MATHEMATICS

M. M. VAINBERG and Ya. L. ENGELSON

ON THE SQUARE ROOT OF A LINEAR OPERATOR IN LOCALLY CONVEX SPACES

(Presented by Academician S. L. Sobolev, 3 VI 1958)

1. In a paper by one of the authors \((^1)\), by studying quadratic forms in the spaces \(L^q\) \((1<q<2)\), propositions were established on the square root of linear completely continuous operators acting from the space \(L^q\) into the space \(L^p\) \((p^{-1}+q^{-1}=1)\). These propositions were extended by the other author in \((^2)\) to a broad class of linear completely continuous operators acting in locally convex spaces. The principal result established in \((^2)\) can be formulated as follows.

Let the following condition be fulfilled:

\((\alpha)\) \(E\) is a real locally convex space; \(E'\) is the space strongly conjugate to \(E\) \((^*)\); \(H\) is a Hilbert space, dense in \(E'\), such that \(E \subset H \subset E'\), and the topologies of \(E\) and \(H\) are compatible, i.e. the topology of the space \(E\) majorizes the topology induced in \(E\) by the space \(H\), and the topologies of \(H\) and \(E'\) are also compatible; the bilinear functional \(\langle x,y\rangle\), where \(y\in E'\) and \(x\in E\) or \(x\in E''\), coincides with the scalar product in \(H\) for \(y\in H\).

Then, if \(E\) is a semicomplete barrelled space \((^3)\) and \(A\) is a linear completely continuous operator from \(E'\) into \(E\), self-adjoint and quasi-positive in \(H\), then the principal square root of \(A\) defined in \(H\),

\[ A^{1/2}u=\sum_k \sqrt{|\lambda_k|}\,(\varphi_k,u)e_k\varphi_k,\quad e_k=\operatorname{sign}\lambda_k, \tag{1} \]

where \(\varphi_k\) and \(\lambda_k\) are the eigenvectors and the corresponding eigenvalues of the operator \(A\) in \(H\), acts completely continuously from \(H\) into \(E\) and quasi-completely continuously from \(E'\) into \(H\).

In this proposition we adhere to the following generally accepted terminology: a linear operator is completely continuous if it maps some neighborhood of zero into a relatively bicompact set; a linear operator is quasi-completely continuous if it maps every bounded set into a relatively bicompact set.

We note that the formulated proposition in fact also contains the assertion that the operator \(A^{1/2}\) is bounded from \(E'\) into \(H\), i.e. maps some neighborhood of zero from \(E'\) into a bounded set of \(H\). This assertion follows from the following proposition, which we shall also use below.

Lemma. If \(A\) is a bounded operator from a locally convex space \(E\) into a locally convex space \(F\), then the adjoint operator \(A'\) is bounded from \(F'\) into \(E'\).

Proof follows from the following two facts: if \(A(M)\subset N\), then \(A'(N^0)\subset M^0\), where \(M\) and \(N\) are sets respectively in \(E\) and \(F\), and \(M^0\) and \(N^0\) are their polars; if \(M\) is a neighborhood of zero in \(E\) and \(N\) is bounded—

bounded set in \(F\), then \(M^0\) is a bounded set in \(E'\) and \(N^0\) is a neighborhood of zero in \(F'\).

In the present paper we establish various propositions on the square root of a linear bounded operator acting from one locally convex space into another, without assuming its complete continuity.

1. Here we shall assume that the spaces \(E\) and \(H\) satisfy condition \((\alpha)\). This condition, in particular, is fulfilled if \(E=L^p(B)\) and \(H=L^2(B)\), where \(B\) is a set of finite measure in an \(s\)-dimensional Euclidean space and \(p>2\). Another such example is \(E=D\) and \(H=L^2(-\infty,+\infty)\), where \(D\) is the space of infinitely differentiable finite functions defined on the line.

Theorem 1. Let \(A\) be a linear bounded operator from \(E'\) into \(E\), self-adjoint and positive in \(H\). Then the positive square root \(A^{1/2}\) of the operator \(A\), considered in \(H\), is a bounded operator from \(H\) into \(E''\) and has a continuous extension \(\widetilde A^{1/2}\) from \(E'\) into \(H\).

