Gruss-Landau Inequalities for Elementary Operators and Inner Product Type Transformers in Q and Q* Norm Ideals of Compact Operators

For a probability measure mu on Omega and square integrable (Hilbert space) operator valued functions {A(t)*}(t is an element of Omega), {B-t}(t is an element of)(Omega), we prove Gruss-Landau type operator inequality for inner product type transformers vertical bar integral(Omega) A(t)XB(t) d mu(t) - integral(Omega) A(t)d mu(t)X integral(Omega) B-t d mu(t)vertical bar(2 eta)<= parallel to integral(Omega) A(t)A(t)* d mu(t) - vertical bar integral(Omega) A(t)*d mu(t)vertical bar(2)parallel to(eta)(integral(Omega) B-t*X*XB(t)d mu(t) -vertical bar X integral(Omega) B(t)d mu(t)vertical bar(2))(eta), for all X is an element of B(H) and for all eta is an element of [0, 1]. Let p >= 2, Phi to be a symmetrically norming (s.n.) function, Phi((p)) to be its p-modification, Phi((p))* is a s.n. function adjoint to Phi((p)) and parallel to.parallel to(Phi)(p)* to be a norm on its associated ideal partial derivative Phi(p)*(H) of compact operators. If X is an element of partial derivative(Phi)|(p)*(H) and {alpha(n)}(n=1)(infinity) is a sequence in (0,1], such that Sigma(infinity)(n=1)alpha(n) = 1 and Sigma(infinity)(n=1) parallel to alpha(-1/2)(n) A(n)f parallel to(2)+parallel to alpha B--1/2(n)n*f parallel to(2) < + infinity for some families {A(n)}(n=1)(infinity) and {B-n}(n=1)(infinity) of bounded operators on Hilbert space H and for all f is an element of H, then parallel to Sigma(infinity)(n=1) alpha(-1)(n) A(n)XB(n) - Sigma(infinity)(n=1) A(n)X Sigma(infinity)(n=1) B-n parallel to(Phi(p))* <= parallel to root Sigma(infinity)(n=1) alpha(-1)(n)vertical bar A(n)vertical bar(2) - vertical bar Sigma(infinity)(n=1) A(n)vertical bar(2) X root Sigma(infinity)(n=1)alpha(-1 )(n)vertical bar B-n*vertical bar(2) -vertical bar Sigma(n=)1(infinity) B-n*vertical bar(2)parallel to(Phi(p))*, if at least one of those operator families consists of mutually commuting normal operators. The related Gruss-Landau type parallel to.parallel to(Phi(p))* norm inequalities for inner product type transformers are also provided.