Matrix Transformations of Double Convergent Sequences with Powers for the Pringsheim Convergence

In 2004-2006, the corresponding double sequence spaces were defined for the Pringsheim and the bounded Pringsheim convergence by Gokhan and Colak. In 2009, Colak and Mursaleen characterized some classes of matrix transformations transforming the space of bounded Pringsheim convergent (to 0) double sequences with powers and the space of uniformly bounded double sequences with powers to the space of (bounded) Pringsheim convergent (to 0) double sequences. But many of their results appeared to be wrong. In 2024, we gave corresponding counterexamples and proved the correct results. Moreover, we gave the conditions for a wider class of matrices. As is well known, convergence of a double sequence in Pringsheim's sense does not imply its boundedness. Assuming, in addition, boundedness for double sequences usually simplifies proofs. In this paper, we characterize matrix transformations transforming the space of Pringsheim convergent (to 0) double sequences with powers or the space of ultimately bounded double sequences with powers without assuming uniform boundedness.