In　this　paper　we　consider　the　initial　value　problem　to　the　fractional　compressible　isentropic　generalized　Navier-Stokes　equations　for　viscous　fluids　with　one　Levy　diffusion　process　in　which　the　viscosity　term　appeared　in　the　fluid　equations　is　described　by　the　nonlocal　fractional　Laplace　operator.　We　give　one　detailed　spectrum　analysis　on　a　linearized　operator　and　the　decay　law　in　time　of　the　solution　semigroup　for　the　linearized　fractional　compressible　isentropic　generalized　Navier-Stokes　equations　around　a　constant　state　by　the　Fourier　analysis　technique,　which　is　shown　that　the　order　of　the　fractional　derivatives　plays　a　key　role　in　the　analysis　so　that　the　spectrum　structure　involved　here　is　more　complex　than　that　of　the　classical　compressible　Navier-Stokes　system.　Based　on　this　and　the　elaborate　energy　method,　the　global-in-time　existence　and　one　optimal　decay　rate　in　time　of　the　smooth　solution　are　obtained　under　the　assumption　that　the　initial　data　are　given　in　a　small　neighborhood　of　a　constant　state.　©　2023　Elsevier　Inc.