UDC 513.88:517.948.35

MATHEMATICS

E. A. LARIONOV

EXTENSION OF DUAL SUBSPACES

(Presented by Academician I. M. Vinogradov, 16 XII 1966)

Let \(\Pi\) be a Hilbert space in which, in addition to the ordinary scalar product \([x,y]\), an indefinite scalar product \((x,y)=[Jx,y]\) is given, where \(J=P_+-P_-\); \(P_+\), \(P_-\) are mutually complementary orthoprojectors in \(\Pi\).

Denote by \(\mathfrak{M}_+\) the set of all maximal subspaces of \(\mathfrak{P}_+=\{x\in\Pi:\ (x,x)\geqslant0\}\), and by \(\mathfrak{K}_+\) the set of all operators acting in a nonexpanding manner from \(\Pi_+=P_+\Pi\) into \(\Pi_-=P_-\Pi\), and analogously introduce the sets \(\mathfrak{M}_-\), \(\mathfrak{P}_-\), \(\mathfrak{K}_-\). There are one-to-one correspondences \(\mathfrak{M}_+\leftrightarrow\mathfrak{K}_+\) and \(\mathfrak{M}_-\leftrightarrow\mathfrak{K}_-\) \((^1)\). An algebra \(R\) of linear bounded operators in \(\Pi\) is called symmetric if from \(A\in R\) it follows that \(A^0\in R\), where \(A^0\) is the operator \(J\)-adjoint to \(A\), defined by the equality \((Ax,y)=(x,A^0y)\), \(x,y\in\Pi\). If \(A=A^0\) \((UU^0=U^0U=I)\), then the operator \(A\) \((U)\) is called \(J\)-self-adjoint (\(J\)-unitary). A pair of subspaces \(\{\mathcal L_1,\mathcal L_2\}\) is called dual if \((\mathcal L_1,\mathcal L_2)=0\) and \(\mathcal L_1\subset\mathfrak{P}_+\), \(\mathcal L_2\subset\mathfrak{P}_-\), and a maximal dual pair if, in addition, \(\mathcal L_1\in\mathfrak{M}_+\) and \(\mathcal L_2\in\mathfrak{M}_-\).

R. S. Phillips posed the problem of extending a dual pair of subspaces \(\{\mathcal L_1^0,\mathcal L_2^0\}\), invariant with respect to an algebra \(R\), to a maximal dual pair of subspaces \(\{\mathcal L_1,\mathcal L_2\}\) invariant with respect to \(R\) \((^2)\). In the commutative case the problem was solved under the additional condition of symmetry of \(R\) with respect to ordinary adjunction, and in the noncommutative case under \(A^0=A^*\) for all \(A\in R\), where \(A^*\) is the ordinary adjoint of \(A\), or if \(\mathcal L_1\oplus\mathcal L_2=\Pi\), which is equivalent in the commutative case to the fact that the group of \(J\)-unitary operators \(G\) generating the algebra \(R\) is bounded in norm by some constant \(C\) \((^2)\). Thus, for \(U\in G\), \(\|U^n\|\leqslant C\), \(n=0,\pm1,\pm2,\ldots\).

We shall solve the extension problem for a commutative algebra \(R\) under the condition \(\|U^n\|\leqslant C_U\), \(n=0,\pm1,\pm2,\ldots\), and \(P_+UP_-\in\gamma_\infty\) for every \(U\in G\), where \(\gamma_\infty\) is the aggregate of all completely continuous operators in \(\Pi\).

Theorem 1. Any dual pair of subspaces \(\{\mathcal L_1,\mathcal L_2\}\), invariant with respect to a commutative algebra \(R\), extends to a maximal dual pair of subspaces \(\{\mathcal L_1,\mathcal L_1\}\) invariant with respect to \(R\), if for every \(U\in G\), \(\|U^n\|\leqslant C_U\), \(n=0,\pm1,\pm2,\ldots\), and \(P_+UP_-\in\gamma_\infty\).

Proof. The operator \(U\), with respect to the decomposition \(\Pi=\Pi_+\oplus\Pi_-\), has the matrix representation

\[ U\sim \begin{pmatrix} U_{11} & U_{12}\\ U_{21} & U_{22} \end{pmatrix} \tag{1} \]

and maps \(\mathfrak{M}_+\) onto itself, inducing a fractional-linear transformation of \(\mathfrak{K}_+\) onto itself

\[ f_U(K)=(U_{21}+U_{22}K)(U_{11}+U_{12}K)^{-1}. \tag{2} \]

The set \(\mathfrak{K}_+\) is bicompact in the weak operator topology, and the transformation (2) is continuous in this topology by virtue of the condition

\(P_+UP_- \in \gamma_\infty\) (³). By the Schauder–Tikhonov principle, the transformation (2) has a fixed point \(K_0\) in \(\mathfrak K_+\), realizing a subspace \(\mathcal L_0 \in \mathfrak M_+\) invariant with respect to the operator \(U\), so that \(\mathcal L=\{x\in\Pi: x=x_+\oplus K_0x_+,\, x_+\in\Pi_+\}\).

