Abstract Cauchy Problem of Conformable Type and Operator Ideals in Hilbert Space

In this paper, we focus our study on the fractional homogeneous abstract Cauchy problems of conformable type, w β ( t ) = U w ( t ), t ∈ [0, T ), T ≤ ∞, w (0) = x , where U is a linear operator that generates a fractional β ‐semigroup on a Hilbert space H . In fact, we gave an answer to the question if U is the infinitesimal generator of a conformable β ‐semigroup { φ ( t )} t ≥0 of bounded linear operators on a Hilbert space H , under what conditions { φ ( t )} t ≥0 can be in the Schatten class C p ( H ). Moreover, we prove the validity of inequality of resolvent operator R ( λ , U ) of the operator U in the Schatten norm.