parenthesis left-parenthesis ModifyingAbove phi With caret Subscript i minus j Baseline right-parenthesis right-parenthesis comma 0 less-than-or-equal-to i comma j less-than-or-equal-to n comma">((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 onto<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="kernel upper S Superscript asterisk Baseline Superscript n plus 1 Baseline equals logical-or StartSet 1 comma e Superscript i theta Baseline comma e Superscript 2 i theta Baseline comma ellipsis comma e Superscript i n theta Baseline EndSet comma">ker⁡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, gand<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction 1 Over f EndFraction comma StartFraction 1 Over g EndFraction element-of upper L i p Subscript beta Baseline intersection upper H Superscript normal infinity Baseline left-parenthesis double-struck upper T right-parenthesis comma one half greater-than beta greater-than 1">1f,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, put<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E left-parenthesis psi right-parenthesis equals exp sigma-summation Underscript k equals 1 Overscript normal infinity Endscripts k left-parenthesis log f right-parenthesis Subscript k Baseline left-parenthesis log g overbar right-parenthesis Subscript negative k">E(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.<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D Subscript n Baseline left-parenthesis upper T Subscript phi Baseline right-parenthesis equals left-parenthesis negative 1 right-parenthesis Superscript left-parenthesis n plus 1 right-parenthesis gamma Baseline upper G left-parenthesis psi right-parenthesis Superscript n plus 1 Baseline upper E left-parenthesis psi right-parenthesis upper G left-parenthesis StartFraction g overbar Over f EndFraction right-parenthesis Superscript gamma Baseline dot det left-parenthesis left-parenthesis upper T Subscript StartFraction f Over g overbar EndFraction z Sub Superscript n plus 1 Subscript Baseline dot left-bracket 1 minus upper H Subscript StartFraction g overbar Over f EndFraction Baseline upper Q Subscript n minus gamma Baseline upper H Subscript left-parenthesis StartFraction f Over g overbar EndFraction right-parenthesis Sub Superscript ModifyingAbove With tilde Subscript Baseline right-bracket Superscript negative 1 Baseline z Superscript alpha minus 1 Baseline comma z Superscript tau minus 1 Baseline right-parenthesis right-parenthesis Subscript gamma times gamma Baseline dot left-bracket 1 plus upper O left-parenthesis n Superscript 1 minus 2 beta Baseline right-parenthesis right-bracket period">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,<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper X equals intersection Underscript n equals 1 Overscript normal infinity Endscripts r a n left-parenthesis upper T Subscript phi Baseline upper T Subscript phi Sub Superscript negative 1 Subscript Baseline right-parenthesis Superscript n">X=⋂n=1∞ran⁡(TϕTϕ−1)n{\mathcal X}= \bigcap _{n=1}^\infty \operatorname {ran} (T_\phi T_{\phi ^{-1}})^nand<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper Y equals union Underscript n equals 1 Overscript normal infinity Endscripts kernel left-parenthesis upper T Subscript phi Baseline upper T Subscript phi Sub Superscript negative 1 Subscript Baseline right-parenthesis Superscript n">Y=⋃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.,<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper T Subscript phi Baseline upper T Subscript phi Sub Superscript negative 1 Subscript Baseline right-parenthesis Superscript m Super Subscript alpha Baseline u Subscript alpha Baseline equals left-parenthesis negative 1 right-parenthesis Superscript m Super Subscript alpha Baseline w Subscript alpha">(TϕTϕ−1)mαuα=(−1)mαwα(T_\phi T_{\phi ^{-1}})^{m_\alpha }u_\alpha =(-1)^{m_\alpha }w_\alphawhere<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m Subscript alpha Baseline equals max left-brace m element-of upper Z Subscript plus Baseline colon there-exists x so that left-parenthesis upper T Subscript phi Baseline upper T Subscript phi Sub Superscript negative 1 Subscript Baseline right-parenthesis Superscript m Baseline x equals w Subscript alpha Baseline right-brace Superscript negative 1">mα=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 =<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction product Underscript lamda element-of sigma left-parenthesis upper T Subscript phi Baseline upper T Subscript phi Sub Superscript negative 1 Subscript Baseline right-parenthesis minus StartSet 0 EndSet Endscripts lamda Over det left-parenthesis u Subscript alpha Baseline comma upper T Subscript StartFraction 1 Over g EndFraction Baseline z Superscript tau minus 1 Baseline right-parenthesis EndFraction">∏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<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket 1 minus upper H Subscript StartFraction g overbar Over f EndFraction Baseline upper Q Subscript n minus gamma Baseline upper H Subscript left-parenthesis StartFraction f Over g overbar EndFraction right-parenthesis Sub Superscript ModifyingAbove With tilde Subscript Baseline right-bracket Superscript negative 1">[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 of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction upper D Subscript n Baseline left-parenthesis upper T Subscript phi Baseline right-parenthesis Over left-parenthesis negative 1 right-parenthesis Superscript left-parenthesis n plus 1 right-parenthesis gamma Baseline upper G left-parenthesis psi right-parenthesis Superscript n plus 1 Baseline det left-parenthesis upper T Subscript StartFraction f Over g overbar EndFraction z Sub Superscript n plus 1 Subscript Baseline z Superscript alpha minus 1 Baseline comma z Superscript tau minus 1 Baseline right-parenthesis EndFraction">Dn(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.