Szegö type limit theorems
If \({\mathcal S}^1\) denotes the unit circle \(P_k\) the orthogonal projection of \(L^2({\mathcal S}^1)\) onto \([e^{\pm inx}]^k_{n=0}\) and \(B\) the operator of multiplication on \(L^2({\mathcal S}^1)\) by a smooth function \(b(x)\), the theorem of Szegö mentioned in the title gives an asymptotic expression for \(\text{tr }\log(P_k BP_k)\) as \(k\to\infty\) in terms of the integral of \(\log b\) around \({\mathcal S}^1\). In this paper, a generalization of this result in which \(\{P_k\}\) is replaced by the spectral projections \(\{P_\lambda\}\) of an elliptic selfadjoint differential operator \(A\) and \(\log(x)\) is replaced by some smooth function \(\Phi(x)\) is introduced. The corresponding result allows a study of the asymptotic behavior form \(P_\lambda BP_\lambda\) for a wide class of differential and pseudo-differential operators in which the remainder estimates given by the classical Szegö result still hold.
differential and pseudo-differential operators
scientific article; zbMATH DE number 908037