On the quasi-similarity of operators with flag structure

Let A denote the operator class in which every nonzero intertwiner between two operators in A has dense range. Utilizing the operators in A as atoms and the flag structure as connection, we introduce an extended operator class Fₙ(A) (n\ and\ n≥2), along with its subclass OFₙ(A). We establish that, under certain conditions, quasi-similarity within the classes Fₙ(A) and OFₙ(A) is equivalent, which provides an approach to describing quasi-similarity and similarity for high-index Fredholm operators. Furthermore, we demonstrate that quasi-similarity implies similarity for a large number of operators in Fₙ(A), thereby yielding a partial answer to the question raised by D.A. Herrero and generalizing existing numerous results. As applications, several examples of quasi-similarity and similarity involving multiplication operators on vector-valued reproducing kernel Hilbert spaces are presented. Lastly, we show that the strong irreducibility is preserved up to quasi-similarity within the class Fₙ(A). This offers a partial solution to C.L. Jiang's question.