# A note on triangular operators on Smooth Sequence Spaces

## TL;DR

This paper investigates conditions under which triangular operators, defined by specific matrices, are linear and continuous between certain nuclear K"othe spaces, extending understanding of their properties and transpose behavior.

## Contribution

It introduces necessary and sufficient conditions for the linearity and continuity of Cauchy Product maps between nuclear K"othe spaces and their transposes.

## Key findings

- Established criteria for linearity of Cauchy Product maps
- Determined conditions for continuity of these operators
- Analyzed the transpose of the operators in this context

## Abstract

For a scalar sequence {(\theta_n)}_{n \in \mathbb{N}}, let C be the matrix defined by c_n^k = \theta_{n-k+1} if n > k, c_n^k = 0 if n < k. The map between K\"{o}the spaces \lambda(A) and \lambda(B) is called a Cauchy Product map if it is determined by the triangular matrix C. In this note we introduced some necessary and sufficient conditions for a Cauchy Product map on a nuclear K\"{o}the space \lambda(A) to nuclear G_1-space \lambda(B) to be linear and continuous. Its transpose is also considered.