Representation of Compact Operators between Banach spaces

In this article, we give a representation for compact operators acting between reflexive Banach spaces, which generalizes the representation given by Edmunds et al. for compact operators between reflexive Banach spaces with strictly convex duals. Further, we give a representation for operators on Banach spaces that are comparable to compact normal operators on Hilbert spaces and illustrate our result with an example.Comment: 16 Pages, submitted to a journal. Comments/suggestions are welcom

Absolutely norm attaining Toeplitz and absolutely minimum attaining Hankel operators

Let H1 and H2 be complex Hilbert spaces. A bounded linear operator T:H1→H2 is called norm attaining if ‖Tx‖=‖T‖ for some unit vector x∈H1. If for every closed subspace M of H1, the restriction T|M:M→H2 is norm attaining, then T is called an absolutely norm attaining operator (or AN-operator). In the above definitions, if we replace the norm of the operator by the minimum modulus m(T)=inf⁡{‖Tx‖:x∈H1,‖x‖=1}, then T is called a minimum attaining and an absolutely minimum attaining operator (or AM-operator), respectively. In this article, we characterize Toeplitz AN-operators and discuss a few results on the minimum modulus of Toeplitz operator Tφ, φ∈L∞(T). We further characterize the minimum attaining Hankel operators and deduce that the only Hankel AM-operators are finite rank operators. While proving our results, we also obtained the following result; If φ∈L∞(T), then m(Lφ)=ess inf|φ| and there exists ψ∈L∞(T) such that γ(Lψ)>ess inf|ψ|, which improves a result from [15]. © 2022 Elsevier Inc

On the closure of absolutely norm attaining operators

Let H-1 and H-2 be complex Hilbert spaces and T : H-1 -> H-2 be a bounded linear operator. We say T is norm attaining if there exists x is an element of H-1 with parallel to x parallel to = 1 such that parallel to Tx parallel to = parallel to T parallel to. If for every non-zero closed subspaceMof H-1, the restriction T|(M) : M -> H-2 is norm attaining, then T is called an absolutely norm attaining operator or ANoperator. If we replace the norm of the operator by the minimum modulus m(T) = inf {parallel to Tx parallel to : x is an element of H-1, parallel to x parallel to = 1} in the above definitions, then T is called a minimum attaining and an absolutely minimumattaining operator orAM-operator, respectively. In this article, we discuss the operator norm closure of AN-operators. We completely characterize operators in this closure and study several important properties. Wemainly give a spectral characterization of positive operators in this class and give a representation when the operator is normal. Later, we also study the analogous properties for AMoperators and prove that the closure ofAM-operators is the same as the closure ofAN-operators. Consequently, we prove similar results for operators in the norm closure ofAM-operators

Antibacterial and Antibiofilm Activity of Cationic Small Molecules with Spatial Positioning of Hydrophobicity: An in Vitro and in Vivo Evaluation

More than 80% of the bacterial infections are associated with biofilm formation. To combat infections, amphiphilic small molecules have been developed as promising antibiofilm agents. However, cytotoxicity of such molecules still remains a major problem. Herein we demonstrate a concept in which antibacterial versus cytotoxic activities of cationic small molecules are tuned by spatial positioning of hydrophobic moieties while keeping positive charges constant. Compared to the molecules with more pendent hydrophobicity from positive centers (MIC = 1–4 μg/mL and HC₅₀ = 60–65 μg/mL), molecules with more confined hydrophobicity between two centers show similar antibacterial activity but significantly less toxicity toward human erythrocytes (MIC = 1–4 μg/mL and HC₅₀ = 805–1242 μg/mL). Notably, the optimized molecule is shown to be nontoxic toward human cells (HEK 293) at a concentration at which it eradicates established bacterial biofilms. The molecule is also shown to eradicate preformed bacterial biofilm <i>in vivo</i> in a murine model of superficial skin infection