# General explicit expressions for intertwining operators and direct rotations of two orthogonal projections

## TL;DR

This paper derives explicit formulas for intertwining operators and direct rotations of orthogonal projections, improving upon existing theoretical results using block operator and spectral theory techniques.

## Contribution

It introduces general explicit expressions for these operators, advancing the theoretical understanding beyond previous theorems by Kato, Avron et al., and Davis and Kahan.

## Key findings

- Explicit formulas for intertwining operators
- Explicit formulas for direct rotations
- Improved theoretical results over prior theorems

## Abstract

In this paper, based on the block operator technique and operator spectral theory, the general explicit expressions for intertwining operators and direct rotations of two orthogonal projections have been established. As a consequence, it is an improvement of Kato's result (Perturbation Theory of Linear operators, Springer-Verlag, Berlin/Heidelberg, 1996); J. Avron, R. Seiler and B. Simon's Theorem 2.3 (The index of a pair of projections, J. Funct. Anal. 120(1994) 220-237) and C. Davis, W.M. Kahan, (The rotation of eigenvectors by a perturbation, III. SIAM J. Numer. Anal. 7(1970) 1-46).