MATHEMATICS

M. A. NAIMARK

ON CONDITIONS FOR THE UNITARY EQUIVALENCE OF COMMUTATIVE SYMMETRIC ALGEBRAS IN THE SPACE \(\Pi_k\)

(Presented by Academician L. S. Pontryagin on 21 IX 1964)

1. In the author’s preceding article \((^1)\), a description was given of commutative symmetric algebras (c.s.a.) in the Pontryagin space \(\Pi_k\). In the present article necessary and sufficient conditions are given for the equivalence of two c.s.a.’s realized in accordance with Theorem 1 in \((^1)\); the terminology and notation of article \((^1)\) are retained. For the case \(k=1\), equivalence conditions were indicated earlier by the author in \((^{2,3})\).

Let \(R,\widetilde R\) be equivalent c.s.a.’s in the spaces \(\Pi_k,\widetilde\Pi_k\), with only real eigenfunctionals (e.f.), realized by means of the decompositions

\[ \Pi_k=(\mathfrak N \dotplus \mathfrak N')\oplus \mathfrak H\oplus \Pi, \tag{1} \]

\[ \widetilde\Pi_k=(\widetilde{\mathfrak N}\dotplus \widetilde{\mathfrak N}')\oplus \widetilde{\mathfrak H}\oplus \widetilde\Pi \tag{1'} \]

of the algebras \(R_1,R_2,\widetilde R_1,\widetilde R_2\) in \(\mathfrak H,\Pi,\widetilde{\mathfrak H},\widetilde\Pi\), respectively, of biorthogonal bases \(\{x_{jl}\}\) in \(\mathfrak N\), \(\{y_{jl}\}\) in \(\mathfrak N'\); \(\{\widetilde x_{jl}\}\) in \(\widetilde{\mathfrak N}\), \(\{\widetilde y_{jl}\}\) in \(\widetilde{\mathfrak N}\), and defining many-valued mappings \(\Xi,\widetilde\Xi\). Let \(U\) be an operator which maps \(\Pi_k\) isometrically onto \(\widetilde\Pi_k\), such that the operators \(\widetilde A=UAU^{-1}\), \(A\in R\), form precisely the algebra \(\widetilde R\). It is not hard to verify that then:

I. If \(\lambda_1(A),\ldots,\lambda_p(A)\) are all the distinct e.f.’s of the algebra \(R\), then \(\widetilde\lambda_1(\widetilde A),\ldots,\widetilde\lambda_p(\widetilde A)\), where \(\widetilde\lambda_j(\widetilde A)=\lambda_j(A)\) for \(\widetilde A=UAU^{-1}\), are all the distinct e.f.’s of the algebra \(\widetilde R\).

II. If \(\mathfrak P,\mathfrak M,\mathfrak N;\widetilde{\mathfrak P},\widetilde{\mathfrak M},\widetilde{\mathfrak N}\) are the principal, basic, and basic null spaces of the algebras \(R\) and \(\widetilde R\), then
\[ \widetilde{\mathfrak P}=U\mathfrak P,\qquad \widetilde{\mathfrak M}=U\mathfrak M,\qquad \widetilde{\mathfrak N}=U\mathfrak N. \]

III. If \(\mathfrak P\) is a nonnegative \(k\)-dimensional subspace in \(\Pi_k\), invariant with respect to all \(A\in R\), and \(\mathfrak P=\mathfrak P_1\oplus\cdots\oplus\mathfrak P_p\) is its decomposition into root lineals in \(\mathfrak P\), corresponding to \(\lambda_1,\ldots,\lambda_p\), then
\[ \widetilde{\mathfrak P}=\widetilde{\mathfrak P}_1\oplus\cdots\oplus\widetilde{\mathfrak P}_p, \]
where \(\widetilde{\mathfrak P}=U\mathfrak P,\ \widetilde{\mathfrak P}_j=U\mathfrak P_j\), is the decomposition of \(\widetilde{\mathfrak P}\) into root lineals in \(\widetilde{\mathfrak P}\), corresponding to \(\widetilde\lambda_1,\ldots,\widetilde\lambda_p\).

An analogous assertion is valid if \(\mathfrak P,\mathfrak P_j\) are replaced by the subspaces \(\mathfrak N,\mathfrak N_j\).

Put

\[ x'_{jl}=U^{-1}\widetilde x_{jl},\qquad y'_{jl}=U^{-1}\widetilde y_{jl},\qquad U^{-1}\widetilde{\mathfrak N}=\mathfrak N'. \tag{2} \]

