UDC 519.4:517:513.88

MATHEMATICS

A. V. ZAFIEVSKII

ON SEMIGROUPS HAVING SINGULARITIES AT ZERO SUMMABLE WITH A POWER WEIGHT

(Presented by Academician S. L. Sobolev on 13 IV 1970)

As is known, semigroups of linear continuous operators, strongly continuous only for \(t>0\), may have arbitrary growth as \(t\to 0\). In this note an attempt is made to extend some results of the theory of semigroups of class \(C_0\) to the classes \(L^{(n)}\) of semigroups \(T(t)\), strongly continuous for \(t>0\), for which the functions \(t^n\|T(t)x\|\) \((x\in X)\) have summable singularities at zero. Semigroups of this type were first considered by Da Prato \((^2)\). Namely, he considered semigroups \(T(t)\) satisfying the estimate \(t^\alpha\|T(t)\|\le C\) for some integer \(\alpha\). The results of the present paper make it possible to carry out a finer classification of semigroups, taking into account the case of a fractional exponent \(\alpha\). Moreover, the results obtained make it possible to consider, besides semigroups for which the functions \(t^\alpha\|T(t)x\|\) \((x\in X)\) are bounded, also semigroups for which the functions \(t^\alpha\|T(t)x\|\) are summable. The principal result of the note is the establishment of a correspondence between semigroups of class \(L^{(n)}\) and their infinitesimal operators.

1. Suppose that \(T(t)\) is a semigroup of linear continuous operators in a Banach space \(X\), strongly continuous for \(t>0\), \(\omega_0\) is its type, and \(A_0\) is its infinitesimal operator, i.e. the operator defined by the equality

\[ A_0x=\lim_{t\to 0}\frac{T(t)x-x}{t} \tag{1} \]

on the set \(D(A_0)\) of all those \(x\in X\) for which the limit (1) exists. Introduce the notation

\[ N_0=\bigcap_{t>0}\{x:T(t)x=0\};\qquad X_0=\overline{\bigcup_{t>0}\{T(t)x:x\in X\}}. \]

Theorem 1. Let \(N_0\cap X_0=\{0\}\). Then the infinitesimal operator \(A_0\) admits a closure.

One of the closed extensions of the operator \(A_0\) can be specified in the following way. Define \(D(B)\) as the set of all \(x\in X\) for which there exists such a \(z\in X_0\) that, for all \(t\) and \(s\), \(0<s<t\),

\[ T(t)x-T(s)x=\int_s^t T(\tau)z\,d\tau . \]

If the assumptions of Theorem 1 are satisfied, then the element \(z\) is determined uniquely by the element \(x\). Therefore the equality \(Bx=z\) defines some operator \(B\). This operator is closed and is an extension of the operator \(A_0\).

Next introduce the operator \(R(\lambda)\), defined for \(\operatorname{Re}\lambda>\omega_0\) by the formula

\[ R(\lambda)x=\lim_{t\to 0}\int_t^\infty e^{-\lambda s}T(s)x\,ds \tag{2} \]

on the set \(D_R\) of all those \(x\in X\) for which the limit (2) exists for arbitrary \(\lambda\), \(\operatorname{Re}\lambda>\omega_0\).

Lemma 1. The operator \(R(\lambda)\) maps \(D_R\) into \(D_R\) for every \(\lambda\). Moreover, the following relations hold:

a) \(R(\lambda)x-R(\mu)x=(\mu-\lambda)R(\lambda)R(\mu)x\quad (x\in D_R)\);

b) \((\lambda I-A_0)R(\lambda)x=x\), if

\[ \lim_{t\to 0}\frac{1}{t}\int_0^t T(s)x\,ds=x; \]

c) \(R(\lambda)(\lambda I-A_0)x=x\quad (x\in D(A_0))\);

d) \(R(\lambda)(\lambda I-B)x=x\), if \(x\in D(B)\) and \(\lim_{t\to 0}T(t)x=x\);

e) \((\lambda I-B)R(\lambda)x=x\quad (x\in D_R\cap X_0)\).

Lemma 2. If \(x\in D_R\), then for any nonnegative integer \(m\) the equality

\[ R^{m+1}(\lambda)x=\frac{1}{m!}\int_0^\infty t^m e^{-\lambda t}T(t)x\,dt \tag{3} \]

holds.

Let us note that, in contrast to the case of a semigroup of class \(L\) \((^3)\), the operator \(R(\lambda)\) need not be bounded.

We shall say that the semigroup \(T(t)\) belongs to the class \(L^{(n)}\) if \(N_0=\{0\}\), \(X_0=X\), and if for every \(x\in X\) the function \(t^n\|T(t)x\|\) is summable on every interval of the form \((0,a)\), \(a<\infty\).

Lemma 3. Let \(T(t)\) be a semigroup of class \(L^{(n)}\). Then the operator \(S_n(\lambda)\), defined on \(X\) by the formula

\[ S_n(\lambda)x=\frac{1}{n!}\int_0^\infty t^n e^{-\lambda t}T(t)x\,dt\quad (\operatorname{Re}\lambda>\omega_0), \tag{4} \]

is continuous for \(\operatorname{Re}\lambda>\omega_0\), and for every \(\omega>\omega_0\) the estimate \(\|S_n(\lambda)\|\le M_\omega\) holds \((\operatorname{Re}\lambda>\omega)\).

