When do triple operator integrals take value in the trace class? (Q2115477)

For a separable Hilbert space \(\mathcal{H},\) the space of Hilbert-Schmidt operators on \(\mathcal{H}\) is denoted by \(S^{2}(\mathcal{H})\). Also the space of trace class operators on \(\mathcal{H}\) is denoted by \(S^{1}(\mathcal{H})\). Following the paper [\textit{C. Coine} et al., J. Funct. Anal. 271, No. 7, 1747--1763 (2016; Zbl 1350.47011)], the authors, in the present paper, investigate an analogue of Peller's Theorem for triple operator integrals. They established the main result which find a necessary and sufficient condition assured that this triple operator integrals map \(S^{2}(\mathcal{H})\times S^{2}(\mathcal{H})\) to \(S^{1}( \mathcal{H})\) and compare it with previous constructions.