Perturbation vectors

Let F denote the Fredholm operators on the Hilbert space H. Let Q --> F be the Quillen det bundle. Form Q + Q* as a bundle over F x F. Let = det(Q + Q*) be the determinant bundle. Let M = {(S, T) : (S,T) epsilon F x F and S - T epsilon L-1(H)} with i(M) : M --> F x F the inclusion map. We prove the existence of a trivialization of the pullback i(M)(*)(Q) by construction of a section, sigma, called the perturbation section. The thereby constructed "perturbation vectors" sigma(S,T) generalize the perturbation determinant det(S-1T) when the operators S and T are singular. Then, for pairs of elements (A, D) and (B, C) in M with AB = CD there is a Koszul complex K(A, B; C, D) with two torsion vectors built from its homology whose tensor product is a scalar multiple of a(A,D) x sigma(B,C). This gives a number tau(A, B, C, D; H) epsilon C*, called the joint torsion. It has previously been shown that if A and B are commuting operators in F then the Steinberg symbol determinant det(*) o partial derivative{A + L-1(H), B + L-1(H)} = tau(A, B, B, A; H). Corresponding results are now proved in the non-commuting case, and the section sigma and joint torsion are used to examine the n x n subdeterminants of Toeplitz matrices with non-zero index as n --> infinity. With P-n denoting the orthogonal projection onto the null space of the shift, ker S*(n+l), the strong Szego limit theorem for the large n behavior of D-n(phi) = PnTphiPn when phi > 0 can be recast as a result about the perturbation section (T. The Szego theorem becomes: \\ sigma(1),Txn+1hT(x) over bar n+1h-1 \\(2) D-n(phi)/ exp n+1/2 pi integral(0)(2 pi) log \phi(e(i0))\d theta --> \\ sigma(1),T\h\ -2 \\ where phi = h (h) over bar is the Fejer-Riesz factorization. Study of the map sigma A,B (L)--> sigma(AL,BL) then links algebraic K theory to the trigonometry of the Hilbert space, and the case, [arg phi(e(i theta))\(2 pi)(0) not equal 0. Thus if psi epsilon K-2,(1/2,1/2)(2), Krein algebra, phi(i) is a finite Blaschke product, and phi = phi(i) . psi where T-psi is invertible, and psi factors as the product of an analytic and anti-analytic function psi = psi(+)psi(-) we prove: Theorem \\sigma(1),TphiTphi-1 \\ = lim(n-->infinity) \D-n(phi)\ / exp n+1/2 pi integral(0)(2 pi) log\phi(e(i theta))\d theta . 1 / det(EFnE)(1/2) II\x\<1 \c(x)(psi(-),phi(i)) / c(x)(psi(+),phi(i))\.det(PTphi i)T\(psi+)\(2) P(T phi(i)))(1/2.). det(P(T-phi i)T\psi-\-2P(T-phi i))(1/2) .\det(*) o partial derivative{Tpsi- + L-1(H), Tpsi+ L-1(H)}\, where c(x)(f, h) = (-1)(orderx h.orderx f.) (f(orderx) (h)/h(orderx) (f)) denotes the tame symbol at x, and P(T-f) denotes orthogonal projection to the finite dimensional ker T-f. The new denominator det(EFnE)(1/2) corresponds to an explicitly determined zero limit sequence of products of cosines of angles between subspaces of the Hilbert space determined by the projections E and F-n. The Steinberg symbol determinant in this relation, det(*) o partial derivative(T psi_ + -L-1(H), Tpsi+ L-1(H)), is equal to det(TpsiTpsi-1) and is evaluated as an integral.