  Inspired by the universal operator growth hypothesis, we extend the formalism
of Krylov construction in dissipative open quantum systems connected to a
Markovian bath. Our construction is based upon the modification of the
Liouvillian superoperator by the appropriate Lindbladian, thereby following the
vectorized Lanczos algorithm and the Arnoldi iteration. This is well justified
due to the incorporation of non-Hermitian effects due to the environment. We
study the growth of Lanczos coefficients in the transverse field Ising model
(integrable and chaotic limits) for boundary amplitude damping and bulk
dephasing. Although the direct implementation of the Lanczos algorithm fails to
give physically meaningful results, the Arnoldi iteration retains the generic
nature of the integrability and chaos as well as the signature of
non-Hermiticity through separate sets of coefficients (Arnoldi coefficients)
even after including the dissipative environment. Our results suggest that the
Arnoldi iteration is meaningful and more appropriate in dealing with open
systems.
