Abstract
Full Text
L. MÁTÉ (L. MÁTÉ)
ON A SEMIGROUP OF OPERATORS IN A FRÉCHET SPACE
(Presented by Academician V. I. Smirnov on 30 X 1961)
§ 1. Introduction
Some Fréchet spaces (\(F\)-spaces) are beginning to play an important role in the study of partial differential equations. For example, such a space is the class of continuous functions on a locally compact space with the usual metric, or the class of infinitely differentiable functions on a compact space, if, in metrizing it, we require that certain differential operators be continuous in this space \((^{2,5})\).
Let us note one of the differences between Banach spaces and Fréchet spaces*. Let \(\{T(t);\, t \geqslant 0\}\) be a semigroup of operators in a Banach space, strongly continuous at \(t=0\). Then there exists an \(\alpha>0\) such that the semigroup**
\[ T^0(t)=e^{-\alpha t}T(t) \]
satisfies the condition (see \((^2)\)):
\[ \text{The set } \{T^0(t)x;\, t \geqslant 0\} \text{ is bounded for every } x \in X. \tag{*} \]
In Fréchet spaces this is, generally speaking, not so; an example is the translation semigroup: \(T(t)x(s)=x(s+t)\) in the class of continuous functions on \((-\infty,\infty)\). (The class of continuous functions on \((-\infty,\infty)\) is a Fréchet space with seminorms \((^5)\)
\(\|x\|_k=\sup_{|s|\leqslant k}|x(s)|\) \((k=1,2,\ldots)\).)
The fundamental Hille—Yosida—Phillips theorem \(((^1), p. 624)\) remains valid for a semigroup in an \(F\)-space if condition \((*)\) is satisfied for every \(x\), but it loses its force if \((*)\) is satisfied only on some everywhere dense subset of the space. The theorems obtained here are, in their nature, close to the results of Feller \((^2)\), which belong to another branch of semigroup theory.
The purpose of the present paper is to generalize the Hille—Yosida—Phillips theorem to those semigroups in an \(F\)-space for which condition \((*)\) is satisfied only on some everywhere dense subset. As an application, in § 4 we shall consider the question of extending a strongly continuous semigroup from a Banach space to a Fréchet space containing it as a subspace.
§ 2. Preliminary lemmas and definitions
I. A Fréchet space is a locally convex metrizable complete space.
II. A strongly continuous semigroup of operators is a family \(\{T(t);\, t \geqslant 0\}\) of bounded operators mapping the \(F\)-space \(X\) into itself and satisfying the following con—
* The definitions will be given in § 2.
** In what follows we shall everywhere replace \(T(t)\) by the equivalent semigroup \(T^0(t)\).
conditions:
A. \(T(t+s)=T(t)T(s),\ s,t\geqslant 0.\)
B. \(T(0)=E\) (\(E\) is the identity operator).
C. \(\lim_{t\to 0}T(t)x=x\) for all \(x\in X\).
III. The principle of uniform boundedness carries over to Fréchet spaces: let \(\{T(t),\,t\geqslant 0\}\) be a family of bounded operators in \(X\) such that the set \(\{T(t)x;\,t\geqslant 0\}\) is bounded for each \(x\in X\). Then the relation
\[ \lim_{x\to 0} T(t)x=\theta \]
holds uniformly in \(t\) (\(\theta\) is the zero element of the space).
IV. Consider the subsets of \(X\)
\[ \Sigma=\{x:\sup_{t\geqslant 0}\|T(t)x\|_k<\infty,\quad k=1,2,\ldots\}, \tag{1} \]
\[ S=\{x:\sup_k\|x\|_k<\infty\} \tag{2} \]
with seminorms, respectively,
\[ P_k(x)=\sup_{t\geqslant 0}\|T(t)x\|_k \quad (k=1,2,\ldots), \tag{3} \]
\[ =\sup_k\|x\|_k. \tag{4} \]
As a result of the topology generated by these seminorms, \(\Sigma\) is an \(F\)-space* and \(S\) is a \(B\)-space.
V. \(S\), as a subset of \(X\), is everywhere dense in \(X\).
§ 3. Generalization of the Hille—Yosida—Phillips theorem to Fréchet spaces. First of all we give a sufficient condition under which the semigroup \(T(t)\) satisfies condition (*) on an everywhere dense subset of the space \(X\).
Theorem 1. If \(T(t)\) is a strongly continuous semigroup of operators and
\[ T(t)S\subseteq S, \tag{5} \]
then \(S\subseteq \Sigma\) and, consequently, (*) is fulfilled on an everywhere dense subset of the space \(X\).
Proof is based on § 2, IV and V.
Example. In the case of the translation semigroup mentioned in § 1, the Banach space \(S\) is \(C(-\infty,\infty)\), and condition (5) is, obviously, fulfilled.
In what follows we shall consider only such semigroups \(\{T(t);t\geqslant 0\}\) for which condition (*) is satisfied on an everywhere dense set \(\Sigma\). First of all, it is easy to see (§ 2, IV) that \(T(t)\) is strongly continuous also in \(\Sigma\), and thus (5) holds:
Theorem 2. Let the semigroup \(\{T(t);\,t\geqslant 0\}\) in \(\Sigma\) satisfy conditions (*), and also A, B, C of § 2, II.
The infinitesimal generating operator** \(A\) is closed in \(\Sigma\), and its domain of definition \(D(A)\) is everywhere dense in \(\Sigma\).
