In this paper, we study the norm and almost everywhere convergence and divergence questions regarding subsequences of the general matrix-based T means \(\sigma _{n}^{T}(f)\) of the Walsh-Fourier series. We give sufficient conditions that are in a certain sense optimal for convergence for every integrable function f, namely \(\begin{aligned} \sum _{k=1}^{n}t_{k,n}^{2}=O\left( \frac{1}{n}\right) . \end{aligned}\) In comparison with previous results in this topic, we do not use the monotonicity of the defining sequences \((t_{k,n},1\le k\le n, k\in \mathbb {P})\) of T. The T summation is a common generalization of several well-known summation methods, such as Fejér, Cesàro, Weierstrass, Riesz, Picard and Bessel, and Nörlund summation methods.
