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<p>This monograph proposes a unified theory of the calculus of fractional and standard derivatives by means of an abstract operator-theoretic approach. By highlighting the axiomatic properties shared by standard derivatives, Riemann-Liouville and Caputo derivatives, the author introduces two new classes of objects. The first class concerns differential triplets and differential quadruplets; the second concerns boundary restriction operators. Instances of &nbsp;boundary restriction operators can be generalized fractional differential operators supplemented with homogeneous boundary conditions. The analysis of these operators comprises:</p>

<ul>
<li>The computation of adjoint operators;</li>
<li>The definition of abstract boundary values;</li>
<li>The solvability of equations supplemented with inhomogeneous abstract linear boundary conditions;</li>
<li>The analysis of fractional inhomogeneous Dirichlet Problems.</li>
</ul>

<p>As a result of this approach, two striking consequences are highlighted: Riemann-Liouville and Caputo operators appear to differ only by their boundary conditions; and the boundary values of functions in the domain of fractional operators are closely related to their kernel.</p>

<p><em>Unified Theory for Fractional and Entire Differential Operators</em> will appeal to researchers in analysis and those who work with fractional derivatives. It is mostly self-contained, covering the necessary background in functional analysis and fractional calculus.</p>