Let \(F_U\) be the set of fixed points of the transformation (2). Since the transformation (2) is weakly continuous, \(F_U\) is a bicompact subset of \(\mathfrak K_+\) in the weak topology. For the group \(G_k\) formed by any finite number of operators \(U_1,U_2,\ldots,U_k\) from \(G\), the assertion of Theorem 1 holds, since, by commutativity, \(G_k\) is bounded by the constant \(\alpha_k=C_1C_2\ldots C_k\), where \(\|U_i^n\|\le C_i,\ n=0,\pm1,\pm2,\ldots;\ 1\le i\le k\).

Let \(\{\mathcal L_1^0,\mathcal L_2^0\}\) be any dual pair invariant with respect to \(R\), and let \(\widehat{\mathfrak M}_+\) be the set of all subspaces from \(\mathfrak M_+\) containing \(\mathcal L_1^0\) and \(J\)-orthogonal to \(\mathcal L_2^0\). It follows from (4) that the set of operators \(\widehat{\mathfrak K}_+\) from \(\mathfrak K_+\) corresponding to the set of subspaces \(\widehat{\mathfrak M}_+\) from \(\mathfrak M_+\) under \(\mathfrak M_+\leftrightarrow\mathfrak K_+\) is a convex bicompact set in the weak operator topology. The set \(\widehat{\mathfrak M}_+\) is mapped by each operator \(U\) from \(G\) onto itself, which induces a transformation of \(\widehat{\mathfrak K}_+\) onto itself of the form (2), continuous in the weak operator topology. As above, the set of fixed points \(\widehat{\mathfrak K}_+^U\) of this transformation is a bicompact subset in the weak operator topology, realizing the set of subspaces \(\widehat{\mathfrak M}_+^U\) from \(\widehat{\mathfrak M}_+\) invariant with respect to the operator \(U\). Since there exists a subspace \(\mathcal L_1^k\) from \(\widehat{\mathfrak M}_+\) invariant with respect to the group \(G_k\), it follows that \(\bigcap_{G_k}\widehat{\mathfrak K}_+^U\ne\varnothing\), which, together with the bicompactness of the sets \(\widehat{\mathfrak K}_+^U\), gives that \(\bigcap_{G_k}\widehat{\mathfrak K}_+^U\ne\varnothing\). The last relation shows that there exists a subspace \(\mathcal L_1\) from \(\widehat{\mathfrak M}_+\), invariant with respect to \(G\). If now \(\mathcal L_2\) is the \(J\)-orthogonal complement in \(\Pi\) to \(\mathcal L_1\), then \(\mathcal L_2\in\mathfrak M_-\), \(\mathcal L_2^0\subseteq\mathcal L_2\), and, moreover, \(\mathcal L_2\) is invariant with respect to \(G\).

Thus, the pair \(\{\mathcal L_1,\mathcal L_2\}\) is a maximal dual pair of subspaces extending the initial dual pair \(\{\mathcal L_1^0,\mathcal L_2^0\}\), invariant with respect to the group \(G\), and therefore also with respect to the algebra \(R\), since \(R\) is the linear span of \(G\).

Let us give some consequences of the theorem proved. Let \(B(H_1,H_2)\) be the space of all bounded linear operators from the Hilbert space \(H_1\) into the Hilbert space \(H_2\); \(\mathfrak K\) the unit ball in \(B(H_1,H_2)\); \(S\) the surface of the ball \(\mathfrak K\), i.e. \(S=\{K\in\mathfrak K:\|K\|=1\}\), and let \(f(K)\) be a transformation of \(\mathfrak K\) into itself of the form

\[ f(K)=(A+BK)(C+DK)^{-1}. \tag{3} \]

Consider in the space \(H=H_1\oplus H_2\) the operator \(U\) which, relative to the decomposition \(H=H_1\oplus H_2\), is given by the matrix

\[ U\sim \begin{pmatrix} C & D\\ A & B \end{pmatrix}. \tag{4} \]

Put \(J=P_1-P_2\), where \(P_iH=H_i,\ i=1,2\). Obviously, if the operator \(U\) is \(J\)-unitary in \(H\), then the transformation (3) induced by it maps \(\mathfrak K\) onto \(\mathfrak K\) in such a way that \(S\) is mapped onto \(S\).

Introduce in \(H\) the sets \(\mathfrak P_+\), \(\mathfrak M_+\) \((\mathfrak P_-,\mathfrak M_-)\) analogously to the corresponding sets in \(\Pi\).

Corollary 1. Let \(\mathfrak U\) be a finite commutative family of transformations of \(\mathfrak K\) into itself of the form (3), and let each operator constructed from the corresponding transformation from \(\mathfrak U\) by means of relation (4) be \(J\)-unitary in the space \(H=H_1\oplus H_2\). If each of the transformations \(f_i(K)\) from \(\mathfrak U\) has a fixed point \(K_i\) such that \(\|K_i\|<1\), then there exists a common fixed point \(K\) of the transformations \(\mathfrak U\) such that \(\|K\|<1\).

