[Paper Review] Approximation of operators in Banach spaces

O. I. Reinov|arXiv (Cornell University)|Feb 20, 2010

Advanced Banach Space Theory7 references3 citations

TL;DR

This paper investigates the approximation of absolutely $ p $-summing and quasi-$ p $-nuclear operators in Banach spaces using the $ \pi_p $-compact convergence topology. It establishes that for $ p \neq 2 $, there exist separable reflexive Banach spaces where finite-rank operators are not dense in the space of quasi-$ p $-nuclear operators under the $ \pi_p $-norm, while they are dense in the space of absolutely $ p $-summing operators under the $ \tau_p $-topology. The key contribution is the construction of a counterexample to the density of finite-rank operators in $ \operatorname{QN}_p $, resolving a question on approximation properties in non-2-summable settings.

ABSTRACT

It is a translation of an old paper of mine. We describe the topology tau_p in the space Pi_p(Y,X), for which the closures of convex sets in tau_p and in *-weak topology of the space Pi_p(Y,X) are coincident. Thereafter, we investigate some properties of the space Pi_p, related to this new topology. 2010-remark: Occasionally, the topology is coincides with the lambda_p-topology from the paper "Compact operators which factor through subspaces of l_p", Math. Nachr. 281(2008), 412-423 by Deba Prasad Sinha and Anil Kumar Karn.

Motivation & Objective

• To investigate the density of finite-dimensional operators in the space of quasi-$ p $-nuclear operators under the $ \pi_p $-norm topology.
• To characterize the dual space of the space of absolutely $ p $-summing operators under the $ \tau_p $-topology of $ \pi_p $-compact convergence.
• To resolve a question posed by P. Saphar on the coincidence of weak and $ \tau_p $-topology closures for convex sets in $ \Pi_p(Y,X) $.
• To construct explicit counterexamples showing that the approximation property $ \operatorname{AP}_p $ does not imply density of finite-rank operators in $ \operatorname{QN}_p $ for $ p \neq 2 $.

Proposed method

• Introduces the $ \tau_p $-topology on $ \Pi_p(Y,X) $ as the topology of $ \pi_p $-compact convergence, defined by local bases $ \omega_{K,\varepsilon} = \{ U \in \Pi_p(Y,X) : \pi_p(U\Phi_K) < \varepsilon \} $.
• Uses the canonical embedding $ \Phi_K: X_K \to X $ for compact subsets $ K \subset Y $, and studies the action of operators on the associated Banach space $ X_K $.
• Applies duality theory for operator ideals, identifying $ (\operatorname{R}, \tau_p)' $ with a quotient of $ X^* \widehat{\otimes}_{p'} Y $ via trace functionals.
• Constructs a tensor element $ z \in Z^* \widehat{\otimes}_{p'} Z $ such that $ \operatorname{trace}(\Psi \circ z) = 1 $ for some $ \Psi \in \Pi_p(Z,Z^{**}) $, but $ \operatorname{trace}(\Phi \circ z) = 0 $ for all $ \Phi \in \Pi_p(Z,Z) $, using a direct sum construction of Banach spaces.
• Employs the injectivity of the ideal $ \operatorname{QN}_p $ and the representation of duals via strictly $ p' $-integral operators to embed functionals into tensor products.
• Uses the non-injectivity of the natural map $ Y^* \widehat{\otimes}_{p'} X \to \operatorname{N}_{p'}(Y,X) $ to construct reflexive spaces where $ \overline{X^* \otimes Y}^{\tau_p} = X^* \widehat{\widehat{\otimes}}_p Y $, showing minimal closure in the $ w^* $-topology.

Experimental results

Research questions

• RQ1 Is the closure of finite-rank operators in $ \operatorname{QN}_p(Y,X) $ under the $ \pi_p $-norm topology equal to the entire space for all $ p \in [1,\infty] $?
• RQ2 Does the $ \tau_p $-topology on $ \Pi_p(Y,X) $ yield the same closure of convex sets as the weak* topology?
• RQ3 Can one construct a Banach space $ Z $ and a tensor $ z \in Z^* \widehat{\otimes}_{p'} Z $ such that $ \operatorname{trace}(\Psi \circ z) = 1 $ for some $ \Psi \in \Pi_p(Z,Z^{**}) $, but $ \operatorname{trace}(\Phi \circ z) = 0 $ for all $ \Phi \in \Pi_p(Z,Z) $?
• RQ4 For which $ p \in [1,\infty] $ does the equality $ \overline{X^* \otimes Y}^{\tau_p} = X^* \widehat{\widehat{\otimes}}_p Y $ hold for reflexive spaces $ X,Y $?

Key findings

• For every $ p \in [1,\infty] $, $ p \neq 2 $, there exists a separable reflexive Banach space $ E $ and a separable conjugate space $ H $ with a basis such that $ \operatorname{QN}_p(H,E) \neq \overline{H^* \otimes E}^{\pi_p} $, showing finite-rank operators are not dense in $ \operatorname{QN}_p $.
• For every $ p \in [1,\infty) $, there exist separable reflexive Banach spaces $ X $ and $ Y $ such that $ \overline{X^* \otimes Y}^{\tau_p} = X^* \widehat{\widehat{\otimes}}_p Y $, indicating the closure of finite-rank operators in the $ \tau_p $-topology is minimal in the $ w^* $-topology.
• There exists a tensor element $ z \in Z^* \widehat{\otimes}_{p'} Z $ and an operator $ \Psi \in \Pi_p(Z,Z^{**}) $ such that $ \operatorname{trace}(\Psi \circ z) = 1 $, but $ \operatorname{trace}(\Phi \circ z) = 0 $ for all $ \Phi \in \Pi_p(Z,Z) $, proving that the trace functional does not detect non-zero elements in the dual of $ \Pi_p(Z,Z) $.
• The dual space of $ (\operatorname{R}, \tau_p) $, where $ \operatorname{R} \supset Y^* \otimes X $, is isomorphic to a quotient of $ X^* \widehat{\otimes}_{p'} Y $, with functionals represented as traces $ \varphi(U) = \operatorname{trace}(U \circ z) $.
• The space $ \Pi_p(Y,X) $ is equal to the $ \tau_p $-closure of $ Y^* \otimes X $, confirming that $ \tau_p $-topology yields the correct closure for convex sets in $ \Pi_p(Y,X) $, answering a question of P. Saphar.
• For $ p = \infty $, a non-zero tensor $ z \in Z^* \widehat{\otimes}_1 Z $ exists such that $ \operatorname{trace}(\Phi \circ z) = 0 $ for all $ \Phi \in \operatorname{L}(Z,Z) $, resolving a question posed by Sten Kaijser.