  Integrable Floquet spin chains are known to host strong zero and $\pi$ modes
which are boundary operators that respectively commute and anticommute with the
Floquet unitary generating stroboscopic time-evolution, in addition to
anticommuting with a discrete symmetry of the Floquet unitary. Thus the
existence of strong modes imply a characteristic pairing structure of the full
spectrum. Weak interactions modify the strong modes to almost strong modes that
almost commute or anticommute with the Floquet unitary. Manifestations of
strong and almost strong modes are presented in two different Krylov subspaces.
One is a Krylov subspace obtained from a Lanczos iteration that maps the
time-evolution generated by the Floquet Hamiltonian onto dynamics of a single
particle on a fictitious chain with nearest neighbor hopping. The second is a
Krylov subspace obtained from the Arnoldi iteration that maps the
time-evolution generated directly by the Floquet unitary onto dynamics of a
single particle on a fictitious chain with longer range hopping. While the
former Krylov subspace is sensitive to the branch of the logarithm of the
Floquet unitary, the latter obtained from the Arnoldi scheme is not. The
effective single particle models in the Krylov subspace are discussed, and the
topological properties of the Krylov chain that ensure stable $0$ and $\pi$
modes at the boundaries are highlighted. The role of interactions is discussed.
Expressions for the lifetime of the almost strong modes are derived in terms of
the parameters of the Krylov subspace, and are compared with exact
diagonalization.