Then \(\{x'_{jl}\}\) is a basis in \(\mathfrak N'\), \(\{y'_{jl}\}\) is a basis in \(\widehat{\mathfrak N}'\), biorthogonal to \(\{x'_{jl}\}\); \(\mathfrak N,\widehat{\mathfrak N}'\) are skew-related. Moreover, \(\{x_{jl}\}\), as well as \(\{x'_{jl}\}\) for fixed \(j\), form a basis in \(\mathfrak N_j\); consequently,

\[ x'_{jl}=\sum_{s=1}^{r_j} a_{jls}x_{js}, \tag{3} \]

where \(a_j=\|a_{jls}\|\), \(l,s=1,\ldots,j\), is a nonsingular matrix. Further putting \(U^{-1}\mathfrak H=\mathfrak H'\), \(U^{-1}\Pi=\Pi'\), we have

\[ \mathfrak H'=\mathfrak M\cap \mathfrak H'^{\perp},\quad \Pi'=\mathfrak L\cap \mathfrak H'^{\perp},\quad \mathfrak M=\mathfrak R\oplus \mathfrak H',\quad \mathfrak L=\mathfrak M\oplus \Pi'; \tag{4} \]

\[ \Pi_k=(\mathfrak R\dotplus \mathfrak R')\oplus \mathfrak H'\oplus \Pi'. \tag{5} \]

Moreover,

\[ \Pi_k=(\mathfrak R\dotplus \mathfrak R')\oplus \mathfrak H\oplus \Pi. \tag{6} \]

By virtue of the third relation in (4), every element \(h\in\mathfrak H\) can be represented in the form

\[ h=\sum_{j=1}^{q}\sum_{l=1}^{r_j}(h,y'_{jl})x'_{jl}+h',\quad h'\in\mathfrak H. \tag{7} \]

On the other hand, applying (6) to \(y'_{jl}\) and taking (3) into account, we obtain

\[ y'_{jl}=\sum_{\mu=1}^{q}\sum_{\nu=1}^{r_\mu}\gamma_{jl\mu\nu}x_{\mu\nu} +\sum_{\nu=1}^{r_j}\bar b_{j\nu l}y_{j\nu}+h^0_{jl}+\pi^0_{jl}, \tag{8} \]

where \(\gamma_{jl\mu\nu}=(y'_{jl},y_{\mu\nu})\), \(h^0_{jl}\in\mathfrak H\), \(\pi^0_{jl}\in\Pi\), \(b_j=\|b_{j\nu l}\|\), \(\nu,l=1,\ldots,r_j\), is the matrix inverse to \(a_j\): \(b_j=a_j^{-1}\). From (8) we conclude that \((h,y'_{jl})=(h,h^0_{jl})\), and therefore (7) is rewritten in the form

\[ h=\sum_{j=1}^{q}\sum_{l=1}^{r_j}(h,h^0_{jl})x'_{jl}+h',\quad h'\in\mathfrak H'. \tag{9} \]

Define the operator \(W_1\) from \(\mathfrak H\) into \(\mathfrak H'\) by putting

\[ W_1h=h'=-\sum_{j=1}^{q}\sum_{l=1}^{r_j}(h,h^0_{jl})x'_{jl}+h. \tag{10} \]

IV. The operator \(W_1\) maps \(\mathfrak H\) isometrically onto \(\mathfrak H'\), and the inverse operator is given by the formula

\[ W_1^{-1}h'=\sum_{j=1}^{q}\sum_{l=1}^{r_j}(h',h^0_{jl})x_{jl}+h'. \tag{11} \]

An analogous assertion is valid for the operator \(W_2\) from \(\Pi\) onto \(\Pi'\), defined by the formula

\[ W_2\pi=\pi'=-\sum_{j=1}^{q}\sum_{l=1}^{r_j}(\pi,\pi^0_{jl})x'_{jl}+\pi. \tag{12} \]

Therefore the formulas \(V_1=UW_1\), \(V_2=UW_2\) define isometric operators from \(\mathfrak H\) and \(\Pi\) onto \(\mathfrak H\) and \(\Pi\), respectively.

Let now the operators \(A\in R\) and \(\widetilde A=U^{-1}AU\in\widetilde R\) be given, with the aid of the systems \(\xi=\{\lambda_{jls},\alpha_{jl\mu s},h_{jl},\pi_{jl},A_1,A_2\}\in\Xi\), \(\widetilde\xi=\{\widetilde\lambda_{jls},\widetilde\alpha_{jl\mu s},\widetilde h_{jl},\widetilde A_1,A_2\}\in\Xi\), by the formulas

\[ Ax_{jl}=\sum_{s=1}^{l}\lambda_{jls}x_{js},\quad Ah=\sum_{j=1}^{q}\sum_{l=1}^{r_j}(h,h_{jl})x_{jl}+A_1h, \]

\[ A\pi=\sum_{j=1}^{q}\sum_{l=1}^{r_j}(\pi,\pi_{jl})x_{jl}+A_2\pi, \tag{13} \]

