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

Abstract

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 [InlineEquation not available: see fulltext.] for some bounded Hilbert space operator X, we have proved ||(∑n=0N(-1)n(Nn)C∗nCn)12(X-∑K=0N-1(nK)Cn--K(∑i=0K(-1)i(Ki)CiXDi)Dn--K)×(∑n=0N(-1)n(Nn)DnD∗n)12||Ψ⩽||(I-AC)12(∑n=0N(-1)n(Nn)CnXDn)(I-AD∗)12||Ψ⩽||∑n=0N(-1)n(Nn)CnXDn||Ψ,where AC=defslimn→∞C∗nCn and AD∗=defslimn→∞DnD∗n. Under the additional convergence conditions, this implies ||(∑n=0N(-1)n(Nn)C∗nCn)12X(∑n=0N(-1)n(Nn)DnD∗n)12||Ψ⩽||∑n=0N(-1)n(Nn)CnXDn||Ψ.Above, [InlineEquation not available: see fulltext.] denotes the ideal of compact operators associated with the s.n. function Ψ.