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The role played by positivity property of differential and difference operators in Hilbert and Banach spaces in the study of various properties of boundary value problems for partial differential equations, of stability of difference schemes for partial differential equations, and of summation Fourier series is well-known (see, [1]-[3]). This is a review paper on results for fractional spaces generated by positive operators. Its scope ranges from theory of differential and difference operators in a space to operators with local and nonlocal boundary conditions. We also discuss their applications to partial differential equations and theory of difference schemes for partial differential equations.