For every \(\lambda>0\) there exists \(R_\lambda\)—a bounded linear operator mapping \(\Sigma\) into itself—and
\[ (\lambda E-A)R_\lambda x=x,\quad \text{if } x\in\Sigma; \]
\[ R_\lambda(\lambda E-A)x=x,\quad \text{if } x\in D(A). \]
* The topology in a Fréchet space can be specified by means of a countable number of seminorms ((4), p. 208). We shall denote these seminorms by \(\|\ldots\|_k\).
** On the concept of an infinitesimal generating operator see (1), p. 614 or (5). We everywhere regard \(A\) as an operator in \(\Sigma\), even if it can be extended to a broader domain.
For every \(x \in \Sigma\) the set
\[ \{\lambda^n R_\lambda^n x;\ \lambda > 0,\ n = 0,1,2,\ldots\} \tag{6} \]
is bounded in \(\Sigma\).
For every \(x \in \Sigma\)
\[ T(t)x=\lim_{\lambda\to\infty} e^{-\lambda t} \sum_{k=0}^{\infty}\frac{\lambda^k t^k}{k!}(\lambda R_\lambda)^k x . \tag{7} \]
It is known that the boundedness of the set (6) is necessary and sufficient for the operator \(A\) to generate a strongly continuous semigroup in a Banach space. This is the content of the Hille—Yosida—Phillips theorem. The necessary conditions in a Fréchet space are as follows:
Theorem 3. Let \(\{T(t);\, t \ge 0\}\) be a strongly continuous semigroup of operators in \(X\), and let condition \((*)\) be satisfied on an everywhere dense subset \(\Sigma \subset X\). Then:
I. If \(x \in \Sigma\), then
\[ \sup_{\lambda>0,\ n\ge 0}\|\lambda^n R_\lambda^n x\|_k = \sup_{t\ge 0}\|T(t)x\|_k . \]
II. If \(K\) is a bounded set in \(X\) and
\[ \sup_{\tau\le t,\ x\in K}\|T(\tau)x\|_k = v_k(t), \]
then for every \(\varepsilon>0\) and \(x\in \Sigma \cap K\)
\[ \|\lambda^n R_\lambda^n x\|_k \le v_k(N)+\varepsilon, \quad \text{provided that } \lambda N>n \text{ and } \lambda>\lambda_0(x,\varepsilon). \tag{8} \]
Proof is based on (6), on the inequality
\[ \frac{\lambda^n}{(n-1)!}\int_N^\infty e^{-\lambda s}s^{n-1}\,ds \le \frac{\lambda N}{\inf_{i<n}|i-\lambda N|^2} \]
and on the well-known relation between \(R_\lambda x\) and \(T(t)x\) for \(x\in\Sigma\) \({}^{(5)}\).
Condition (8) is not only necessary, but in a certain sense also sufficient. Namely, the following holds:
Theorem 4. Let:
-
\(R_\lambda\) be a linear, but not necessarily bounded, operator in \(X\) for every \(\lambda>0\).
-
The set
\[ \left\{x:\ \sup_{\lambda>0,\ n\ge 0}\|\lambda^n R_\lambda^n x\|_k<\infty\right\} \tag{9} \]
be everywhere dense in \(X\).
- The set (9) be a Fréchet space with seminorms
\[ \pi_k(x)=\sup_{\lambda>0,\ n\ge 0}\|\lambda^n R_\lambda^n x\|_k \]
(we denote this space by \(\Sigma^0\)).
- \(R_\lambda\) satisfy condition (8).
Then:
\(1'\). The set (6) is bounded, and \(R_\lambda\) is a bounded operator in \(\Sigma^0\) for every \(\lambda>0\).
\(2'\). \(A\) generates a strongly continuous semigroup of operators \(T(t)\) in \(X\) such that
\[ \|T(t)x\|_k < v_k(N), \]
if \(t<N\) and \(x\in K\).
\(3'\). \(\Sigma^0\) contains all and only those elements for which
\[ \sup_{t\ge 0}\|T(t)x\|_k<\infty . \]
4°. The seminorms in \(\Sigma^{0}\) are
\[ \mathfrak{p}_{k}(x)=\sup_{t\geq 0}|T(t)x|_{k}. \]
§ 4. Extension theorem
The theorems formulated in the preceding paragraph can be applied to the study of the following problem.
Let \(B\) be a Banach space. Introduce in \(B\) a new topology, weaker than the original one, by means of a countable family of seminorms. Let \(F\) be the closure of this new locally convex space; \(F\) is a Fréchet space. Let \(\{T(t); t\geq 0\}\) be a strongly continuous semigroup of operators in \(B\). The question is as follows: under what conditions can \(T(t)\) be extended, preserving the property of strong continuity, to the whole space \(F\)? The answer follows directly from Theorems 3 and 4.
Theorem 5. Let \(T(t)\) be a strongly continuous semigroup of operators in the Banach space \(B\); let \(F\) be a Fréchet space containing \(B\) as a subset, and let \(B\) (regarded as a subset of \(F\)) be everywhere dense in \(F\).
Then, in order that \(T(t)\) admit an extension to a strongly continuous semigroup in \(F\) (not necessarily satisfying condition ()), it is necessary and sufficient that condition (8) hold for \(\Sigma=B\) and \(X=F\).*
Corollary. In order that \(T(t)\) admit an extension from \(B\) to \(F\) with condition (*) fulfilled, it is necessary and sufficient that condition (8) be satisfied for \(\vartheta_{k}(t)=c_{k}\;(=\mathrm{const})\).
Polytechnic Institute
Budapest, Hungarian People’s Republic
Received
27 X 1961
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