Comparison of formulas (3) and (4) shows that for semigroups of class \(L^{(n)}\) the operator \(R^{n+1}(\lambda)\), defined on \(D_R\), is bounded, and the operator \(S_n(\lambda)\) is its continuous extension to all of \(X\).

For any \(x\in X\) the function \(S_n(\lambda)x\) is the Laplace transform of the function \(t^n(n!)^{-1}T(t)x\), continuous for \(t>0\). Therefore the usual theorems on the Laplace transform are applicable, in particular the uniqueness theorem. In the case of a semigroup of class \(L^{(n)}\), it follows from this theorem that \(S_n(\lambda)x=0\) \((\operatorname{Re}\lambda>\omega_0)\) if and only if \(x=0\). The same can be said about \(R(\lambda)\): \(R(\lambda)x=0\) \((x\in D_R)\) if and only if \(x=0\).

1. Let \(A\) be a closable operator, and suppose that \(D(A^m)\) is dense in \(X\) for every \(m\). An analytic function \(S_n(\lambda,A)\), defined in some domain \(\rho_n(A)\) of the complex plane and taking values in the space of linear continuous operators, will be called the resolvent of order \(n\) of the operator \(A\), if from \(S_n(\lambda,A)x=0\) \((\lambda\in\rho_n(A))\) it follows that \(x=0\), and if for \(\lambda\in\rho_n(A)\)

\[ S_n(\lambda,A)Ax=AS_n(\lambda,A)x\quad (x\in D(A)); \]

\[ S_n(\lambda,A)(\lambda I-A)^{n+1}x=x\quad (x\in D(A^{n+1})). \tag{5} \]

We shall say that the operator \(A\) belongs to the class \(L_\omega^{(n)}\) if \(\rho_n(A)\supseteq\{\lambda:\operatorname{Re}\lambda>\omega\}\), with \(\|S_n(\lambda,A)\|\le M\) \((\operatorname{Re}\lambda>\omega)\), and if there exist a nonnegative function \(\varphi(t,x)\) \((x\in X,\ t>0)\), continuous jointly in its variables, and a nonnegative function \(\varphi(t)\), bounded on every interval of the form \((a,b)\), \(0<a<b\), for which

\[ \lim_{t\to\infty} t^{-1}\ln\varphi(t)<\omega, \]
that

\[ \varphi(t,x)\leqslant\varphi(t)\|x\|;\qquad \int_0^\infty t^n\varphi(t,x)e^{-\omega t}\,dt<\infty, \tag{6} \]

\[ \|S_n^{(m)}(\tau,A)x\|\leqslant \frac1{n!}\int_0^\infty e^{-\tau t}t^{n+m}\varphi(t,x)\,dt \quad(\tau>\omega,\ m=0,1,\ldots). \tag{7} \]

Theorem 2. The infinitesimal operator \(A_0\) of a semigroup \(T(t)\) of class \(L^{(n)}\) belongs to the class \(L_\omega^{(n)}\) for every \(\omega>\omega_0\), and \(S_n(\lambda,A_0)=S_n(\lambda)\) \((\operatorname{Re}\lambda\geqslant\omega)\).

Theorem 3. If \(A\) is an operator of class \(L_\omega^{(n)}\), then there exists a unique semigroup \(T(t)\) of class \(L^{(n)}\) whose infinitesimal operator \(A_0\) satisfies the relation \(S_n(\lambda,A_0)=S_n(\lambda,A)\). Moreover \(\omega_0\leqslant\omega\), \(\|T(t)x\|\leqslant\varphi(t,x)\), and \(\|T(t)\|\leqslant\varphi(t)\).

Let the operator \(A\) have, for \(\operatorname{Re}\lambda\geqslant\omega\), a resolvent \(S_n(\lambda,A)\) of order \(n\), and let there exist a family \(\{T(t):t>0\}\) of linear continuous operators, strongly continuous in \(t\), such that

\[ \int_0^\infty t^n e^{-\omega_1 t}\|T(t)x\|\,dt<\infty \]

for some \(n\) and \(\omega_1\) and for every \(x\). Suppose, in addition, that

\[ S_n(\lambda,A)x=\frac1{n!}\int_0^\infty t^n e^{-\lambda t}T(t)x\,dt \quad(\operatorname{Re}\lambda>\max(\omega,\omega_1)). \]

Then it follows from Theorem 3 that \(T(t)\) is a semigroup of class \(L^{(n)}\).

If we fix the functions \(\varphi(t,x)\) and \(\varphi(t)\), then the estimates (7) give us a necessary and sufficient condition for the function \(S_n(\tau,A)\) to correspond to a semigroup \(T(t)\) satisfying the estimates \(\|T(t)x\|\leqslant\varphi(t,x)\) and \(\|T(t)\|\leqslant\varphi(t)\). By choosing various functions \(\varphi(t,x)\) and \(\varphi(t)\), we shall obtain conditions for a semigroup to belong to one or another class of semigroups. In the case \(n=0\) this path is considered in detail in (³).

