On the quasi-similarity of operators with flag structure

Let $\mathcal{A}$ denote the operator class in which every nonzero intertwiner between two operators in $\mathcal{A}$ has dense range. Utilizing the operators in $\mathcal{A}$ as atoms and the flag structure as connection, we introduce an extended operator class $\mathcal{F}_{n}(\mathcal{A}) (n\in\mathbb{N}\ \mbox{and}\ n\ge2)$, along with its subclass $\mathcal{OF}_{n}(\mathcal{A})$. We establish that, under certain conditions, quasi-similarity within the classes $\mathcal{F}_{n}(\mathcal{A})$ and $\mathcal{OF}_{n}(\mathcal{A})$ 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 $\mathcal{F}_{n}(\mathcal{A})$, 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 $\mathcal{F}_{n}(\mathcal{A})$. This offers a partial solution to C.L. Jiang's question.