  We consider a perturbative expansion of the Lanczos coefficients and the
Krylov complexity for two-dimensional conformal field theories under integrable
deformations. Specifically, we explore the consequences of $T{\bar{T}}$,
$J{\bar{T}}$, and $J{\bar{J}}$ deformations, focusing on first-order
corrections in the deformation parameter. Under $T\bar{T}$ deformation, we
demonstrate that the Lanczos coefficients $b_n$ exhibit unexpected behavior,
deviating from linear growth within the valid perturbative regime. Notably, the
Krylov exponent characterizing the rate of exponential growth of complexity
surpasses that of the undeformed theory for positive value of deformation
parameter, suggesting a potential violation of the conjectured operator growth
bound within the realm of perturbative analysis. One may attribute this to the
existence of logarithmic branch points along with higher order poles in the
autocorrelation function compared to the undeformed case. In contrast to this,
both $J{\bar{J}}$ and $J{\bar{T}}$ deformations induce no first order
correction to either the linear growth of Lanczos coefficients at large-$n$ or
the Krylov exponent and hence the results for these two deformations align with
those of the undeformed theory.
