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

Consider three normal operators A,B,C on a separable Hilbert space ? as well scalar-valued spectral measures ? A ?(A), B ?(B) and C ?(C). For any ??L ? (? ×? ) X,Y?S 2 (?), the of Hilbert–Schmidt ?, we provide general definition triple operator integral ? (?)(X,Y) belonging to S (?) in such way that (?) belongs (S (?)×S (?),S (?)) bounded bilinear resulting mapping :L )?B is w * -continuous isometry. Then show function has property maps into 1 trace class if only it following factorization property: there exist H two functions a?L ;H) b?L ?(t ,t 3 )=a(t ),b(t for a.e. (t )??(A)×?(B)×?(C). This version Peller’s Theorem characterizing double mappings (?)?S (?). In passing Banach spaces E,F, -measurable esssentially valued (E,F from E F factoring through admits factorization.