Proof. From the compatibility of the topologies it follows that \(A\) is a bounded operator from \(H\) into \(H\), and therefore \(A^{1/2}\) is a continuous self-adjoint operator in \(H\). Consequently, for \(x\in H\) we have \(\|A^{1/2}x\|^2=(A^{1/2}x,A^{1/2}x)=(Ax,x)\). Considering \(x\) as an element of \(E'\) and taking condition \((\alpha)\) into account, from the last equality we obtain

\[ \|A^{1/2}x\|^2=\langle Ax,x\rangle . \tag{2} \]

Let \(\varepsilon>0\) be given. By the boundedness of \(A\) from \(E'\) into \(E\) there is a neighborhood of zero \(U_1\subset E'\) such that \(A(U_1)\) is a bounded set in \(E\); hence the polar \((A(U_1))^0\) is a neighborhood of zero in \(E'\), so that for every
\(x\in U=\varepsilon[(A(U_1))^0\cap U_1]\) one has \(|\langle Ax,x\rangle|\leqslant \varepsilon^2\). Hence, from (2), it follows that for all \(x\in U\cap H\) the inequality

\[ \|A^{1/2}x\|\leqslant \varepsilon \tag{3} \]

holds.

Inequality (3) shows that the operator \(A^{1/2}\), defined on the set \(H\), everywhere dense in the locally convex space \(E'\), with values in the complete space \(H\), is continuous; hence, by linearity, it is uniformly continuous on the set \(H\) of \(E'\). Consequently (see (4), Chap. II, § 3, Theorem 1), the operator \(A^{1/2}\) has a unique continuous extension \(\widetilde A^{1/2}\) to all of \(E'\), which is linear. This proves the second assertion of the theorem.

Consider the bilinear form \(l_x(y)=(A^{1/2}x,y)=(x,A^{1/2}y)\), where \(x,y\in H\). By inequality (3) we have
\(|l_x(y)|\leq \|x\|\,\|A^{1/2}y\|\leq \varepsilon\|x\|\) for every \(x\in H\) and arbitrary \(y\in H\cap U\). Hence, as above, we conclude that for each \(x\in H\) there exists a unique continuous extension \(\widetilde l_x(y)\) to all \(E'\) of the functional \(l_x(y)\), and therefore \(\widetilde l_x(y)=\langle z_x,y\rangle\), where \(z_x\in E''\). By the compatibility of the topologies of the spaces \(H\) and \(E'\) we have \(E''\subset H\). Since for every \(y\in H\) the functionals \(\widetilde l_x(y)\) and \(l_x(y)\) coincide, it follows from the preceding and by condition \((\alpha)\) that \((A^{1/2}x,y)=(z_x,y)\), so that \(A^{1/2}x=z_x\in E''\), i.e. the operator \(A^{1/2}\) acts from \(H\) into \(E''\).

In view of the fact that \(A^{1/2}x\in E''\) for every \(x\in H\) and \(\widetilde A^{1/2}\) is a continuous operator from \(E'\) into \(H\), the functionals \(\langle A^{1/2}x,y\rangle\) and \((x,\widetilde A^{1/2}y)\) are continuous functionals of \(y\) on \(E'\) for every \(x\in H\). These functionals coincide on the set \(H\), dense in \(E'\); hence, according to the principle of extension of identities (4), it follows that
\(\langle A^{1/2}x,y\rangle=(x,\widetilde A^{1/2}y)\) for all \(x\in H\) and \(y\in E'\). Taking into account the domains of definition and the ranges of the operators \(A^{1/2}\) and \(\widetilde A^{1/2}\), from the last equality we conclude that \(A^{1/2}=(\widetilde A^{1/2})'\). Hence, since by what has been proved \(\widetilde A^{1/2}\) is a bounded operator, according to the lemma it follows that \(A^{1/2}\) is a bounded operator from \(H\) into \(E''\). The theorem is proved.

