M. A. NAIMARK

ON COMMUTATIVE ALGEBRAS OF OPERATORS IN THE SPACE \(\Pi_k\)

(Presented by Academician L. S. Pontryagin on 7 IX 1964)

1. Let \(R\) be a commutative algebra of bounded linear operators in the Pontryagin space\(^*\) \(\Pi_k\). The algebra \(R\) is called symmetric if from \(A \in R\) it follows that \(A^* \in R\), where \(A^*\) is the operator in \(\Pi_k\) defined by the condition \((Ax,y)=(x,A^*y)\), and \((x,y)\) is the indefinite scalar product in \(\Pi_k\). Two algebras \(R, R'\) in the spaces \(\Pi_k\) and \(\Pi_k'\) are called equivalent if there exists a one-to-one linear mapping of \(\Pi_k\) onto \(\Pi_k'\) that preserves the scalar product and maps \(R\) onto \(R'\). The aim of the present paper is to describe, up to equivalence, commutative symmetric algebras (c.s.a.) in \(\Pi_k\). For \(k=1\) this problem was solved by the author in \((^2)\).

2. Let \(R\) be a c.s.a. in \(\Pi_k\). On the basis of Corollary 2 of Theorem 1 in \((^3)\) (see also \((^4)\)) there exists a \(k\)-dimensional nonnegative subspace \(\mathfrak P\), invariant with respect to all \(A \in R\), and in \(\mathfrak P\) there exists a vector \(x \ne 0\) which is a common eigenvector for all \(A \in R\), i.e.

\[ Ax=\lambda(A)x \quad \text{for all } A\in R, \tag{2,1} \]

where \(A \to \lambda(A)\) is a homomorphism of the algebra \(A\) into the field \(C\) of complex numbers. Every homomorphism \(\lambda \to \lambda(A)\) of the algebra \(R\) into \(C\) is called an eigenfunctional (e.f.) of the algebra \(R\), if there exists a nonnegative vector \(x\ne0\) for which (2,1) holds; the set
\[ S_\lambda=\{x:x\in\Pi_k,\ (A-\lambda(A)1)^l x=0 \text{ for some } l=l(x) \text{ and all } A\in R\} \]
is then called the root lineal of the e.f. \(\lambda\). In view of the preceding, an e.f. always exists; the corresponding root lineals are invariant with respect to all \(A\in R\). An e.f. \(\lambda\) is called real if \(\lambda(A^*)=\overline{\lambda(A)}\), and nonreal otherwise.

I. The nonreal eigenfunctionals, if they exist, form a finite set \(\lambda_1,\mu_1,\ldots,\lambda_\sigma,\mu_\sigma\), where \(\mu_j(A)=\overline{\lambda_j(A^*)}\); the corresponding root lineals \(S_{\lambda_j}, S_{\mu_j}\) are finite-dimensional zero coskew-associated subspaces and
\[ S_{\lambda_j}+S_{\mu_j}\perp S_{\lambda_l}+S_{\mu_l}\quad \text{for } j\ne l. \]

The space
\[ H=\sum_{j=1}^{\sigma}\oplus(S_{\lambda_j}+S_{\mu_j}) \]
is called the hyperbolic space of the algebra \(R\).

It is obvious that \(H\) is a finite-dimensional space of type \(\Pi_k\), invariant with respect to all \(A\in R\); therefore its orthogonal complement \(H^\perp\) is also invariant with respect to all \(A\in R\), \(\Pi_k=H\oplus H^\perp\), and the restriction of \(R\) to \(H^\perp\) no longer has nonreal e.f. Therefore in what follows we shall assume that \(R\) has no nonreal e.f. in \(\Pi_k\). Let \(\mathfrak P\) be a nonnegative \(k\)-dimensional subspace invariant with respect to all \(A\in R\); \(\lambda_1,\ldots,\lambda_p\) are all the distinct (real) e.f. of \(R\) with eigenvectors in \(\mathfrak P\). Put \(\rho_j=\dim(S_{\lambda_j}\cap\mathfrak P)\).

