Image pansharpening can generate a high-resolution hyperspectral (HS) image by combining a high-resolution panchromatic image and a HS image. In this paper, we propose a variational pansharpening method for HS imagery constrained by spectral shape and Gram-Schmidt (GS) transformation. The main novelties of the proposed method are the additional spectral and correlation fidelity terms. First, we design the spectral fidelity term, which utilizes the spectral shape feature of the neighboring pixels with a new weight distribution strategy to reduce spectral distortion caused by the change in spatial resolution. Second, we consider that the correlation fidelity term uses the result of GS adaptive (GSA) to constrain the correlation, thereby preventing the low correlation between the pansharpened image and the reference image. Then, the pansharpening is formulized as the minimization of a new energy function, whose solution is the pansharpened image. In comparative trials, the proposed method outperforms GSA, guided filter principal component analysis, modulation transfer function, smoothing filter-based intensity modulation, the classic and the band-decoupled variational methods. Compared with the classic variation pansharpening, our method decreases the spectral angle from 3.9795 to 3.2789, decreases the root-mean-square error from 309.6987 to 228.6753, and also increases the correlation coefficient from 0.9040 to 0.9367.