On diagonal operators between the sequence (LF)-spaces $$l_p({\mathcal {V}})$$

Mele, Claudio

doi:10.1007/s13398-022-01284-8

Cited by 2 publications

(3 citation statements)

References 21 publications

(33 reference statements)

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“…The characterization of the continuity of operators between (LF)‐spaces is well‐known and due to Grothendieck. The characterization of boundedness as well as the compactness of operators acting between (LF)‐spaces has been given in [17] as follows (see also [7, Proposition 5], where the (LB)‐case is considered). We include also the weakly compact case.…”

Section: Definitions and General Results On (Lf)‐ And (Plb)‐spacesmentioning

confidence: 99%

“…We refer the reader to [17] for analogous results on diagonal (multiplication) operators acting on the sequence (LF)‐space

l_p(V)

, with

1\le p\le \infty

p=0

.…”

Section: Examplesmentioning

confidence: 99%

“…While, in [8] it has been characterized when continuous multiplication operators on a weighted inductive limit of Banach spaces of continuous functions are power bounded, mean ergodic or uniformly mean ergodic. Moreover, in [17] the second author has analyzed the action of the multiplication (diagonal) operators between weighted sequence (LF)-spaces, studying general properties like boundedness, compactness, and dynamics.…”

Section: Introductionmentioning

confidence: 99%

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On composition operators between weighted (LF)‐ and (PLB)‐spaces of continuous functions

Albanese

Mele

2023

Mathematische Nachrichten

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Let X be a locally compact Hausdorff topological space, let be a system of positive continuous functions on X and let φ be a continuous self‐map on X. The composition operators on the weighted function (LF)‐spaces (, resp.) and on the weighted function (PLB)‐spaces (, resp.) are studied. We characterize when the operator acts continuously on such spaces in terms of the system and the map φ, as well as we determine conditions on and φ which correspond to various basic properties of the composition operator , like boundedness, compactness, and weak compactness. Our approach requires a study of the continuity, boundedness, (weak) compactness of the linear operators between (LF)‐spaces and (PLB)‐spaces.

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Section: Definitions and General Results On (Lf)‐ And (Plb)‐spacesmentioning

confidence: 99%

“…We refer the reader to [17] for analogous results on diagonal (multiplication) operators acting on the sequence (LF)‐space

l_p(V)

, with

1\le p\le \infty

p=0

.…”

Section: Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On composition operators between weighted (LF)‐ and (PLB)‐spaces of continuous functions

Albanese

Mele

2023

Mathematische Nachrichten

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Topological Properties of Weighted Composition Operators in Sequence Spaces

Albanese,

Mele

2023

Results Math

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For fixed sequences $$u = (u_i)_{i\in {{\mathbb {N}}}}, \varphi =(\varphi _i)_{i\in {{\mathbb {N}}}}$$ u = ( u i ) i ∈ N , φ = ( φ i ) i ∈ N , we consider the weighted composition operator $$W_{u,\varphi }$$ W u , φ with symbols $$u, \varphi $$ u , φ defined by $$x=(x_i)_{i\in {{\mathbb {N}}}}\mapsto u(x\circ \varphi )= (u_ix_{\varphi _i})_{i\in {{\mathbb {N}}}}$$ x = ( x i ) i ∈ N ↦ u ( x ∘ φ ) = ( u i x φ i ) i ∈ N . We characterize the continuity and the compactness of the operator $$W_{u,\varphi }$$ W u , φ when it acts on the weighted Banach spaces $$l^p(v)$$ l p ( v ) , $$1\le p\le \infty $$ 1 ≤ p ≤ ∞ , and $$c_0(v)$$ c 0 ( v ) , with $$v=(v_i)_{i\in {{\mathbb {N}}}}$$ v = ( v i ) i ∈ N a weight sequence on $${{\mathbb {N}}}$$ N . We extend these results to the case in which the operator $$W_{u,\varphi }$$ W u , φ acts on sequence (LF)-spaces of type $$l_p(\mathcal {V})$$ l p ( V ) and on sequence (PLB)-spaces of type $$a_p(\mathcal {V})$$ a p ( V ) , with $$p\in [1,\infty ] \cup \{0\}$$ p ∈ [ 1 , ∞ ] ∪ { 0 } and $$\mathcal {V}$$ V a system of weights on $${{\mathbb {N}}}$$ N . We also characterize other topological properties of $$W_{u,\varphi }$$ W u , φ acting on $$l_p(\mathcal {V})$$ l p ( V ) and on $$a_p(\mathcal {V})$$ a p ( V ) , such as boundedness, reflexivity and to being Montel.

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On diagonal operators between the sequence (LF)-spaces $$l_p({\mathcal {V}})$$

Cited by 2 publications

References 21 publications

On composition operators between weighted (LF)‐ and (PLB)‐spaces of continuous functions

On composition operators between weighted (LF)‐ and (PLB)‐spaces of continuous functions

Topological Properties of Weighted Composition Operators in Sequence Spaces

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