Properties of analogues of Frobenius powers of ideals

Let $R=\mathbb{K}[X_1,\dots ,X_n]$ be a polynomial ring over a field $\mathbb{K}$. We introduce an endomorphism ${\mathcal{F}}^{[m]}:R\to R$ and denote the image of an ideal $I$ of $R$ via this endomorphism as $I^{[m]}$ and call it to be the $m$ $\mathit{\text{-th}}\mathit{\text{square}}\mathit{\text{power}}$ of $I$. In this article, we study some homological invariants of $I^{[m]}$ such as regularity, projective dimension, associated primes and depth for some families of ideals e.g. monomial ideals.