Multiplication operators on S2($$\mathbb{D}$$)

Composition and multiplication operators on the derivative Hardy space

“…The next proposition is similar to Proposition 2.2 in [4], but here our estimates are tight which leads to a sharp constant.…”

“…The fact S 2 2 (D) and the analogous D n for an integer n ≥ 1 are algebras were also mentioned in [22] by referring to literature in Russian. The recent paper [4] proved that S 2 (D) is an algebra, and another recent paper [11] also proved S 2 2 (D) is an algebra. As a consequence, as mentioned both in [30] and [22], the maximal ideal space of D α for α > 1 (more precisely, the norm-equivalent Banach algebra) is the closed unit disk.…”

“…Now we can show that M ψ is an m-isometry on S 2 (D) if and only if ψ is a constant. The case for m = 1 was proved in [4]. Notice that Lemma 3.6 shows that if T is a strict m-isometry, then for some h, β m−1 (T )h, h = 0, so T n h 2 grows like a polynomial in n of degree m − 1 as n → ∞.…”

“…It is known that the only inner functions belong to Dirichlet space D 2 (D) are finite Blaschke products, so this is also true for the space S 2 1 (D). It has been observed in [4] that the only isometric multiplication operators on S 2 (D) are the constant multiples of the identity operator. Here we prove a stronger result that M f on S 2 1 (D) is a 2-isometry if and only if it is a constant multiple of the identity operator.…”

“…see [4], [17], [18], [24]. Composition operators and linear isometries on Banach space S p (D) were studied in 1978 [28] and 1985 [20].…”