II. The functionals \(\lambda_1,\ldots,\lambda_p\) and the numbers \(\rho_1,\ldots,\rho_p\) do not depend on the choice of the \(k\)-dimensional subspace \(\mathfrak P\), invariant with respect to all \(A\in R\);

\(^*\) For the definition and basic properties of the spaces \(\Pi_k\) and of operators in them, see \((^1)\).

and every l.f. \(R\) coincides with one of these functionals \(\lambda_j\), \(j=1,\ldots,p\). Put

\[ \mathfrak D_j=\{x:x\in \Pi_k,\ (A-\lambda_j(A)1)^{\rho_j}x=0 \quad \text{for all } A\in R\}; \tag{2,2} \]

\[ \mathfrak D=\sum_{j=1}^{p}\oplus \mathfrak D_j . \tag{2,3} \]

\[ \mathfrak M=\mathfrak D^\perp,\qquad \mathfrak N=\mathfrak D\cap \mathfrak M; \tag{2,4} \]

\(\mathfrak D_j,\mathfrak D,\mathfrak M,\mathfrak N\) are closed subspaces, invariant with respect to all \(A\in R\); \(\mathfrak N\) is the null subspace; \(\mathfrak D\) contains every \(k\)-dimensional nonnegative subspace invariant with respect to all \(A\in R\). Hence it follows that \(\mathfrak M\) is nonpositive; \(\mathfrak D,\mathfrak M,\mathfrak N\) are called, respectively, the principal, fundamental, and fundamental null subspace of the algebra \(R\).

Let \(\mathfrak N'\) be a subspace skew-connected with \(\mathfrak N\). Put \(\mathfrak H=\mathfrak M\cap {\mathfrak N'}^\perp\), \(\Pi=\mathfrak D\cap {\mathfrak N'}^\perp\). Then \(\mathfrak H\) is negative, i.e., essentially a Hilbert space, while \(\Pi\) is positive, negative, or of type \(\Pi_k\),

\[ \mathfrak M=\mathfrak N\oplus \mathfrak H,\qquad \mathfrak D=\mathfrak N\oplus \Pi; \tag{2,5} \]

\[ \Pi_k=(\mathfrak N+\mathfrak N')\oplus \mathfrak H\oplus \Pi \tag{2,6} \]

(see \((^1)\), Theorem 4.1). If in this case \(\mathfrak N=(0)\), then one should take \(\mathfrak N'=(0)\), \(\mathfrak H=\mathfrak M\), \(\Pi=\mathfrak D\), and (2,6) passes into \(\Pi_k=\mathfrak M\oplus \mathfrak D\).

1. Put \(\mathfrak N_j=\mathfrak N\cap S_{\lambda_j}\). Let \(j=1,\ldots,q\); \(q\leq p\), be all those values of \(j\) for which \(\mathfrak N_j\ne(0)\) (if there are no such values of \(j\), then, by definition, \(q=0\) and \(\mathfrak N=(0)\)); then \(\mathfrak N=\sum_{j=1}^{q}\cdot\mathfrak N_j\). In \(\mathfrak N_j\) there exists a basis \(x_{jl}\), \(l=1,\ldots,r_j\); \(j=1,\ldots,q\), such that the matrix of each operator \(A\in R\) in \(\mathfrak N_j\) is triangular with \(\lambda_j(A)\) on the diagonal, i.e.,

\[ Ax_{jl}=\sum_{s=1}^{l}\lambda_{jls}(A)x_{js}, \tag{3,1} \]

where \(\lambda_{jll}(A)=\lambda_j(A)\). Let \(y_{jl}\), \(l=1,\ldots,r_j\); \(j=1,\ldots,q\), be a basis in \(\mathfrak N'\), biorthogonal to \(x_{jl}\), so that \((x_{jl},y_{j'l'})=\delta_{jj'}\delta_{ll'}\). Applying to \(A^*y_{jl}\) the relation (2,6), we conclude that

\[ A^*y_{jl}= \sum_{\mu=1}^{q}\sum_{s=1}^{r_\mu}\alpha_{j l\mu s}(A)x_{\mu s} + \sum_{\mu=l}^{r_j}\overline{\lambda_{j\mu l}(A)}\,y_{j\mu} +h_{jl}(A)+\pi_{jl}(A), \tag{3,2} \]

where \(\alpha_{j l\mu s}(A)\) are numerical functions on \(R\), and \(h_{jl}(A)\), \(\pi_{jl}(A)\) are vector-functions on \(R\) with values in \(\mathfrak H\) and \(\Pi\), respectively. Further, applying to \(Ah\), \(h\in\mathfrak H\), the first relation (2,5) and using (3,2), we obtain:

\[ Ah=\sum_{j=1}^{q}\sum_{l=1}^{r_j}(h,h_{jl}(A))x_{jl}+A_1h, \tag{3,3} \]

where \(A_1\) is an operator in \(\mathfrak H\).

III. The correspondence \(A\to A_1\) is a symmetric homomorphism, continuous in the operator norm, of the algebra \(R\) onto the symmetric commutative algebra \(R_1=\{A_1:A\in R\}\) of operators in the Hilbert space \(\mathfrak H\), where \(A_1^*\) is the ordinary adjoint operator to \(A_1\).

