UDC 513.882

MATHEMATICS

R. S. ISMAGILOV

ON RINGS OF OPERATORS IN A SPACE WITH INDEFINITE METRIC

(Presented by Academician L. S. Pontryagin on 25 I 1966)

A well-known theorem of L. S. Pontryagin asserts that a self-adjoint operator in a space of type \(\Pi_n\) \((0<n<\infty)\) has an \(n\)-dimensional nonnegative invariant subspace \((^1)\). This result was extended by M. A. Naimark to a family of mutually commuting normal operators \((^2)\). In the present article a generalization of these results is obtained for symmetric representations in a space of type \(\Pi_n\) of rings with involution having a complete system of representations whose dimensions do not exceed the number \(m<\infty\). Another (very simple) proof of L. S. Pontryagin’s theorem is also obtained.

Theorem 1. Let \(T_x\) be a symmetric representation in a space \(H\) of type \(\Pi_n\) of a ring \(R\) with involution, having a complete system of representations whose dimensions do not exceed \(m<\infty\). Then in \(H\) there exists a subspace \(H_1\), invariant with respect to all \(T_x\) \((x\in R)\), such that \(\dim H_1\le mn\), and the subspace \(H_2=\{x:(x,y)=0\) for all \(y\in H_1\}\) is nonnegative.

Theorem 2. An irreducible ring of operators in a space \(H\) of type \(\Pi_n\), closed in the strong topology and containing, together with each operator, also its adjoint (in the indefinite metric), coincides with the ring of all bounded operators in \(H\).

We pass to the proof.

Lemma 1*. Let \(C\) be a family of bounded operators in a space \(H\) of type \(\Pi_n\), and suppose that from \(A\in C\) it follows that \(A^*\in C\), and that \(C\) has no null invariant subspaces in \(H\). Then

\[ H=H^0\oplus H^1\oplus\cdots\oplus H^m, \tag{1} \]

where \(H^0\) is positive, \(H^i\) \((1\le i\le m)\) is either a space of type \(\Pi_{k_i}\) or a \(k_i\)-dimensional negative subspace, with \(\sum k_i=n\); moreover \(AH_j\subseteq H_j\) \((A\in C,\ 0\le j\le m)\), and the restriction of \(C\) to \(H^i\) \((1\le i\le m)\) is irreducible. The summand \(H^0\) in (1) may be absent.

Proof. Note that, under the conditions of the lemma, the family \(C\) cannot have degenerate invariant subspaces. (Indeed, if \(H_0\) is a degenerate invariant subspace, then \(H_{0,0}=\{x:x\in H_0,\ (x,y)=0\) for all \(y\in H_0\}\) will be an invariant null subspace.)

Suppose that there exist positive invariant subspaces in \(H\). The set \(L\) of such subspaces is partially ordered by inclusion. If \(\{H_\gamma\}\) is a chain in \(L\), then the subspace \(\mathcal H=\bigcup H_\alpha\) is invariant with respect to \(C\) and nonnegative and, by the preceding remark, nondegenerate; hence \(\mathcal H\in L\), so that the chain \(\{H_\gamma\}\) has an upper bound \(\mathcal H\). By Zorn’s lemma, there exists in \(H\) a maximal positive invariant subspace \(H^0\).

* This lemma was obtained by the author in the article \((^6)\); at the same time an analogous assertion was obtained by M. A. Naimark.

In the subspace \(H_1=H\oplus H^0\) (invariant with respect to \(C\)) there are neither positive nor degenerate subspaces. Hence it follows that \(C\) in \(H_1\) is either irreducible, or there is a decomposition \(H_1=H_{1,1}\oplus H_{1,2}\), where \(H_{1,i}\) \((i=1,2)\) is invariant and is either a space of type \(\Pi_{s_i}\) or an \(s_i\)-dimensional negative subspace, with \(0<s_i<n\), \(s_1+s_2=n\). If the restriction of \(C\) to \(H_{1,i}\) \((i=1,2)\) is reducible, then analogous reasoning leads to a decomposition \(H_{1,i}=H_{1,i,1}\oplus H_{1,i,2}\).

Continuing this reasoning, we arrive (in no more than \(n\) steps) at the required decomposition.

Lemma 2. A self-adjoint operator \(A\) in a space \(H\) of type \(\Pi_n\) \((0<n<\infty)\) has a nontrivial invariant subspace.

Proof. If \((x,y)\) is the indefinite scalar product in \(H\), then
\[ (x,y)=[(E-2P)x,y], \]
where \([x,y]\) is the usual scalar product in \(H\), and \(P\) is a projection operator, with \(\dim PH=n\).

If \(A^*=A\) in the \(\Pi_n\)-metric and \(A^+\) is the operator adjoint to \(A\) with respect to the scalar product \([x,y]\), then, obviously,
\[ A^+-A=2A^*P-2PA, \]
whence it follows that the operator \(\operatorname{Im} A=i/2(A^+-A)\) is finite-dimensional; it is known \((^3)\) that such an operator has a nontrivial invariant subspace.

The following lemma is equivalent to L. S. Pontryagin’s theorem.

Lemma 3. A self-adjoint operator \(A\) in a space \(H\) of type \(\Pi_n\) has a nonpositive eigenvector.

Proof. If \(A\) has a zero invariant subspace \(\mathscr H_0\), then \(\dim \mathscr H_0\le n\), and the assertion of the lemma is obvious. If there are no such subspaces in \(H\), then \(H\) admits the decomposition indicated in Lemma 1, and the restriction of \(A\) to \(H_i\) \((1\le i\le m)\) has no nontrivial invariant subspaces. But then, by Lemma 2, \(\dim H_i=1\), as was required.

