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UDC 519.4:517.513.88
MATHEMATICS
P. P. ZABREIKO, A. V. ZAFIEVSKII
ON A CERTAIN CLASS OF SEMIGROUPS
(Presented by Academician I. G. Petrovskii on 5 V 1969)
In the present article an attempt is made to construct a general theory of the class \(L\) of strongly continuous on \((0,\infty)\) semigroups \(T(t)\) of bounded linear operators in a Banach space \(X\), for which the functions \(T(t)x\) \((x \in X)\) have a summable singularity at zero (the precise definition of the class \(L\) is given below). Conditions are clarified under which a semigroup \(T(t)\) from \(L\) has a generator, and the properties of this generator are investigated. The main result consists of necessary and sufficient conditions under which a linear operator \(U\) is the generator of a semigroup of class \(L\). The class \(L\) naturally decomposes into narrower classes \(E\) of semigroups \(T(t)\), for which the functions \(T(t)x\) \((x \in X)\) are not only summable but also belong to a certain ideal space of functions \(E\) (for example, \(L_p\), Orlicz, Lorentz, etc.); the conditions found for membership of the semigroup \(T(t)\) in the class \(E\) in particular cases turn into known theorems on generators (see \((^{1-3})\), where a detailed bibliography is given).
1. Let \(T(t)\) be a semigroup of bounded linear operators in a Banach space \(X\), strongly continuous on \((0,\infty)\). Recall that for every such semigroup there exists the limit
\[ \omega_0=\lim_{t\to\infty}\frac{\ln \|T(t)\|}{t}, \tag{1} \]
called the type of the semigroup. The equality
\[ A_0x=\lim_{t\to 0}\frac{T(t)x-x}{t} \tag{2} \]
defines the infinitesimal operator \(A_0\) of the semigroup; if the operator \(A_0\) admits a closure, then one says that the semigroup has the generator \(A=\overline{A}_0\).
We shall say that the semigroup \(T(t)\) belongs to the class \(L\) if the following conditions are satisfied:
a) The subspace
\[ E_0=\bigcup_{0<t<\infty} R[T(t)] \tag{3} \]
(\(R(C)\) is the range of the operator \(C\)) is dense in \(X\).
b) For every \(x \in X\) the function \(T(t)x\) is summable on every finite interval.
For every semigroup \(T(t)\) of class \(L\), the domain \(D(A_0)\) of the infinitesimal operator \(A_0\) is dense in \(X\). Thus the subspace \(H_0\) of elements \(x \in X\) for which
\[ \lim_{t\to\infty}\frac{1}{t}\int_0^t T(s)x\,ds=x. \tag{4} \]
is also dense in \(X\).
Let \(T(t)\) be a semigroup of class \(L\), and let \(\omega_0\) be its type. Then for \(\operatorname{Re}\lambda>\omega_0\) a linear continuous operator is defined in \(X\) by
\[ R(\lambda)x=\int_0^\infty e^{-\lambda t}T(t)x\,dt. \tag{5} \]
Lemma 1. The following equalities hold
\[ R(\lambda)-R(\mu)=(\mu-\lambda)R(\lambda)R(\mu) \qquad (\operatorname{Re}\lambda,\operatorname{Re}\mu>\omega_0); \tag{6} \]
\[ (\lambda I-A_0)R(\lambda)x=x \qquad (x\in H_0,\ \operatorname{Re}\lambda>\omega_0); \tag{7} \]
\[ R(\lambda)(\lambda I-A_0)x=x \qquad (x\in D(A_0),\ \operatorname{Re}\lambda>\omega_0). \tag{8} \]
It is natural to make the supposition that \(R(\lambda)\) is the resolvent of the generating operator \(A\) of the semigroup \(T(t)\). Indeed, the following is true.
Theorem 1. A semigroup \(T(t)\) of class \(L\) has a generating operator if and only if \(T(t)x=0\) \((0<t<\infty)\) implies \(x=0\). If \(T(t)\) has generating operator \(A\), then \(R(\lambda,A)=R(\lambda)\) for \(\operatorname{Re}\lambda>\omega_0\).
