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The boundary behaviour of \({\mathfrak S}_ p\)-valued functions analytic in the half-plane with nonnegative imaginary part. (English) Zbl 0815.47021

Zemánek, Jaroslav (ed.), Functional analysis and operator theory. Proceedings of the 39th semester at the Stefan Banach International Mathematical Center in Warsaw, Poland, held March 2-May 30, 1992. Warsaw: Polish Academy of Sciences, Banach Cent. Publ. 30, 277-285 (1994).
An R-function \(T(\lambda)\) is one with values in the space of bounded linear operators on a separable Hilbert space, analytic on \(C_ += \{\lambda\in \mathbb{C}\mid \text{Im } \lambda>0\}\), and for which \(\text{Im} (T(\lambda)) \geq 0\). The following general problem forms the focus for this paper:
If \(T(\lambda)\) is an R-function whose values lie in a given space of operators \(A\), do the non-tangential boundary values of \(T(\lambda)\) exist a.e.?
Answers to this and related problems are given when \(A\) is one of the Schatten \(p\)-classes \(C_ p\), \(p>0\) (e.g. “Yes” when \(0<p<1\), “no” when \(p>1\)).
For the entire collection see [Zbl 0792.00007].

MSC:

47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
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