Abstract
Let A=(aj,k)j,k=−∞∞ be a bounded linear operator on l2(Z) whose diagonals Dn(A)=(aj,j−n)j=−∞∞∈l∞(Z) are almost periodic sequences. For certain classes of such operators and under certain conditions, we are going to determine the asymptotics of the determinants detAn1,n2 of the finite sections An1,n2=(aj,k)j,k=n1n2−1 as their size n2−n1 tends to infinity. Examples of such operators include block Toeplitz operators and the almost Mathieu operator.
Highlights
1.1 BackgroundAsymptotics of determinants of finite sections
all sequences (An) operator acting on l2(Z) whose diagonals are periodic sequences with period N can be identified with a block Laurent operator, which is defined by
The classical strong Szegö-Widom limit theorem [31] describes the asymptotics of the determinants of the finite sections Pn1,n2L(a)Pn1,n2 of block Laurent operators in the case when n1 = 0 and n2 = nN as n → ∞
Summary
An operator acting on l2(Z) whose diagonals are periodic sequences with period N can be identified with a block Laurent operator, which is defined by (1.1.2) with a being an N × N matrix-valued function whose Fourier coefficients an are N × N matrices. The classical strong Szegö-Widom limit theorem [31] describes the asymptotics of the determinants of the finite sections Pn1,n2L(a)Pn1,n2 of block Laurent operators in the case when n1 = 0 and n2 = nN as n → ∞. Let us recall this result for generating functions in BN×N , where B = W ∩ F.
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