Non-Isolation of weighted composition operators on L^p(μ)(1 ≤ p ≤ ∞ ) spaces

Assuming X is a compact Hausdorff space and (X) denotes the Banach algebra of continuous complex-valued functions on X with the supremum norm, comp (X) refers to the collection of all composition operators on (X). We show that each composition operator is isolated in comp (X) under the norm topology as well as the strong operator topology. In addition, we demonstrate that every weighted composition ope Assuming X is a compact Hausdorff space and (X) denotes the Banach algebra of continuous complex-valued functions on X with the supremum norm, comp (X) refers to the collection of all composition operators on (X). We show that each composition operator is isolated in comp (X) under the norm topology as well as the strong operator topology. In addition, we demonstrate that every weighted composition operator on Lp( (1 p ) spaces is non-isolated under the norm topology. Mathematics subject classification (2010): 47L25; 47A30; 47A10 rator on Lp( (1 p ) spaces is non-isolated under the norm topology. Mathematics subject classification (2010): 47L25; 47A30; 47A10