Some inequalities involving traces of operators

Abstract

We prove that for arbitrary completely continuous operators A₁, A₂, ···, A_n and for positive numbers p₁, p₂, ···, p_n with ∑_{k = 1} ⁿp_k^-1 = 1, the inequalities, |Tr(A₁A₂···A_n)|≤ ∏ⁿ_k=1[Tr(A_kA_k)^½Pk]^Pk-1|Trexp(A₁ + A₂ +···+A_n)|≤∏ⁿ_k=1[Trexp{½Pk(A_k + A_k)}]^Pk-1 hold. Further if a₁, a₂, ···, a_n are the annihilation operators of an N-dimensional harmonic oscillator, m and n are any positive integers, and ρ is a nonnegative definite operator, we prove the inequality |Tr(pa_i1···a_im a_j1···a_jn| ≤ ∏^m_k=1[Trp(a_ik a_ik)^m]^½m∏ⁿ_i=1[Trp(a_ji a_ji)ⁿ]^½n .Some consequences of these inequalities, related results, and some applications to correlation functions of the quantized electromagnetic field are discussed.