On the boundary values of analytic operator-valued functions with positive imaginary parts

Let $\mathfrak Y_p$ $(0<p<\infty)$ be the Schatten-von-Heumann class of operators on a Hilbert space. We prove that for $\mathfrak Y_p$-valued $(0<p<1)$ $\mathbb R$-functions the nontangential limits exist a. e. on $\mathbb R$ and belong to $\mathfrak Y_p$. For $p>1$ the “boundary values” can even be unbounded everywhere on $\mathbb R$. Finally, for $p=1$ the nontangential limits on $\mathfrak Y_q$, exist in the norm of $q>1$. However, they belong, in general, only to the symmetric ideal $\mathfrak Y_\Omega$, which is adjoint to Matsaev's class.