An isomorphic property in spaces of compact operators and some classes of operators on C(K,X)

A Study on Dunford–Pettis Completely Continuous Like Operators

“…Furthermore, if Ω is a dispersed compact Hausdorff space and T : C(Ω, X) → Y is strongly bounded, then we show that T * is Dunford-Pettis p-convergent if and only if m(A) * : Y * → X * is Dunford-Pettis p-convergent, for each A ∈ Σ. Note that our results in this section are motivated by results in [3,6,17,21].…”

“…• Suppose that T : C(Ω, X) → Y is a strongly bounded operator with representing measure m : Σ → L(X, Y ) and T : B(Ω, X) → Y is the restriction of T * * to B(Ω, X), then is T Dunford-Pettis p-convergent if and only if T is Dunford-Pettis p-convergent? These kind of researches have done by many authors for different operators, see [3,4,6,7,17,21]. Here, we try answer to the above questions.…”

“…Suppose that T : C(Ω, X) → Y is a strongly bounded operator with representing measure m : Σ → L(X, Y ) and T : B(Ω, X) → Y is its extension. The authors in [3,4,5,6,7,17,21] have found that study of T is more suitable than T. Our aim in this section is to obtain some characterizations of Dunford-Pettis p-convergent operators in terms of their representing measure. By using the same argument as in the proof of ([3, Theorem 3]), we obtain the following result.…”

“…The proof of the following result is similar to the proof of Theorem 4.4, and will be omitted. Let us recall from [21], that if T : C(Ω, X) → Y is a bounded linear operator, Ω is a metrizable compact space, and π : Ω → Ω a continuous map which is onto, we will call Ω a quotient of Ω. The map π : C(Ω) → C(Ω) given by π(f ) = f • π defines an isometric embedding of C(Ω) into C(Ω).…”

“…5. 3])), by using Lemmas 18 and 20 in [21], we can suppose without loss of generality that Ω is metrizable and so, Ω is countable (see ([26, 8. 5.…”