The property of a semigroup \(T(t)\) of class \(L\) that “from \(T(t)x=0\) \((0<t<\infty)\) it follows that \(x=0\)” is equivalent to the fact that, for some \(\lambda_0\) \((\operatorname{Re}\lambda_0>\omega_0)\), from \(R(\lambda_0)x=0\) it follows that \(x=0\).
Theorem 2. Let \(T(t)\) be a semigroup of class \(L\) with generating operator \(A\). Then:
a) If \(x\in D(A)\), then
\[ T(t)x-x=\int_0^t T(s)Ax\,dx, \tag{9} \]
and, consequently, the function \(T(t)x\) is continuous on \([0,\infty)\).
b) If
\[ T(t)x-x=\int_0^t T(s)z\,ds, \tag{10} \]
then \(x\in D(A)\) and \(Ax=z\).
c) \(D(A^2)\subseteq D(A_0)\).
It is of interest to determine when the infinitesimal operator \(A_0\) of a semigroup \(T(t)\) of class \(L\) is closed, i.e., when it coincides with the generating operator \(A\) of the semigroup.
Theorem 3. Let \(T(t)\) be a semigroup of class \(L\). Then \(A_0=A\) if and only if \(X=H_0\).
- We now pass to the question of what conditions a given operator \(U\) must satisfy in order to be the generating operator of a semigroup of class \(L\). Naturally, it is necessary to consider only closed operators with domain dense in \(X\).
Theorem 4. In order that a linear closed operator \(U\) with domain dense in \(X\) be the generating operator of a semigroup \(T(t)\) of class \(L\) of type \(\omega_0\), it is necessary and sufficient that, for some \(\omega\) \((\omega>\omega_0)\), the following conditions hold:
a) \(\|R(\lambda,U)\|\leq M\) for \(\operatorname{Re}\lambda>\omega\);
b) there exist a nonnegative function \(\varphi(t,x)\) \((0<t<\infty,\ x\in X)\), continuous in the aggregate of variables, and a nonnegative function \(\varphi(t)\) \((0<t<\infty)\), satisfying the conditions
\[ \varphi(t_1t_2)\leq \varphi(t_1)\varphi(t_2),\qquad \lim_{t\to\infty}\frac{\ln\varphi(t)}{t}<\omega, \tag{11} \]
such that
\[ \varphi(t,x) \leqslant \varphi(t)\|x\|,\qquad \int_0^\infty \varphi(t,x)e^{\omega t}\,dt < \infty \quad (x\in X), \tag{12} \]
\[ \|R^{(n)}(\lambda,U)x\| \leqslant \int_0^\infty e^{-\lambda t}t^n\varphi(t,x)\,dt \quad (x\in X,\ \operatorname{Re}\lambda>\omega;\ n=0,1,\ldots). \tag{13} \]
Moreover,
\[ \|T(t)x\| \leqslant \varphi(t,x)\quad (x\in X),\qquad \|T(t)\| \leqslant \varphi(t). \tag{14} \]
- A semigroup of class \(L\) will be called a semigroup of class \(L_0\) if the function \(T(t)\) is also summable on every finite interval.
Theorem 5. In order that a linear closed operator \(U\) with domain dense in \(X\) be the generating operator of a semigroup \(T(t)\) of class \(L_0\) of type \(\omega_0\), it is necessary and sufficient that, for some \(\omega\) \((\omega>\omega_0)\) and for some continuous nonnegative function \(\varphi(t)\) \((0<t<\infty)\) satisfying conditions (11), such that
\[ \int_0^\infty \varphi(t)e^{\omega t}\,dt < \infty, \tag{15} \]
the inequalities
\[ \|R^{(n)}(\lambda,U)\| \leqslant \int_0^\infty e^{-\lambda t}t^n\varphi(t)\,dt \quad (\lambda>\omega;\ n=0,1,\ldots). \tag{16} \]
hold. Moreover,
\[ \|T(t)\| \leqslant \varphi(t). \tag{17} \]
Semigroups of class \(L_0\) were considered, in another connection, by P. E. Sobolevskii (4).
