Steinberg symbols modulo the trace class, holonomy, and limit theorems for Toeplitz determinants期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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

Authors:	Richard W. Carey Joel D. Pincus

Affiliation:	Department of Mathematics, University of Kentucky, Lexington, Kentucky 40511 ; 806 Hunt Lane, Manhasset, New York 11030

Abstract:	Suppose that where and $psi in text{rm Lip}_beta,,{1over 2}<beta<1$ , and the Toeplitz operator is invertible. Let be the determinant of the Toeplitz matrix $((hatphi _{i,j}))=((hatphi _{i-j})),quad 0leq i,jleq n ,$ where $hat phi_k={1over 2pi}int_0^{2pi} phi(theta)e^{-iktheta}, dtheta$ . Let be the orthogonal projection onto $ker {S^}^{n+1}=bigvee{1,e^{itheta}, e^{2itheta},ldots, e^{intheta}},$ where ; set , let denote the Hankel operator associated to , and set for . For the Wiener-Hopf factorization where and ${1over f },{1over g}in text{rm Lip}_betacap H^infty(mathbb{T} ), {1over 2}<beta<1$ , put $E(psi)=expsum_{k=1}^infty k(log f)_k(log bar g)_{-k}$ , Theorem A.* $D_n(T_phi)=(-1)^{(n+1)gamma} G(psi)^{n+1}E(psi) G({bar gover f})^gamma$ $cdot detbigg((T_{{fover bar g}z^{n+1}}cdot [1-H_{bar gover f} Q_{n-gam... ...^{alpha-1},z^{tau-1})bigg)_{gamma times gamma} cdot [1+O(n^{1-2beta})].$ Let $H^2(mathbb{T} )= {mathcal X}dotplus {mathcal Y}$ be a decomposition into $T_phi T_{phi^{-1}}$ invariant subspaces, ${mathcal X}= bigcap_{n=1}^inftyoperatorname{ran} (T_phi T_{phi^{-1}})^n$ and ${mathcal Y}=bigcup _{n=1}^inftyker (T_phi T_{phi^{-1}})^n$ , so that $T_phi T_{phi^{-1}}$ restricted to is invertible, is finite dimensional, and $T_phi T_{phi^{-1}}$ restricted to 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 be the top vector in a Jordan root vector chain of length lying over $(-1)^{m_alpha}w_alpha$ , i.e., $(T_phi T_{phi^{-1}})^{m_alpha}u_alpha =(-1)^{m_alpha}w_alpha$ where $m_alpha=max{min Z_+:exists x,text{rm so that} (T_phi T_{phi^{-1}})^mx=w_alpha}^{-1}$ . Theorem B. $E( psi) G({bar gover f})^gamma=$ ${prod_{lambdainsigma(T_{phi} T_{phi^{-1}})setminus {0}},lambda}over det( u_alpha,T_{1over g}z^{tau-1})$ $=left (bar gcup ftimes {bar gover f}cup z^gammaright )(mathbb{T} )$ , the holonomy of a Deligne bundle with connection defined by the factorization . Note that the generalizations of the Szegö limit theorem for which have appeared in the literature with instead of $[1-H_{bar gover f} Q_{n-gamma} H_{({fover bar g})^{tilde{}}}]^{-1}$ have the defect that the limit of ${D_n(T_phi)over (-1)^{(n+1)gamma} G(psi)^{n+1} det(T_{{fover bar g}z^{n+1}}z^{alpha-1},z^{tau-1})}$ does not exist in general. An example is given with yet $D_{gamma-1}(T_{{fover bar g}z^{n+1}})=0$ for infinitely many .

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