Spectral Properties of the Operators $AB$ and $BA$

For linear bounded operators $A$, $B$ from the Banach algebra of linear bounded operators acting in a Banach space, we prove a number of statements on the coincidence of the properties of the operators $I_{Y}-AB$, $I_{X}-BA$ related to their kernels and images. In particular, we establish the identical dimension of the kernels, their simultaneous complementability property, the coincidence of the codimensions of the images, their simultaneous Fredholm property and the coincidence of their Fredholm indices. We construct projections onto the image and the kernel of these operators. We establish the simultaneous nonquasianalyticity property of the operators $AB$ and $BA$.