Lemma 4. If \(C\) is an irreducible set of operators in a space \(H\) of type \(\Pi_n\), and from \(A\in C\) it follows that \(A^*\in C\), and the operator \(B\) commutes with all \(A\in C\), then \(B=\lambda E\).

Proof. We may assume that \(B=B^*\) (otherwise we consider the operators \(B+B^*\), \(i(B-B^*)\)). If \(\lambda\) is an eigenvalue of \(B\) (see Lemma 3) and \(H_\lambda=\{x:Bx=\lambda x\}\), then, by the conditions of the lemma, \(AH_\lambda\subseteq H_\lambda\) \((A\in C)\). By the irreducibility of \(C\), \(H_\lambda=H\), i.e. \(B=\lambda E\), as was required.

Proof of Theorem 2. Let \(C\) be an irreducible ring of operators in \(H\), and suppose that from \(A\in C\) it follows that \(A^*\in C\), \(B\) is a bounded operator in \(H\), \(\{f_j^0,\ 1\le j\le N\}\) are elements of \(H\), and \(\varepsilon>0\). We shall show that there exists an \(A\in C\) such that
\[ \|Af_i^0-Bf_i^0\|<\varepsilon \qquad (1\le i\le N). \]
Obviously, we may assume that the vectors \(f_j^0\) are linearly independent.

Let \(\mathscr H=H\oplus\cdots\oplus H\) (\(N\) summands). In a natural way \(\mathscr H\) becomes a \(\Pi_{nN}\)-space. Further, let \(P_i\) be the operator acting by the formula
\[ P_i(f_1\ldots f_N)=f_i. \]
The operators
\[ B'(f_1\ldots f_N)=(Bf_1,\ldots,Bf_N) \]
form in \(\mathscr H\) a ring \(C'\) with a natural involution.

Let \(\mathscr H_0\) be the closure in \(\mathscr H\) of the set of vectors of the form
\[ (Bf_1^0,\ldots,Bf_N^0)\qquad (B\in C). \tag{2} \]
Obviously, \(B'\mathscr H_0\subseteq \mathscr H_0\) \((B'\in C')\), and \(\mathscr H_0\) is a nondegenerate subspace (otherwise the subspace \(\mathscr H_0^0=\{x:\ x\in\mathscr H_0,\ (x,y)=0\ \text{for all }y\in\mathscr H_0\}\) would be invariant with respect to \(B'\in C'\) and \(0<\dim\mathscr H_0^0\le nN\); but then the subspace \(P_i\mathscr H_0^0\subset H\) would be invariant with respect to \(C\) and finite-dimensional for all \(1\le i\le N\), whence \(P_i\mathscr H_0^0=0\), i.e. \(\mathscr H_0^0=0\)).

Define in \(\mathcal H\) the operator \(Q\) as follows:

\[ Qx=\begin{cases} x, & x\in \mathcal H_0,\\ 0, & x\in \mathcal H\ominus \mathcal H_0. \end{cases} \]

Then \(QA=AQ\) \((A\in C)\). Let \(\{Q_{ij},\,1\le i,j\le N\}\) be the matrix corresponding to \(Q\). Clearly, \(Q_{ij}A=AQ_{ij}\) \((A\in C)\), whence, by Lemma 4, \(Q_{ij}=\lambda_{ij}E\). At the same time
\[ Q(f_1^0\ldots f_N^0)=(f_1^0\ldots f_N^0), \]
i.e. \(\sum \lambda_{ij}f_j=f_i\) \((1\le i\le N)\), whence, by the linear independence of \(\{f_j\}\), \(\lambda_{ij}=0\) for \(i\ne j\), \(\lambda_{ii}=1\), i.e. \(Q=E\). This means that the system of vectors of the form (2) is dense in \(\mathcal H\), which proves the theorem.

Proof of Theorem 1. We shall use the following result of Godement (4): if a ring \(R\) has a complete system of representations of dimension \(\le m\), then every completely irreducible representation of \(R\) has dimension \(\le m\). Complete irreducibility means that every bounded operator in the representation space is a strong limit of operators \(T_x\) \((x\in R)\). However, Godement’s proof remains valid if the condition of complete irreducibility is replaced a priori by the weaker one: the strong closure of the ring \(\{T_x\}\) coincides with the ring of all operators in the representation space.

Returning to Theorem 1, suppose first that there are no null invariant subspaces in \(H\). Then the decomposition indicated in Lemma 1 holds, and, by Godement’s theorem and Theorem 2, \(\dim H_i\le m\), which proves the theorem.

If there are null invariant subspaces in \(H\), and \(H_0\) is some maximal one among them, then \(\dim H_0=k\le N\) and the subspace
\[ H_0^\perp\{x\in H,\;x\perp H_0\} \]
is invariant with respect to \(T_x\) \((x\in R)\), with \(H_0\subset H_0^\perp\). The quotient space
\[ \widehat H_0=H_0^\perp/H_0 \]
is naturally transformed into a space of type \(\Pi_{n-k}\) (5), and in \(\widehat H_0\) there arises a representation \(\widehat T_x\) that no longer contains null invariant subspaces. But then in \(\widehat H_0\) there exists an invariant subspace of dimension \(\le m(n-k)\). Thus in \(H_0^\perp\) there exists an invariant subspace of dimension
\[ k+(n-k)m\le nm, \]
satisfying the requirements of Theorem 1.

Moscow Institute of Electronic
Machine Building

Received
3 XI 1965

REFERENCES

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3. J. Schwartz, Commun. Pure and Appl. Math., 15, No. 2, 159 (1962).
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6. R. S. Ismagilov, Izv. Akad. Nauk SSSR, Ser. Mat., 30, No. 3 (1964).