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We consider spectral projectors associated to the Euclidean Laplacian on the two-dimensional torus, in the case where the spectral window is narrow. Bounds for their L2 to Lp operator norm are derived, extending the classical result of Sogge; a new question on the convolution kernel of the projector is introduced. The methods employed include 
$\ell^2$ decoupling, small cap decoupling and estimates of exponential sums.
We investigate norms of spectral projectors on thin spherical shells for the Laplacian on tori. This is closely related to the boundedness of resolvents of the Laplacian and the boundedness of 
$L^{p}$ norms of eigenfunctions of the Laplacian. We formulate a conjecture and partially prove it.
