Entropy numbers of diagonal operators on Orlicz sequence spaces

Let M 1 and M 2 be functions on [0,1] such thatM 1 ( t 1 / p ) andM 2 ( t 1 / p ) are Orlicz functions for some p ∈ ( 0 , 1 ] . Assume thatM 2 − 1( 1 / t ) / M 1 − 1( 1 / t )is non‐decreasing for t ≥ 1 . Let( α i ) i = 1 ∞ be a non‐increasing sequence of nonnegative real numbers. Under some conditions on( α i ) i = 1 ∞ , sharp two‐sided estimates for entropy numbers of diagonal operatorsT α : ℓM 1→ ℓM 2generated by( α i ) i = 1 ∞ , where ℓM 1and ℓM 2are Orlicz sequence spaces, are proved. The results generalise some works of Edmunds and Netrusov in [8] and hence a result of Cobos, Kühn and Schonbek in [6].