1. Here we shall assume that the space \(E\) is quasi-barrelled, i.e., every barrel absorbing any bounded set from \(E\) is a neighborhood of zero. As is known \((^5)\), this requirement is necessary and sufficient for the strong topology of the space \(E''\) to induce on \(E\) a topology coinciding with the original topology of the space \(E\).

Theorem 2. Suppose the conditions of Theorem 1 are fulfilled. Then

\[ A=A^{1/2}\widetilde A^{\prime 1/2}. \]

Indeed, the operator \(C=A^{1/2}\widetilde A^{\prime 1/2}\), by Theorem 1, is continuous from \(E'\) into \(E''\), while the operator \(A\), continuous from \(E'\) into \(E\), in view of the quasi-barrelledness of \(E\), is also continuous from \(E'\) into \(E''\). Since the operators \(A\) and \(C\) coincide on the set \(H\), dense in \(E'\), it follows, by the principle of extension of identities, that \(A=A^{1/2}\widetilde A^{\prime 1/2}\), i.e., \(A\) is representable as the product of two operators, of which one—\(\widetilde A^{\prime 1/2}\)—is continuous from \(E'\) into \(H\), and the other—\(A^{1/2}\)—is bounded from the set \(\widetilde A^{\prime 1/2}(E')\subset H\) into the space \(E\).

Theorem 3. Suppose the conditions of Theorem 1 are fulfilled. Then the operator \(A\) has a continuous extension \(\widetilde A\) from \(E'''\) into \(E''\), representable in the form \(\widetilde A=A^{1/2}(A^{1/2})'\), where \((A^{1/2})'\) is a continuous operator from \(E'''\) into \(H\) and \(A^{1/2}\) is a bounded operator from \(H\) into \(E''\).

Indeed, since \(A^{1/2}\) is a bounded operator from \(H\) into \(E''\), by the lemma the operator \((A^{1/2})'\) is continuous from \(E'''\) into \(H\). Then from the equality \(\langle A^{1/2}x,y\rangle=(x,(A^{1/2})'y)\), where \(x\in H,\ y\in E'''\), according to the preceding, in particular when \(y\in E'\), we have \((x,\widetilde A^{\prime 1/2}y)=\langle A^{1/2}x,y\rangle=(x,(A^{1/2})'y)\) for every \(x\in H\). Consequently, \((A^{1/2})'y=\widetilde A^{\prime 1/2}y\) for every \(y\in E'\), i.e., \((A^{1/2})'\) is a continuous extension to \(E'''\) of the operator \(\widetilde A^{\prime 1/2}\). Hence it follows that \(A^{1/2}(A^{1/2})'\) is also a continuous extension of the operator \(A^{1/2}\widetilde A^{\prime 1/2}=A\).

The theorem just proved generalizes a theorem of V. I. Sobolev \((^6)\), established by him for Banach spaces.

1. Let a linear bounded operator \(A\) from \(E'\) into \(E\) admit in the space \(H\) a representation \(A=BB^*\), where \(B\) is a linear operator continuous in \(H\), and \(B^*\) is its adjoint. Repeating the preceding arguments, we arrive at the following proposition.

Theorem \(1'\). Suppose condition \((\alpha)\) is fulfilled. Then \(B\) is a bounded operator from \(H\) into \(E\), and \(B^*\) has a continuous extension \(\widetilde B^*\) from \(E'\) into \(H\).

We note that, for the operator \(A\) considered here, Theorems 2 and 3 remain valid if in their formulations the operator \(A^{1/2}\) is everywhere replaced by the operator \(B\), the operator \(\widetilde A^{\prime 1/2}\) by the operator \(\widetilde B^*\), and the operator \((A^{1/2})'\) by the operator \(B'\), adjoint to the operator \(B\) from \(H\) into \(E''\).

