Laplace transformers in norm ideals of compact operators

Abstract

Let p⩾ 2 , Φ a symmetrically norming (s.n.) function (resp. Φ(p) its p-modification and Φ(p)∗ a s.n. function adjoint to Φ(p)) and CΦ(H) (resp. CΦ(p)(H) and CΦ(p)∗(H)) be its associated ideals of compact operators acting on a Hilbert space H and let f, g, h: [0 , + ∞) → C be Lebesgue measurable functions. Some recently obtained Cauchy–Schwarz-type norm inequalities were used to systematically explore a class of Laplace transformers of the form Lf(Δ A,B) : X↦ ∫ [,+∞)e-tAXe-tBf(t)dt(=∫[0,+∞)e-tΔA,BXf(t)dt), acting on the B(H) , CΦ(H),CΦ(p)(H) or CΦ(p)∗(H), induced by a generalized derivation Δ A,B: B(H) → B(H) : X↦ AX+ XB and B(B(H)) valued Laplace transform Lf of a function f. If ∫[0,+∞)(||e-tAx||2|f(t)|2+||e-tBx||2|g(t)|2)dt<+∞ for all x∈ H and both A and B are normal, then for all X∈CΦ(H)||∫[0,+∞)e-tAXe-tBf(t)g(t)dt||Φ⩽||(∫[0,+∞)e-t(A∗+A)|f(t)|2dt)12X(∫[0,+∞)e-t(B∗+B)|g(t)|2dt)12||Φ.If α, β∈ [0 , 1] , then for all X∈CΦ(p)∗(H)||L(f∗g)(ΔA,B)X||Φ(p)∗⩽||(L(|f|2-2α∗|g|2-2β)(ΔA∗,A)(I))12X(L(|f|2α∗|g|2β)(ΔB,B∗)(I))12||Φ(p)∗,whenever ∫[0,+∞)(||e-tAx||2|f|2-2α∗|g|2-2β(t)+||e-tB∗x||2|f|2α∗|g|2β(t))dt< + ∞ for all x∈ H and at least one of operators A or B is normal, where Lh(C)=def∫[0,+∞)e-tCh(t)dt denotes the operator valued Laplace transform of a function h and f∗ g denotes a convolution function f∗g(t)=def∫[0,t]f(t-s)g(s)ds for all t⩾ 0. Applications of Laplace transformers to norm inequalities include the norm inequality ||(A∗2+2A∗A+A2)1/2X(B2+2BB∗+B∗2)1/2||Φ(p)∗⩽||A2X+2AXB+XB2||Φ(p)∗,if A, B, X∈ B(H) are such that A, B∗ are 2-hyper-accretive and at least one of them is normal, satisfying A2X+2AXB+XB2∈CΦ(p)∗(H).