Banach Compactness and Banach Nuclear Operators
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Results in Mathematics
Banach Compactness and Banach Nuclear Operators Ju Myung Kim, Keun Young Lee, and Bentuo Zheng Abstract. In this paper, we introduce the notion of (uniformly weakly) Banach-compact sets, (uniformly weakly) Banach-compact operators and (uniformly weakly) Banach-nuclear operators which generalize p-compact sets, p-compact operators and p-nuclear operators, respectively. Fundamental properties are investigated. Factorizations and duality theorems are given. Injective and surjective hulls are used to show the spaces of (uniformly weakly) Banach-compact operators and (uniformly weakly) Banach-nuclear operators are quasi Banach operator ideals. Mathematics Subject Classification. 47L05, 47L20. Keywords. Quasi Banach operator ideal, nuclear operator, compact operator, injective hull, surjective hull.
1. Introduction Compactness is a topological property which also has geometric characterizations in Banach spaces. One of the most famous such characterizations is probably the Grothendieck’s criterion [6] which can be stated as the following: A subset K of a Banach space X is relatively compact if and only if for every ε > 0, there exists (xn )n ∈ c0 (X) with (xn )n ∞ ≤ supx∈K x + ε such that ∞ αn xn : (αn )n ∈ B1 , K⊂ n=1
Ju Myung Kim was supported by NRF-2018R1D1A1B07043566 (Korea) Keun Young Lee was supported by NRF-2017R1C1B5017026 (Korea) Bentuo Zheng was supported in part by Simons Foundation Grant 585081. 0123456789().: V,-vol
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where c0 (X) is the space of sequences in X which converge to 0 in norm. We denote by BZ the closed unit ball of a Banach space Z. Motivated by this, Sinha and Karn [12] introduced the notion of pcompact sets. Let 1 ≤ p < ∞ and let 1/p + 1/p∗ = 1. A subset K of X is said to be p-compact if there exists (xn )n ∈ p (X), which is the Banach space with the norm · p of all X-valued absolutely p-summable sequences, such that ∞ αn xn : (αn ) ∈ Bp∗ . K ⊂ p-co(xn )n := n=1
Every p-compact set is relatively compact. One goal of this paper is to develop the notion of a generalized compactness for subsets of Banach spaces which we will call Banach-compactness. Such compactness will be used to generalize p-compact operators. An operator T : Y → X is called p-compact if there exists (xn )n ∈ p (X) such that T (BY ) ⊂ p-co(xn )n . The set of p-compact operators from Y into X is denoted by Kp (Y, X) and the Kp -norm is given by T Kp := inf (xn )n p : (xn )n ∈ p (X), T (BY ) ⊂ p-co(xn )n . Delgado et al. [3,4] showed that [Kp , · Kp ] is a Banach operator ideal. Let 1 ≤ p ≤ ∞. An operator T : X → Y is called p-nuclear if there exist (x∗n )n ∈ p (X ∗ ) (c0 (X ∗ ) when p = ∞) and (yn )n ∈ w p∗ (Y ) (c0 (Y ) when p = 1) such that ∞ x∗n ⊗yn , T = n=1 ∗ where w p∗ (Y ) is the Banach space of all Y -valued weakly p -summable sew ∗ quences with the norm · p∗ . Here xn ⊗yn is an operator from X to Y defined by (x∗n ⊗yn )(x) = x∗n (x)yn . The space of all p-nuclear operators from X to Y is denoted by Np
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