Grüss-Landau inequalities for elementary operators and inner product type transformers in Q and Q* norm ideals of compact operators

Abstract

For a probability measure ? on ? and square integrable (Hilbert space) operator valued functions {A*t}t??, {Bt}t??, we prove Gr?ss-Landau type operator inequality for inner product type transformers |?? AtXBtd?(t)- ?? At d?(t)X ?? Btd?(t)|2? ? ||??AtA*td?(t)- |??A*td?(t)|2||? (?? B*tX*XBtd?(t)- |X?? Btd?(t)|2)?, for all X ? B(H) and for all ? ? [0,1]. Let p ? 2, ? to be a symmetrically norming (s.n.) function, ? (p) to be its p-modification, ? (p)* is a s.n. function adjoint to ?(p) and ||?||?(p)* to be a norm on its associated ideal C?(p)*(H) of compact operators. If X ? C?(p)*(H) and {?n}?n=1 is a sequence in (0,1], such that ??n=1 ?n = 1 and ??n=1 ||?n-1/2 An f||2+||?-1/2n B*nf||2 < +? for some families {An}? n=1 and {Bn}? n=1 of bounded operators on Hilbert space H and for all f ? H, then ||?? n=1 ?-1n AnXBn-?? n=1 AnX ?? n=1 Bn||?(p)* ? ||???n=1 ?-1n |An|2-|??n=1 An|2 X ? ??n=1 ?-1n |B*n|2-|??n=1 B*n|2||?(p)+, if at least one of those operator families consists of mutually commuting normal operators. The related Gr?ss-Landau type ||?||?(p) norm inequalities for inner product type transformers are also provided.