Norm inequalities for hypercontractive quasinormal operators and related higher order Sylvester–Stein equations in ideals of compact operators

Amongst others, for N ∈ N , some Q ∗ symmetrically norming (s.n.) functions Ψ and N -hypercontractive operators C and D ∗ , such that at least one of C , D ∗ is quasinormal and for some bounded Hilbert space operator X , we have proved | | ( ∑ n = 0 N ( - 1 ) n N n C ∗ n C n ) 1 2 ( X - ∑ K = 0 N - 1 n K C n -- K ( ∑ i = 0 K ( - 1 ) i K i C i X D i ) D n -- K ) × ( ∑ n = 0 N ( - 1 ) n N n D n D ∗ n ) 1 2 | | Ψ ⩽ | | ( I - A C ) 1 2 ( ∑ n = 0 N ( - 1 ) n N n C n X D n ) ( I - A D ∗ ) 1 2 | | Ψ ⩽ | | ∑ n = 0 N ( - 1 ) n N n C n X D n | | Ψ , where A C = def s lim n → ∞ C ∗ n C n and A D ∗ = def s lim n → ∞ D n D ∗ n . Under the additional convergence conditions, this implies | | ( ∑ n = 0 N ( - 1 ) n N n C ∗ n C n ) 1 2 X ( ∑ n = 0 N ( - 1 ) n N n D n D ∗ n ) 1 2 | | Ψ ⩽ | | ∑ n = 0 N ( - 1 ) n N n C n X D n | | Ψ . Above, denotes the ideal of compact operators associated with the s.n. function Ψ .