The algebra \(R_1\), up to equivalence, does not depend on the choice of the subspace \(\mathfrak N'\) skew-connected with \(\mathfrak N\).

Similarly we find that for \(\pi \in \Pi\)

\[ A\pi=\sum_{j=1}^{q}\sum_{l=1}^{r_j}(\pi,\pi_{jl}(A))x_{jl}+A_2\pi, \tag{3,4} \]

where \(A_2\) is an operator in \(\Pi\).

IV. The correspondence \(A\to A_2\) is a continuous, in the operator norm, symmetric homomorphism of the algebra \(R\) onto a commutative symmetric algebra
\[ R_2=\{A_2:\ A\in R\} \]
of operators in \(\Pi\), and this algebra, up to equivalence, does not depend on the choice of the subspace associated with \(\mathfrak N\).

From (2,2) and (2,3) it follows that

\[ \Pi=\sum_{j=1}^{p}\oplus \Pi^j, \tag{3,5} \]

where each \(\Pi^j\) is a subspace, positive, negative, or of type \(\Pi_k\), invariant with respect to all \(A_2\in R_2\), and, moreover,

\[ (A_2-\lambda_j(A)1)^{\rho_j}=0 \quad \text{on } \Pi^j \text{ for all } A\in R. \tag{3,6} \]

We shall call an algebra \(R\) in a Hilbert space or in a space of type \(\Pi_k\) degenerate if there exists a homomorphism \(A\to \lambda(A)\) of the algebra \(R\) into \(C\) and a natural number \(\rho\) such that \((A-\lambda(A)1)^\rho=0\) for all \(A\in R\). From (3,6) it follows easily that \(R_2\) is degenerate on each \(\Pi^j\). It is also easy to see that a degenerate algebra in a Hilbert space is simply an algebra of operators of multiplication by a number. The structure of degenerate algebras in a space \(\Pi_k\) will be described below in § 4.

Formulas (3,2) with \(A^*\) instead of \(A\), and (3,1), (3,3), (3,4), completely determine the operators \(A\in R\). To obtain now a description of all possible c.s.a. in \(\Pi_k\), it remains only to rewrite these formulas in the form

\[ Ax_{jl}=\sum_{s=1}^{l}\lambda_{jls}x_{js}; \tag{3,7} \]

\[ Ah=\sum_{j=1}^{q}\sum_{l=1}^{r_j}(h,h_{jl})x_{jl}+A_1h; \tag{3,8} \]

\[ A\pi=\sum_{j=1}^{q}\sum_{l=1}^{r_j}(\pi,\pi_{jl})x_{jl}+A_2\pi; \tag{3,9} \]

\[ Ay_{jl}=\sum_{\mu=1}^{q}\sum_{s=1}^{r_\mu}\alpha^{*}_{jl\mu s}x_{\mu s} +\sum_{\mu=l}^{r_j}\overline{\lambda}^{\,*}_{j\mu l}y_{j\mu} +h^{*}_{jl}+\pi^{*}_{jl}, \tag{3,10} \]

where the systems
\[ \xi=\{\lambda_{jls},\alpha_{jl\mu s},h_{jl},\pi_{jl},A_1,A_2\}, \]
which determine the operators \(A\in R\), range over a certain linear manifold \(\Xi\), on which an involution
\[ \xi\to \xi^*=\{\lambda^{*}_{jls},\alpha^{*}_{jl\mu s},h^{*}_{jl},\pi^{*}_{jl},A^{*}_1,A^{*}_2\} \]
is defined and which satisfies a system of axioms expressing the fact that the corresponding operators \(A\) range over a c.s.a. Every such linear manifold \(\Xi\) will be called a determining manifold associated with the decomposition (2,6) and with the algebras \(R_1,R_2\).

Thus, the following holds.

Theorem 1. Every c.s.a. in \(\Pi_k\) is determined by a decomposition of the form (2,6), c.s.-algebras \(R_1,R_2\) in \(\mathfrak H\) and \(\Pi\), and a determining manifold \(\Xi\) associated with them. Here \(\mathfrak N,\mathfrak N'\) are associated null subspaces of dimension \(k_0\le k\); \(\mathfrak H\) is negative; \(\Pi\) is negative, positive of dimension \(k-k_0\), or a space \(\Pi_{k-k_0}\), when \(k>k_0\);

\[ \Pi=\sum_{j=1}^{p}\oplus \Pi^j, \]

where each \(\Pi^j\) is nondegenerate, invariant with respect to all \(A_2\in R_2\), narrowed-

… \(R_2\) on \(\Pi^j\) is a degenerate algebra, and \(R\) consists of all operators \(A\) in \(\Pi_k\) defined by formulas (3,7)—(3,10), where \(\xi\) ranges over \(\Xi\); \(x_{jl}\) is a basis in \(\mathfrak N\), and \(y_{jl}\) is a biorthogonal basis in \(\mathfrak N'\).