- Let \(E\) be an ideal space of functions \(\xi(t)\) defined on \([0,\infty)\); suppose that \(E\) contains all bounded functions that vanish outside some interval \([0,a]\), and that each function from \(E\) is summable on any interval \([0,a]\). By \(E_\omega\) (\(\omega\) a real number) we denote the ideal space of functions \(\xi(t)\) for which \(e^{\omega t}\xi(t)\in E\), with norm
\[ \|\xi\|_\omega=\|e^{\omega t}\xi(t)\|. \]
A semigroup \(T(t)\) of class \(L\) (class \(L_0\)) will be called a semigroup of class \(E\) (class \(E_0\)) if \(\|T(t)x\|\in E_\omega\) \((x\in X)\) (if \(\|T(t)\|\in E_\omega\)) for some \(\omega\). In the case \(E=L_1\), the class \(E\) (\(E_0\)) coincides with the whole class \(L\) (\(L_0\)). In the case \(E=L_\infty\), the classes \(E\) and \(E_0\) coincide with the class \(C_0\) of semigroups bounded at zero and, consequently, strongly continuous on \([0,\infty)\).
Theorems 4 and 5 make it possible to formulate conditions under which a given linear operator is the generating operator of some semigroup of class \(E\) or class \(E_0\).
Theorem 6. In order that a linear closed operator \(U\) with domain dense in \(X\) be the generating operator of a semigroup of class \(E\) \((E_0)\), it is necessary and sufficient that conditions a) and b) of Theorem 4 (Theorem 5) be satisfied; moreover, the function \(\varphi(t,x)\) (the function \(\varphi(t)\)) can be chosen so that \(\varphi(t,x)\in E_\omega\) for \(x\in X\) \((\varphi(t)\in E_\omega)\).
Let us consider, in particular, the case where \(E\) is the space \(C_\alpha\) of all measurable functions on \([0,\infty)\) for which the norm
\[ \|\xi(t)\|=\sup_{0<t<\infty}|t^\alpha \xi(t)|; \]
here \(0 \leq \alpha < 1\). From the principle of uniform boundedness it follows that the classes \(C_\alpha\) and \((C_\alpha)_0\) coincide; in the case under consideration, as the function \(\varphi(t)\) one may always take the function \(\varphi(t)=Mt^{-\alpha}e^{\omega_1 t}\), where \(M\) and \(\omega_1\) \((\omega_1<\omega)\) are certain constants. For this function \(\varphi(t)\), the condition of Theorem 5, in view of
\[ \int_0^\infty e^{-\lambda t} t^n \varphi(t)\,dt = \frac{M\Gamma(n-\alpha+1)}{(\lambda-\omega_1)^{\,n-\alpha+1}}, \]
takes the quite simple form:
\[ \left\| R^{(n)}(\lambda,U)\right\| = M\Gamma(n-\alpha+1)/(\lambda+\omega)^{\,n-\alpha+1} \qquad (\lambda>\omega;\ n=0,1,\ldots). \tag{18} \]
Thus we arrive at the following assertion: a linear closed operator \(U\) with domain dense in \(X\) is the generating operator of a semigroup \(T(t)\) of class \(C_\alpha\) of type \(\omega_0\) if and only if, for some \(M\) and \(\omega\) \((\omega>\omega_0)\), the inequalities (18) hold. For \(\alpha=0\) this assertion coincides with the assertion of the classical Hille—Phillips—Miyadera theorem; for \(\alpha>0\) a close assertion was proved by P. E. Sobolevskii.
In conclusion we note that already for semigroups of class \(C_\alpha\) \((\alpha>0)\) the “positivity” inequality
\[ \lim_{\lambda\to\infty}\|\lambda R(\lambda)\|<\infty, \]
generally speaking, does not hold.
The authors express their gratitude to M. A. Krasnosel’skii for a detailed discussion of the results of the work.
Voronezh State
University
Received
29 IV 1969
CITED LITERATURE
- E. Hille, R. Phillips, Functional Analysis and Semi-Groups, IL, 1963.
- N. Dunford, J. T. Schwartz, Linear Operators. General Theory, IL, 1962.
- S. G. Krein, Linear Differential Equations in Banach Space, “Nauka,” 1967.
- P. E. Sobolevskii, Zap. Voronezhsk. sel’skokhozyaistvenn. inst., 28, 2, 399 (1959).