Properties of Multiplication Operators on K\"othe spaces

TL;DR

This paper provides a comprehensive analysis of multiplication operators on K"othe spaces, characterizing their properties, spectra, and algebraic structure, and calculating key operator norms and spectra.

Contribution

It offers a complete characterization of various properties of multiplication operators on K"othe spaces, including boundedness, invertibility, spectrum, and algebraic structure, which was not previously fully explored.

Findings

Characterized bounded, injective, onto, bijective, Fredholm, compact, and closed-range multiplication operators.

Computed the spectrum, powers, and essential norm of these operators.

Analyzed the algebraic structure of the set of multiplication operators.

Abstract

We make an exhaustive study of the properties of multiplication operator $M_u$ acting on K\"othe spaces. We characterize the multiplication operators, acting on K\"othe spaces, which are: bounded, injective, onto, bijective, Fredholm, compact, with closed range and with range finite. We also study the spectrum, the $n$-power and calculate the essential norm of $M_u$ and we study the algebra of multiplication operators.