About the Legendre type operators

The article considers Legendre type operators acting in the corresponding weight separable Hilbert spaces. The choice of these spaces is due to the fact that these operators preserve all properties of the Legendre operator acting in a space with a summable square on the segment [-1,1]. In particular, 1) the Legendre type operators, operating in respective weight separable Hilbert spaces, remain discrete, 2) the spectrum does not change relative to the classical Legendre operator, 3) the corresponding eigenfunctions are compositions of Legendre polynomials and some functions.