It is natural to supplement Theorem 3 with the following assertion.

Theorem 4. Let a closed operator \(A\) belong to the class \(L_\omega^{(n)}\) and have a resolvent \(R(\lambda,A)\) for \(\operatorname{Re}\lambda>\omega\). Then the closure \(\overline A_0\) of the operator \(A_0\) coincides with the operator \(A\), and \(S_n(\lambda,A)=R^{\,n+1}(\lambda,A)\).

1. We shall call a semigroup \(T(t)\) of class \(L^{(n)}\) a semigroup of class \(L_0^{(n)}\) if the function \(t^n\|T(t)\|\) is summable on every interval of the form \((0,a)\), \(a<\infty\). Obviously, if \(T(t)\) is a semigroup of class \(L_0^{(n)}\), then the estimates

\[ \|S_n^{(m)}(\lambda)\|\leqslant \frac1{n!}\int_0^\infty t^{n+m}e^{-t\operatorname{Re}\lambda}\|T(t)\|\,dt \leqslant \frac1{n!}\int_0^\infty t^{n+m}e^{-t\operatorname{Re}\lambda}\varphi(t)\,dt, \]

are valid if \(\|T(t)\|\leqslant\varphi(t)\) and the function \(t^n\varphi(t)e^{-\omega t}\) is summable on \((0,\infty)\). The converse assertion is also valid:

Theorem 5. Let an operator \(A\) have a resolvent \(S_n(\lambda,A)\) of order \(n\) for real \(\tau>\omega\), and suppose the estimates

\[ \|S_n^{(m)}(\tau,A)\|\leqslant \frac1{n!}\int_0^\infty t^{n+m}e^{-\tau t}\varphi(t)\,dt \quad(\tau>\omega,\ m=0,1,\ldots), \tag{8} \]

where the function \(\varphi(t)\) is nonnegative and \(t^n\varphi(t)e^{-\omega t}\) is summable on \((0,\infty)\). Then there exists a semigroup \(T(t)\) of class \(L_0^{(n)}\) for which \(S_n(\tau)=S_n(\tau,A)\) for \(\tau>0\).

1. As an application, let us consider semigroups \(T(t)\) with \(N_0=\{0\}\) and \(X_0=X\), satisfying the estimate \(\|T(t)\|\le Ct^{-\alpha}e^{\omega t}\). We shall call these semigroups semigroups of class \(C_\alpha\). They belong to the classes \(L_0^{(n)}\) for \(n\ge[\alpha]\). The estimates (8) take for them a simpler form.

Theorem 6. Let \(T(t)\) be a semigroup of class \(C_\alpha\). Then its infinitesimal operator \(A_0\) has, for \(\operatorname{Re}\lambda>\omega\), a resolvent \(S_n(\lambda,A_0)\) of order \(n\) \((n\ge[\alpha])\), and

\[ \left\|S_n^{(m)}(\lambda,A_0)\right\| \le \frac{C\Gamma(m+n+1-\alpha)} {n!(\operatorname{Re}\lambda-\omega)^{m+n+1-\alpha}} \qquad (\operatorname{Re}\lambda>\omega,\ m=0,1,\ldots). \tag{9} \]

Conversely, suppose an operator \(A\) has, for real \(\tau>\omega\), a resolvent \(S_n(\tau,A)\) of order \(n\), and the estimate

\[ \left\|S_n^{(m)}(\tau,A)\right\| \le \frac{C\Gamma(m+n+1-\alpha)} {n!(\tau-\omega)^{m+n+1-\alpha}} \qquad (\tau>\omega,\ m=0,1,\ldots) \tag{10} \]

holds. Then there exists a unique semigroup \(T(t)\) of class \(L_0^{(n)}\), satisfying the estimate \(\|T(t)\|\le Ct^{-\alpha}e^{\omega t}\), for which

\[ S_n(\tau,A)x=\frac{1}{n!}\int_0^\infty t^n e^{-\tau t}T(t)x\,dt. \]

Theorem 6 is a generalization of the Da Prato theorem \((^2)\) to the case of fractional \(\alpha\). Another theorem of a similar kind for semigroups satisfying the estimate \(\|T(t)\|\le Ct^{-\alpha}e^{\omega t}\) was proved by P. E. Sobolevskii \((^4)\).

The author expresses gratitude to P. P. Zabreiko for proposing the problem and for his constant attention to the work.

Voronezh State University
named after the Lenin Komsomol

Received
6 IV 1970

CITED LITERATURE

\(^1\) E. Hille, R. Phillips, Functional Analysis and Semigroups, IL, 1962.
\(^2\) G. Da Prato, C. R., AB 262, No. 18, A 996 (1966).
\(^3\) P. P. Zabreiko, A. V. Zafievskii, DAN, 189, No. 5 (1969).
\(^4\) P. E. Sobolevskii, Functional Analysis and Its Applications, 5, issue 2 (1970).