1. Here we shall assume that condition \((\alpha')\) is fulfilled, which differs from condition \((\alpha)\) in that \(E\) is replaced by \(E'\) and conversely, so that \(E'\subset H\subset E\). In addition, we shall assume that the space \(E\) is quasi-barrelled. Repeating then the preceding arguments with some change in the exposition, we arrive at the following propositions.

Theorem 4. Let \(A\) be a linear bounded operator from \(E\) into \(E'\), self-adjoint and positive in \(H\). Then the positive square root \(A^{1/2}\) of the operator \(A\), considered in \(H\), is a bounded operator from \(H\) into \(E'\) and has a continuous extension \(\widetilde A^{1/2}\) from \(E\) into \(H\).

Theorem 5. Suppose the conditions of Theorem 4 are fulfilled. Then the operator \(A\) is representable in the form \(A=A^{1/2}\widetilde A^{1/2}\), where \(\widetilde A^{1/2}\) is continuous from \(E\) into \(H\) and \(A^{1/2}\) is continuous from \(H\) into \(E'\).

Theorem 6. Let a linear bounded operator \(A\) from \(E\) into \(E'\) admit in \(H\) the representation \(A=BB^*\). Then \(B\) is a bounded

operator from \(H\) to \(E'\), and \(B^*\) has a continuous extension \(\widetilde{B}^*\) from \(E\) to \(H\).

Let us note that Theorem 6 generalizes Theorem 1 of (7), established for Banach spaces.

Let us also note that Theorem 5 remains valid for an operator \(A\) from \(E\) to \(E'\), having in \(H\) the representation \(A=BB^*\), if in its formulation \(A^{1/2}\) is replaced by \(B\) and \(\widetilde{A}^{1/2}\) by \(\widetilde{B}^*\).

1. Suppose condition \((\alpha)\) is satisfied. Consider a linear continuous operator \(A\) from \(E'\) to \(E\), self-adjoint in \(H\). Denote by \(A_+\) and \(A_-\) the positive and negative parts of \(A\) in \(H\), and suppose that \(A_+\) and \(|A_-|\) have extensions \(\widetilde{A}_+\) and \(|\widetilde{A}_-|\), bounded from \(E'\) to \(E\). Then, using Theorems 1 and 2, one can prove that the operator \(A\) has the representation
   \[ A=A^{1/2}|\widetilde{A}|^{1/2}, \]
   where \(|\widetilde{A}|^{1/2}\) is the extension of the square root of
   \[ |A|=A_+ + |A_-|, \]
   continuous from \(E'\) to \(H\), and
   \[ A^{1/2}=A_+^{1/2}-|A_-|^{1/2} \]
   is the principal square root of \(A\) in \(H\), bounded from \(|\widetilde{A}|^{1/2}(E')\subset H\) to \(E\). There is also an analogue of Theorem 3; i.e., the operator \(A\) has a continuous extension \(\widetilde{A}\) from \(E'''\) to \(E''\), representable in the form
   \[ \widetilde{A}=A^{1/2}(|A|^{1/2})', \]
   where \((|A|^{1/2})'\) is continuous from \(E'''\) to \(H\) and \(A^{1/2}\) is bounded from \(H\) to \(E''\). If condition \((\alpha')\) is satisfied, then propositions analogous to those of item 5 hold.

Moscow Regional Pedagogical Institute
named after N. K. Krupskaya

Latvian State University
named after P. Stuchka

Received
2 VI 1958

REFERENCES

¹ M. M. Vainberg, DAN, 100, No. 5, 845 (1955). ² Ya. L. Engelson, Uch. zap. Latv. gos. univ., 8, issue 2, 73 (1956). ³ N. Bourbaki, Espaces vectoriels topologiques, ch. III—V, 1955. ⁴ N. Bourbaki, Topologie générale, ch. I, II, 1951. ⁵ J. A. Dieudonné, Bull. Am. Math. Soc., 59, 495 (1953). ⁶ V. I. Sobolev, DAN, 111, No. 5, 951 (1956). ⁷ M. A. Krasnosel’skii, S. G. Krein, Tr. seminara po funktsional’n. analizu Voronezhsk. gos. univ., issue 5, 98 (1957).