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A Fourier multiplier operator $T_m$ is defined by $\widehat{T_{m}f}(\xi )=m(\xi )\hat{f}(\xi )$. On $L^2$, a bounded symbol gives a bounded operator by the Plancherel theorem, while an unbounded symbol defines a closed linear operator on the functions for which $m\hat{f}$ remains square-integrable.

The graph norm of $A:D(A)\subseteq X\to Y$ is $\Vert x{\Vert }_A=\Vert x{\Vert }_X+\Vert Ax{\Vert }_Y$. If $X$ and $Y$ are Banach spaces, then $A$ is a closed linear operator exactly when its domain is complete in the graph norm.

A nonautonomous abstract Cauchy problem has the form $u^{\prime }(t)=A(t)u(t)$, with a time-dependent generally closed linear operator. Under stability, common-domain, and regularity hypotheses, it is propagated by an evolution family.

The infinitesimal generator of a semigroup is the linear operatorwith generator domainFor example, the Bochner integralbelongs to $D(A)$ for every $u\in H$ and $t>0$, since $A{u}_t=U(t)u-u$.

To prove that $A$ is a closed linear operator, suppose $u_n\in D(A)$, $u_n\to u$, and $A{u}_n\to v$. For vectors in the generator domain,Passing to the limit in the Banach space givesAfter division by $t$, strong continuity makes the right side converge to $v$ as $t\downarrow 0$. Hence $u\in D(A)$ and $Au=v$, so $A$ is closed.

For $\mathrm{Re}z>\omega$, define the Bochner integralIt converges absolutely becauseIntegrating the semigroup difference quotient shows that $R(z)u\in D(A)$ and $(zI-A)R(z)u=u$. The same computation for $u\in D(A)$ gives $R(z)(zI-A)u=u$. Thus the Laplace-transform formula for a semigroup resolvent proves

Let $(u_n)$ be a Cauchy sequence in the graph normThen $(u_n)$ and $(A{u}_n)$ are Cauchy sequences in the Banach space $H$, so for some $u,v\in H$,Because $A$ is a closed linear operator, $u\in D(A)$ and $Au=v$. It follows thatThus every Cauchy sequence converges in $Y=(D(A),\Vert \cdot {\Vert }_Y)$, and

The familyinherits the identity, semigroup property, and strong continuity from $U$, whileIts difference quotient satisfiesfor $u\in D(A)$. Conversely, existence of this limit implies existence of the generator limit for $U$, so $D(\hat{A})=D(A)$ andThis is the exponentially shifted semigroup construction.

The Hille-Yosida theorem in the uniformly bounded case says that a linear operator $B$ on a Banach space generates a C0-semigroup with $\Vert e^{tB}\Vert \le M$ if and only if: