Essential norms and weak compactness of integration operators

Laitila, Jussi; Miihkinen, Santeri; Nieminen, Pekka

doi:10.1007/s00013-011-0272-z

Cited by 27 publications

(24 citation statements)

References 11 publications

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“…In section 5 we study an integral operator T g on BM OA and B and we show in particular that its compactness and weak compactness are equivalent (Theorem 6). This result for the case of BM OA was also obtained independently by different methods in [13]. In section 6 we apply these results for T g for a special choice of the symbol g = γ (defined later in Definition 4) to obtain a characterization of the cases of equality [ϕ t , BM OA] = V M OA and [ϕ t , B] = B 0 in terms of γ (Corollary 2).…”

Section: F ∈ H(d)mentioning

confidence: 53%

Semigroups of composition operators and integral operators in spaces of analytic functions

Contreras¹,

Díaz-Madrigal²,

Martínez³

et al. 2013

Ann. Acad. Sci. Fenn. Math.

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Abstract. We study the maximal spaces of strong continuity on BM OA and the Bloch space B for semigroups of composition operators. Characterizations are given for the cases when these maximal spaces are V M OA or the little Bloch B 0 . These characterizations are in terms of the weak compactness of the resolvent function or in terms of a specially chosen symbol g of an integral operator T g . For the second characterization we prove and use an independent result, namely that the operators T g are weakly compact on the above mentioned spaces if and only if they are compact.

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Section: F ∈ H(d)mentioning

confidence: 53%

Semigroups of composition operators and integral operators in spaces of analytic functions

Contreras¹,

Díaz-Madrigal²,

Martínez³

et al. 2013

Ann. Acad. Sci. Fenn. Math.

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show abstract

“…It remains to see that (6) implies (7). Suppose that ε, N > 0 are such that (6) holds and let x ∈ M(X, L).…”

Section: Results and Proofsmentioning

confidence: 99%

“…This result is classical for M 0 = c 0 , and has also been proven for the case when M 0 = V MO [12], the latter fact which has been used in [7] and [8] to characterize the weak compactness of Volterra-type integral operators and composition operators on the analytic BMO-space.…”

Section: Introductionmentioning

confidence: 90%

“…The pair (M 0 , M) was first introduced in [14], and the quoted examples are given there. This paper is inspired by recent works on the compactness properties of composition and integral operators acting on specific examples of spaces M 0 and M [3], [7], [8], [10]. It often turns out that weak compactness and compactness are equivalent for these classes of operators, a phenomenon which can be readily understood given the main results of this article.…”

Section: Introductionmentioning

confidence: 99%

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Weak compactness of operators acting on o–O type spaces

Perfekt

2015

Bull. London Math. Soc.

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Abstract. We consider operators T : M 0 → Z and T : M → Z, where Z is a Banach space and (M 0 , M ) is a pair of Banach spaces belonging to a general construction in which M is defined by a "big-O" condition and M 0 is given by the corresponding "little-o" condition. Prototype examples of such spaces M are given by ℓ ∞ , weighted spaces of functions or their derivatives, bounded mean oscillation, Lipschitz-Hölder spaces, and many others. The main result characterizes the weakly compact operators T in terms of a certain norm naturally attached to M , weaker than the M -norm, and shows that weakly compact operators T : M 0 → Z are already quite close to being completely continuous. Further, we develop a method to extract c 0 -subsequences from sequences in M 0 . Applications are given to the characterizations of the weakly compact composition and Volterra-type integral operators on weighted spaces of analytic functions, BM OA, V M OA, and the Bloch space.

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“…Define h n = f n+1 − f n . By the proof of Theorem 2 in [8], it holds that h n * ≃ 1 and h n 2 → 0, as n → ∞. By Lemma 4.1, we can pick a subsequence (h n k ) ⊂ (h n ) which is equivalent to the standard basis {e k } of c 0 .…”

Section: N By Condition (Ii) Thus It Always Holds Thatmentioning

confidence: 96%

Strict singularity of a Volterra-type integral operator on $H^p$

Miihkinen

2016

Proc. Amer. Math. Soc.

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Abstract. We prove that a Volterra-type integral operatordefined on Hardy spaces H p , 1 ≤ p < ∞, fixes an isomorphic copy of ℓ p , if the operator T g is not compact. In particular, this shows that the strict singularity of the operator T g coincides with the compactness of the operator T g on spaces H p . As a consequence, we obtain a new proof for the equivalence of the compactness and the weak compactness of the operator T g on H 1 .

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Essential norms and weak compactness of integration operators

Cited by 27 publications

References 11 publications

Semigroups of composition operators and integral operators in spaces of analytic functions

Semigroups of composition operators and integral operators in spaces of analytic functions

Weak compactness of operators acting on o–O type spaces

Strict singularity of a Volterra-type integral operator on $H^p$

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