Ideals of compact operators with Nakano type norms in a Hilbert space

Let H be a separable Hilbert space with a norm {double pipe}.{double pipe} _H. For a compact linear operator A acting in H, let λ _k(A) be the eigenvalues, s _k(A) (k = 1,2, ...) singular values and {double pipe}A{double pipe} _H = sup _x∈H{double pipe}Ax{double pipe} _H{double pipe}x{double pipe} _H. Let π = {p _k} ^∞ _k=1 be a nondecreasing sequence of numbers p _k ≥ 1. Put We investigate the ideal X _π of operators satisfying γ _π (tA) < ∞ for all t > 0. In particular, it is proved that for any A ∈ X _π we have where _A = {double pipe}A{double pipe} _H if {double pipe}A{double pipe} _H > 1 and ν _A = 1 if {double pipe}A{double pipe} _H ≤ 1.