Köthe Sets and Köthe Sequence Spaces

Abstract. We investigate several properties of operator-weighted composition mapsspaces of vector-valued analytic functions on the unit disc D. Here ϕ is an analytic self-map of D and ψ is an analytic operator-valued function on D. We characterize when the operator is continuous, maps a neighbourhood into a bounded set or maps bounded sets into relatively compact sets. In this way we extend results due to Laitila and Tylli for the case of Banach valued functions. This more general setting permits us to compare the results in the unweighted and weighted cases. New examples are provided, especially when the spaces X and Y are Köthe echelon spaces. They show the differences between the present setting and the case of functions taking values in Banach spaces.

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“…We refer the reader to [21] and [3] for a detailed study of Köthe echelon spaces. We recall here a few definitions.…”

Section: Analytic Dependence Of Diagonal Operators In Values Of Köthementioning

confidence: 99%

“…The following essentially well-known lemma follows from the characterization of bounded sets in a Köthe echelon space in [3].…”

Section: It Is Easy To See That G Is a Weakmentioning

confidence: 99%

Operator-weighted composition operators between weighted spaces of vector-valued analytic functions

Bonet

Gómez-Collado

Jornet

et al. 2012

Ann. Acad. Sci. Fenn. Math.

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“…The purpose of this note is to show that for coechelon spaces k ∞ (v) of order ∞ the condition v regularly decreasing [BMS,Definition 3.1] is necessary and sufficient for every compact set to be contained in a closed absolutely convex hull of some null sequence.…”

Section: On Compact Subsets In Coechelon Spaces Of Infinite Ordermentioning

confidence: 99%

“…[PB,Proposition 8.3.12]); hence, every compact set in K ∞ V is also a compact set in k ∞ (v). Finally, we recall that the matrix v is called regularly decreasing [BMS,Definition 3.1] if, and only if, given n ∈ N there is m ≥ n such that…”

Section: Angela a Albanesementioning

confidence: 99%

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On compact subsets in coechelon spaces of infinite order

Albanese

1999

Proc. Amer. Math. Soc.

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Abstract. For coechelon spaces k∞(v) of infinite order it is proved that every compact subset of k∞(v) is contained in a closed absolutely convex hull of some null sequence if and only if the matrix v is regularly decreasing.In connection with the study of some interesting problems on Montel maps which are closely connected to the classical Grothendieck question on completeness of regular LB-spaces [PB, Problem 13.8.6] and the problem of bornologicity of C (K, E) with E an LB-space [S, Chapter IV], Dierolf and Domański [DD2, Example 3.1] gave an example of a coechelon LB-Montel space of infinite order which has compact sets not contained in closed absolutely convex hulls of any null sequence. Consequently, the well-known characterization of compact sets in Fréchet spaces [J, Theorem 9.4.2] turns out to be not generally true in the LB setting.The purpose of this note is to show that for coechelon spaces k ∞ (v) of order ∞ the condition v regularly decreasing [BMS, Definition 3.1] is necessary and sufficient for every compact set to be contained in a closed absolutely convex hull of some null sequence. For more information on Montel maps and related questions the reader is referred to [DD1], [DD2], [DD3] and [D].In what follows we recall some notation. Let E be a Fréchet space with a fundamental system of seminorms ( n ) n ; then the inductive dual E i is defined to be ind n E n , where the E n are the completions of the normed spaces (E/ ker n , n ). It is known that algebraically E i = E , the inclusion map E i → E β is continuous and E i is the bornological space associated with E β , i.e., E i = (E , β (E , E )) [J, Theorem 13.4.2] (E and E β denote the topological dual and the strong dual of E, resp.).We also recall that an LB-space E = ind n E n is called boundedly retractive (respectively, compactly regular) if, and only if, for each bounded (respectively, compact) subset B of E there is n ∈ N such that B ⊂ E n and E n and E induce the same topology on B (see [PB,, -(iii)]). It is clear that a boundedly retractive LB-space is compactly regular. On the other hand, in [N] it is proved that these conditions are also equivalent.

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On Completeness and Projective Descriptions of Weighted Inductive Limits of Spaces of Fréchet-Valued Continuous Functions

Albanese

2000

Math. Nachr.

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Köthe Sets and Köthe Sequence Spaces

Cited by 91 publications

References 5 publications

Operator-weighted composition operators between weighted spaces of vector-valued analytic functions

Operator-weighted composition operators between weighted spaces of vector-valued analytic functions

On compact subsets in coechelon spaces of infinite order

On Completeness and Projective Descriptions of Weighted Inductive Limits of Spaces of Fréchet-Valued Continuous Functions

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