Compactness of classical operators on weighted Banach spaces of holomorphic functions

Invariant Subspaces of the Integration Operators on Hörmander Algebras and Korenblum Type Spaces

“…We refer the reader to the introduction of [4] for classical results about invariant subspaces of the integration operators and more recent ones in [7], [19] and [20]. The continuity of the integration operator on weighted Banach spaces of holomorphic functions was investigated by Harutyunyan and Lusky [22]; see also [2] and [5]. Other aspects, like spectrum and ergodic or dynamical properties, were considered by Beltrán, Fernández and the first author in [8].…”

“…These (LB)-spaces have been investigated by many authors; see e.g. [5], [13] and [14] and the references therein. On the other hand, if A = (a n ) n is an increasing sequence of weights on G = D or G = C, the weighted Fréchet space of holomorphic functions on G is defined by A 0 H(G) := proj n H 0 an (G).…”

Integration and Differentiation in Hardy Growth Spaces

“…The bounded and compact operators J : H ∞ w (C) → H ∞ u (C) are studied in [2,3] under specific restrictions on w or for w = u. We show that the problems in question have quite elementary solutions if w is assumed to be equivalent to a power series 2010 Mathematics Subject Classification.…”

“…Integration operator on H ∞ w (D) As mentioned in the introduction, the bounded or compact integration operators between growth spaces H ∞ w (C) and H ∞ u (C) are characterized in [2] under specific restrictions on w or for w = u. So, in the present section, we are primarily interested in the case D = C.…”

“…It is know that a weight w on [0, 1) is equivalent to a power series with positive coefficients if and only if w is equivalent to a log-convex weight (see [6,Proposition 1.3]). Therefore, Proposition 2.1 for D = D is essentially proved in [2,Proposition 4], where the corresponding compact operators are characterized under assumption that w is a log-convex weight. So, below in the present section we assume that D = C.…”