Some remarks on C-semigroups and integrated semigroups.

*(English)*Zbl 0642.47034Let X be a Banach space and let B(X) be the set of all bounded linear operators from X into itself. Let C be an injective operator in B(X) and the range R(C) be dense in X.

A family \(\{\) S(t);t\(\geq 0\}\) in B(X) is called C-semigroup, if

I) \(S(t+s)C=S(t)S(s)\) for t,s\(\geq 0\), \(S(0)=C,\)

II) \(S(\cdot)X:(0,+\infty)\to X\) is continuous for \(x\in X,\)

III) there are \(M\geq 0\) and \(a\in R\) such that \(\| s(t)\| Me^{at}\) for \(t\geq 0.\)

If \(G(x)=\lim_{t\to 0^+}(C^{-1}S(t)x-x)/t\) for \(x\in D(G)\), it is known that G is densely defined and closable, the closure \(\bar G\) is called the C-c.i.g of \(\{\) S(t);t\(\geq 0\}.\)

Let n be a positive integer, a family \(\{\) U(t):t\(\geq 0\}\) in B(X) is called n-times integrated semigroup, if

I) \(U(\cdot)x:\{0,+\infty)\to X\) is continuous for \(x\in X.\)

II) \(U(t)U(s)x=1/(n-1)!(\int^{t+s}_{t}(s+t-r)^{n-1}U(r)xdr- \int^{s}_{0}(s+t-r)^{n-1}U(r)xdr)\) for \(x\in X\) and s,t\(\geq 0\) and \(U(0)=0,\)

III) \(U(t)x=0\), for all \(t>0\), implies \(x=0,\)

IV) there are \(M\geq 0\) and \(\omega\in R\) such that \(\| U(t)\| \leq Me^{\omega t}\) for \(t\geq 0.\)

A very good result about the relation between these two kinds of semigroups is the following:

Theorem. Let A be a densely defined closed linear operator in X with \(\rho\) (A)\(\neq \emptyset\). Let \(c\in \rho (A)\) and \(n\geq 0\) be an integer. The following (i)-(iii) are equivalent:

i) A is the generator of an n-times integrated semigroup \(\{\) U(t);t\(\geq 0\},\)

ii) A is the C-c.i.g of a C-semigroup \(\{\) S(t);t\(\geq 0\}\) with \(C=R(C;A)^ n,\)

iii) There exist \(M\geq 0\) and \(a\in R\) such that (a,\(\infty)\subset \rho (A)\) and \(\| R((\lambda;A)^ mR(c;A)^ n\| \leq M/(\lambda -a)^ m\) for \(m\geq 1\) and \(\lambda >a.\)

In this case, we have \[ U(t)x=(c-A)^ n\int^{t}_{0}\int^{t_ 1}_{0}...\int^{t_{n-1}}_{0}s(t_ n)xdt_ n...dt_ 2dt_ 1 \] for \(x\in X\) and \(t\geq 0\).

A family \(\{\) S(t);t\(\geq 0\}\) in B(X) is called C-semigroup, if

I) \(S(t+s)C=S(t)S(s)\) for t,s\(\geq 0\), \(S(0)=C,\)

II) \(S(\cdot)X:(0,+\infty)\to X\) is continuous for \(x\in X,\)

III) there are \(M\geq 0\) and \(a\in R\) such that \(\| s(t)\| Me^{at}\) for \(t\geq 0.\)

If \(G(x)=\lim_{t\to 0^+}(C^{-1}S(t)x-x)/t\) for \(x\in D(G)\), it is known that G is densely defined and closable, the closure \(\bar G\) is called the C-c.i.g of \(\{\) S(t);t\(\geq 0\}.\)

Let n be a positive integer, a family \(\{\) U(t):t\(\geq 0\}\) in B(X) is called n-times integrated semigroup, if

I) \(U(\cdot)x:\{0,+\infty)\to X\) is continuous for \(x\in X.\)

II) \(U(t)U(s)x=1/(n-1)!(\int^{t+s}_{t}(s+t-r)^{n-1}U(r)xdr- \int^{s}_{0}(s+t-r)^{n-1}U(r)xdr)\) for \(x\in X\) and s,t\(\geq 0\) and \(U(0)=0,\)

III) \(U(t)x=0\), for all \(t>0\), implies \(x=0,\)

IV) there are \(M\geq 0\) and \(\omega\in R\) such that \(\| U(t)\| \leq Me^{\omega t}\) for \(t\geq 0.\)

A very good result about the relation between these two kinds of semigroups is the following:

Theorem. Let A be a densely defined closed linear operator in X with \(\rho\) (A)\(\neq \emptyset\). Let \(c\in \rho (A)\) and \(n\geq 0\) be an integer. The following (i)-(iii) are equivalent:

i) A is the generator of an n-times integrated semigroup \(\{\) U(t);t\(\geq 0\},\)

ii) A is the C-c.i.g of a C-semigroup \(\{\) S(t);t\(\geq 0\}\) with \(C=R(C;A)^ n,\)

iii) There exist \(M\geq 0\) and \(a\in R\) such that (a,\(\infty)\subset \rho (A)\) and \(\| R((\lambda;A)^ mR(c;A)^ n\| \leq M/(\lambda -a)^ m\) for \(m\geq 1\) and \(\lambda >a.\)

In this case, we have \[ U(t)x=(c-A)^ n\int^{t}_{0}\int^{t_ 1}_{0}...\int^{t_{n-1}}_{0}s(t_ n)xdt_ n...dt_ 2dt_ 1 \] for \(x\in X\) and \(t\geq 0\).

Reviewer: Wu Liangsen

##### MSC:

47D03 | Groups and semigroups of linear operators |

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\textit{N. Tanaka} and \textit{I. Miyadera}, Proc. Japan Acad., Ser. A 63, 139--142 (1987; Zbl 0642.47034)

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##### References:

[1] | W. Arendt: Resolvent positive operator. Proc. London Math. Soc. (to appear). · Zbl 0617.47029 |

[2] | W. Arendt: Vector valued Laplace transforms and Cauchy problems (preprint 1986). · Zbl 0637.44001 |

[3] | E. B. Davies and M. M. H. Pang: The Cauchy problem and a generalization of the Hille-Yosida theorem (preprint 1986). · Zbl 0651.47026 |

[4] | I. Miyadera: On the generators of exponentially bounded C-semigroups. Proc. Japan Acad., 62A, 239-242 (1986). · Zbl 0617.47032 |

[5] | I. Miyadera and N. Tanaka: Exponentially bounded C-semigroups and generation of semigroups (preprint 1987). · Zbl 0697.47039 |

[6] | N. Tanaka: On the exponentially bounded C-semigroups. Tokyo J. Math, (to appear). · Zbl 0631.47029 |

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