Some new properties of composition operators associated with lens maps

Approximation numbers of weighted composition operators

“…Proof. We begin with the upper bound, following the strategy of [25]. If γ(t) = ϕ(e it ), one easily checks [25, Lemma 2.5] that (4.4) |t| ≤ π 2 ⇒ 1 − |γ(t)| ≈ |1 − γ(t)| ≈ |t| λ (4.5) π 2 ≤ |t| ≤ π ⇒ 1 − |γ(t)| ≈ |1 + γ(t)| ≈ (π − |t|) λ .…”

“…More recently, also the membership of C ϕ in various smaller ideals I of bounded operators on H (such as the p-Schatten class), and more precisely the behavior of the approximation numbers a n (C ϕ ) of C ϕ , was studied in depth in several papers (see e.g. [27,25,26,28]).…”

Some examples of composition operators and their approximation numbers on the Hardy space of the bidisk

“…On the other hand, we have a n (T k ) ≤ c k a n (C λ θ ) ≤ a n (C λ θ ) e −β θ √ n ( [16], Theorem 2.1). Taking n 0 = n 1 = · · · = n K = K 2 in Lemma 5.3, we get: max 0≤k≤K a n k (T k ) e −β θ K .…”

“…The lens map λ θ : D → D is defined, for θ ∈ (0, 1), by: [26], p. 27, or [16], for more information), and corresponds to u → u θ in the right half-plane.…”

“…For sake of completeness, we recall the following elementary fact (see [26], p. 28, or also [16], Lemma 2.5)). Lemma 4.3.…”