Corollary 2. The assertion of Corollary 1 remains valid without the requirement that the family \(\mathfrak U\) be finite, if the transformations \(f(K)\) from \(\mathfrak U\) are continuous in the weak operator topology. In this case the norm of the common fixed pointך

$K \leqslant 1$. For weak continuity of a transformation of the form (3), one may require the full continuity of the operator $D$.

Theorem 2. If the transformation $f(K)$ from $\mathfrak U$ has a fixed point $K_0$ such that $\|K_0\|<1$, then only the following cases are possible:

a) $K_0$ is the unique fixed point of the transformation $f(K)$;

b) there exists an uncountable set of fixed points of the transformation $f(K)$, and among them there is necessarily a point on the set $S$.

Proof. Let $\mathcal L_0$ from $\mathfrak M_+$ be the invariant subspace of the operator $U$ corresponding to the fixed point $K_0$. Since $\|K_0\|<1$, the decomposition

\[ \Pi=\mathcal L_0\oplus\mathcal F_0, \tag{5} \]

is valid, where $\mathcal F_0$ is the subspace from $\mathfrak M_-$ (1) invariant with respect to the operator $U$. With respect to the decomposition (5), the operator $U$ has the matrix representation

\[ U\sim \begin{pmatrix} \widetilde U_{11} & 0\\ 0 & \widetilde U_{22} \end{pmatrix}. \tag{6} \]

Introduce a new definite scalar product

\[ [x,y]_1=(x_+,y_+)-(x_-,y_-), \tag{7} \]

where $x_+,y_+\in\mathcal L_0$, and $x_-,y_-\in\mathcal F_0$. Let $\widetilde P_1(\widetilde P_2)$ be the orthoprojector in $H$ onto $\mathcal L_0(\mathcal F_0)$, and let $\widehat{\mathfrak K}$ be the totality of all operators acting contractively (with respect to the new norm) from $\mathcal L_0$ into $\mathcal F_0$. The new indefinite scalar product

\[ (x,y)=[\widetilde Jx,y]_1, \tag{8} \]

where $\widetilde J=\widetilde P_1-\widetilde P_2$, coincides with the scalar product $(x,y)$, whence it follows that there exists a one-to-one correspondence $\mathfrak M_+\leftrightarrow \widehat{\mathfrak K}$, and therefore also $\mathfrak K\leftrightarrow\widehat{\mathfrak K}$, with $\widetilde S\leftrightarrow S$, where $\widetilde S$ is the surface of the ball $\widehat{\mathfrak K}$ (i.e., $\widetilde S=\{\widetilde K\in\widehat{\mathfrak K}:\|\widetilde K\|_1=1\}$). The operator $U$ induces on the ball $\widehat{\mathfrak K}$ the linear transformation

\[ f_U(\widetilde K)=\widetilde U_{22}\widetilde K\widetilde U_{11}^{-1}, \tag{9} \]

which proves the assertion of Theorem 2.

Let $f(K)\in\mathfrak U$. Define in the usual way the degree of the transformation $f(K)$: $f^n(K)=f(f^{n-1}(K))$, where $(n-1)$ is any natural number. Obviously, $f^n(K)\in\mathfrak U$.

Theorem 3. Let the operator constructed from $f(K)$ by means of relation (4) be $J$-unitary in $H=H_1\oplus H_2$. If there exists a fixed point $\widetilde K_0$ of the transformation $f^n(K)$ such that $\|\widetilde K_0\|<1$, then there also exists a fixed point $K_0$ of the transformation $f(K)$ such that $\|K_0\|<1$.

Proof. It is obvious that the transformation $f^n(K)$ is induced by the operator $U^n$, where $U$ is the operator in $H$ defined above. Further, from $f^n(\widetilde K_0)=\widetilde K_0$ and from the fact that $\|\widetilde K_0\|<1$, we have, by (2),

\[ \|(U^n)^l\|\leqslant C,\qquad l=0,\pm1,\pm2,\ldots \tag{10} \]

The $J$-unitary operators $U,U^2,\ldots,U^{n-1}$ are bounded, which together with (10) gives

\[ \|U^m\|\leqslant C,\qquad m=0,\pm1,\pm2,\ldots \tag{11} \]

It follows from (2) that (11) is equivalent to the assertion of Theorem 3.

In conclusion I express my sincere gratitude to Prof. M. A. Naimark for his constant attention to this work.

Moscow Institute of Physics and Technology

Received
7 XII 1966

REFERENCES

1. Yu. P. Ginzburg, I. S. Iokhvidov, Uspekhi Mat. Nauk, 17, 4 (106), 3 (1962).
2. R. S. Phillips, Proc. Intern. Sympos. on Linear Spaces, Jerusalem, 1960, 1961.
3. M. G. Krein, DAN, 154, No. 5, 1023 (1964).
4. G. K. Langer, DAN, 169, No. 1, 12 (1966).