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).