We present a tomographic weak lensing analysis of the Kilo Degree Survey Data Release 4 (KiDS-1000), using a new pseudo angular power spectrum estimator (pseudo-Cℓ) under development for the ESAEuclidmission. Over 21 million galaxies with shape information are divided into five tomographic redshift bins, ranging from 0.1 to 1.2 in photometric redshift. We measured pseudo-Cℓusing eight bands in the multipole range 76 < ℓ < 1500 for auto- and cross-power spectra between the tomographic bins. A series of tests were carried out to check for systematic contamination from a variety of observational sources including stellar number density, variations in survey depth, and point spread function properties. While some marginal correlations with these systematic tracers were observed, there is no evidence of bias in the cosmological inference.B-mode power spectra are consistent with zero signal, with no significant residual contamination fromE/B-mode leakage. We performed a Bayesian analysis of the pseudo-Cℓestimates by forward modelling the effects of the mask. Assuming a spatially flat ΛCDM cosmology, we constrained the structure growth parameterS8 = σ8(Ωm/0.3)1/2 = 0.754−0.029+0.027. When combining cosmic shear from KiDS-1000 with baryon acoustic oscillation and redshift space distortion data from recent Sloan Digital Sky Survey (SDSS) measurements of luminous red galaxies, as well as the Lyman-αforest and its cross-correlation with quasars, we tightened these constraints toS8 = 0.771−0.032+0.006. These results are in very good agreement with previous KiDS-1000 and SDSS analyses and confirm a ∼3σtension with early-Universe constraints from cosmic microwave background experiments.