Zygmund's theorem and the boundary behavior of operator \(R\)-functions
An important class of operator \(R\)-functions of the form \(T(\lambda)= V^{1/2}(L-\lambda I)^{-1} V^{1/2}\), \(\lambda\in\mathbb{C}_+\), is considered, where \(V\) is a compact nonnegative operator and \(L\) is a selfadjoint operator in a Hilbert space \(H\). The boundary behaviour of such \(R\)-functions is considered, and in particular the problem when the boundary values \(T(k+ i\cdot 0)\) belong to the Schatten class \({\mathfrak S}_1\) for almost all \(k\in R\). The author first considers the Zygmund condition \[ \sum_n s_n(V)\log^+(1/s_n(V))< \infty,\tag{1} \] which is equivalent to the condition \[ \sum_n s_n(V)\ln n< \infty,\tag{2} \] and proves that this condition does not guaranty that the boundary values \(T(k+ i\cdot 0)\) belong to \({\mathfrak S}_1\) for almost all \(k\in R\). Hence the operator analog of the scalar Zygmund theorem fails. Then he describes the Marcinkiewicz classes \({\mathfrak S}_\pi\) (the duals of the Lorentz classes) to which the boundary values \(T(k+i\cdot 0)\) of such \(R\)-functions can belong for almost all \(k\in R\).
operator analog scalar Zygmund theorem fails
scientific article; zbMATH DE number 1023828
scientific article