Conversely, every decomposition of the form (2,6), a.c.a. \(R_1, R_2\), and the defining manifold \(\Xi\) associated with them determine in this way a c.c. algebra in \(\Pi_k\).

1. The preceding construction can be simplified if, leaving everything else unchanged, \(\mathfrak L\) is replaced by any \(k\)-dimensional nonnegative subspace \(\mathfrak P\), invariant with respect to all \(A \in R\), which we shall again denote here by \(\mathfrak L\). Then \(\Pi\) will be finite-dimensional positive and therefore the corresponding degenerate algebras in \(\Pi^j\) will be algebras of multiplication by a scalar. However, this modified construction will now depend on the choice of \(\mathfrak P\).

This modified construction can be applied to a degenerate algebra \(R\) in \(\Pi_k\). In this case \(q \leqslant 1\), \(R_1, R_2\) are algebras of multiplication by a scalar.

1. Suppose now that \(\Pi_k\) is separable and that \(R\) is separable in the operator norm and contains the identity operator. Then one can obtain a final realization of the restriction of the algebra \(R\) to the principal space \(\mathfrak N\). Let \(\overline R_1\) be the closure of \(R_1\) in the operator norm; \(T\) the bicompact space of maximal ideals \(t\) of the algebra \(\overline R_1\); \(A(t)\) the value of the element \(A_1 \in \overline R_1\) at the ideal \(t\). As is known (see, for example, \((^5)\), § 41 and Supplement 11, or \((^6)\), Ch. I, § 7 and Ch. II), \(R_1\), up to equivalence, is realized in the following way. There exist a Borel measure \(\sigma\) and at most a countable family of closed sets
   \[ F_1 = T \supset F_2 \supset F_3 \supset \cdots \]
   such that
   \[ \mathfrak L = \int_T \mathfrak L(t)\,d\sigma, \]
   where \(\mathfrak L(t)\) is a subspace of \(l^2\) consisting of all such \(h=\{h_1,h_2,\ldots\}\in l^2\) that \(h_k=h_{k+1}=\cdots=0\) for \(t\in F_k\). Here \(\overline R_1\) is the set of all operators \(A_1\{h(t)\}=\{A(t)h(t)\}\), \(A(t)\in C(T)\), \(\{h(t)\}\in\mathfrak L\), \(\{A(t), A_1\in \overline R_1\}\) is dense in \(C(T)\), and
   \[ A_1^*\{h(t)\}=\{\overline{A(t)}h(t)\}. \]
   We shall call this realization of \(\overline R_1\) and \(R_1\) canonical; by III, it does not, up to equivalence, depend on the choice of \(\mathfrak N'\).

Cf. \(\lambda_j\) will be called special if there exists a point \(t_j\in T\) such that \(\lambda_j(A)=A(t_j)\) for all \(A\in R\); \(t_j\) is called the corresponding special point of \(R\). Put \(K_j=\mathfrak L(t_j)\), if \(t_j\) is a special point and \(\sigma_j=\sigma(\{t_j\})>0\), and \(K_j=(0)\) otherwise. \(K_j\) is a Hilbert space; it is called a special space of \(R\).

V. In the canonical realization of the algebra \(R_1\)
\[ h_{jl}(A)=h_{jl}(A,t)=(A(t)-\lambda_j(A))\xi_{jl}(t)-\sum_{\mu=l+1}^{r_j}\overline{\lambda_{j\mu l}(A)}\,\xi_{j\mu}(t) \quad \text{for } t\ne t_j, \tag{4,1} \]
\[ h_{jl}(A,t_j)=k_{jl}(A)\in K_j, \]
where \(\xi_{jl}(t)\in \mathfrak L(t)\); \(\xi_{jl}(t)\) is a \(\sigma\)-measurable function of \(t\) such that the right-hand side of (4,1) belongs to \(\mathfrak L\) for all \(A_1\in R_1\).

Theorem 1 and Proposition V give a realization, up to equivalence, of the s.c. algebras, which we shall call their canonical model. The condition for the equivalence of two canonical models will be indicated in a subsequent communication.

Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
27 VIII 1964

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