Invariant Subspaces for Classical Operators on Weighted Spaces of Holomorphic Functions

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

“…Our main results are Theorem 4.1 and its Corollary 4.2, which describe the invariant subspaces when the integration operator acts on Korenblum type spaces, and Theorem 5.1, which explains the situation in case of some Hörmander algebras of entire functions. The proofs of these results depend on some abstract Theorems 3.2 and 3.3 and they rely heavily on Theorems 3.8 and 3.16 due to Abanin and Tien [4]. A different method permits us to handle the (LB)-algebra of entire functions of exponential type in Theorem 5.…”

“…Abanin and Tien describe in [4] the closed invariant subspaces of the integration operator on various scales of weighted Banach spaces of holomorphic functions. As mentioned above, some of their results are very important for our theorems below.…”

“…As mentioned above, some of their results are very important for our theorems below. 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].…”

“…Our first Lemma is stated in page 412 of [4]. We include a proof for the sake of completeness, because it is very relevant in our considerations below.…”

“…We include a proof for the sake of completeness, because it is very relevant in our considerations below. As in [4], given a space of holomorphic functions E on the open unit disc D or the complex plane C, and N ∈ N, we set…”