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

\(h,h_{jl},h^*_{jl}\in\mathfrak H;\ \pi,\pi_{jl},\pi^*_{jl}\in\Pi\) and

\[ \widetilde A\widetilde x_{jl}=\sum_{s=1}^{b}\widetilde\lambda_{jls}\widetilde x_{js},\quad \widetilde A\widetilde h=\sum_{j=1}^{q}\sum_{l=1}^{r_j}(\widetilde h,\widetilde h_{jl})\widetilde x_{jl} +\widetilde A_1\widetilde h,\quad \widetilde A\widetilde\pi=\sum_{j=1}^{q}\sum_{l=1}^{r_j}(\widetilde\pi,\widetilde\pi_{jl})\widetilde x_{jl}+ \]
\[ +\widetilde A_2\widetilde\pi,\quad \widetilde A y_{jl}=\sum_{\mu=1}^{q}\sum_{s=1}^{r_\mu}\widetilde\alpha_{jl\mu s}^{*}x_{\mu s} +\sum_{\mu=1}^{r_j}\overline{\widetilde\lambda}_{j\mu l}y_{j\mu} +\widetilde h_{jl}^{*}+\widetilde\pi_{jl}^{*}, \tag{14} \]

\(\widetilde h,\widetilde h_{jl},\widetilde h_{jl}^{*}\in\widetilde{\mathfrak H};\ \widetilde\pi,\widetilde\pi_{jl},\widetilde\pi_{jl}^{*}\in\widetilde\Pi\) (see (3.8)—3.10) in \((^{1})\). Putting in (14) \(\widetilde A=UAU^{-1}\), \(h'=U^{-1}\widetilde h\), \(\pi'=U^{-1}\widetilde\pi\), \(h=W_1^{-1}h'\), \(\pi=W_2^{-1}\pi'\) and using (2), (3), (8), (10)—(13), we arrive at the following result:

Theorem 1. Let \(R,\widetilde R\) be commutative symmetric algebras with only real eigenfunctions in the spaces \(\Pi_k,\widetilde\Pi_k\), given by means of the decompositions (1), \((1')\), the bases \(\{x_{jl}\}\), \(\{y_{jl}\}\), \(\{\widetilde x_{jl}\}\), \(\{\widetilde y_{jl}\}\), \(l=1,\ldots,r_j;\ j=1,\ldots,q\), in \(\mathfrak R,\mathfrak R',\widetilde{\mathfrak R},\widetilde{\mathfrak R}'\), the algebras \(R_1,\widetilde R_1,R_2,\widetilde R_2\) in \(\mathfrak H,\widetilde{\mathfrak H},\Pi,\widetilde\Pi\), respectively, and the determining manifolds \(\Xi,\widetilde\Xi\) associated with them. The algebras \(R,\widetilde R\) are equivalent if and only if there exist: a) nonsingular matrices \(a_j=\|a_{jls}\|\), \(l,s=1,\ldots,r_j;\ j=1,\ldots,q\); b) numbers \(\gamma_{jl\mu\nu}\), \(l=1,\ldots,r_j;\ \nu=1,\ldots,r_\mu;\ j,\mu=1,\ldots,q\); c) elements \(h_{jl}^{0}\in\mathfrak H\), \(\pi_{jl}^{0}\in\Pi\), \(l=1,\ldots,r_j;\ j=1,\ldots,q\); d) isometric mappings \(V_1,V_2\) of the spaces \(\mathfrak H,\Pi\) onto \(\widetilde{\mathfrak H},\widetilde\Pi\) such that:

\[ 1)\quad \sum_{\nu=1}^{r_\mu}\gamma_{jl\mu\nu}b_{\mu\nu s} +\sum_{\nu=1}^{r_j}\overline b_{j\nu l}\overline\gamma_{\mu s j\nu} +(h_{jl}^{0},h_{\mu s}^{0})+(\pi_{jl}^{0},\pi_{\mu s}^{0})=0, \]
where
\[ b_j=\|b_{jll'}\|=a_j^{-1},\quad l,l'=1,\ldots,r_j;\ s=1,\ldots,r_\mu;\ j,\mu=1,\ldots,q; \]

2) the formulas

\[ \widetilde\lambda_{jls}=\sum_{\mu=1}^{r_j}\sum_{\nu=1}^{\mu}a_{jl\mu}\lambda_{j\mu\nu}b_{j\nu s},\quad \widetilde A_1=V_1A_1V_1^{-1},\quad \widetilde A_2=V_2A_2V_2^{-1}, \tag{15} \]

\[ \widetilde h_{js}=V_1\left(A_1^{*}h_{js}^{0}-\sum_{l=s}^{r_j}\overline{\widetilde\lambda}_{jls}h_{jl}^{0} +\sum_{l=1}^{r_j}\overline b_{jls}h_{jl}\right), \tag{16} \]

\[ \widetilde\pi_{js}=V_2\left(A_2^{*}\pi_{js}^{0}-\sum_{l=s}^{r_j}\overline{\widetilde\lambda}_{jls}\pi_{jl}^{0} +\sum_{l=1}^{r_j}\overline b_{jls}\pi_{jl}\right), \tag{17} \]

\[ \widetilde\alpha_{jl\mu\nu} =\sum_{s=1}^{q}\sum_{\tau=1}^{q}\overline b_{jsl}\alpha_{js\mu\tau}b_{\mu\tau\nu} +\sum_{\tau=1}^{q}\sum_{s=1}^{\tau}\gamma_{jl\mu\tau}\lambda_{\mu\tau s}b_{\mu s\nu} +\sum_{\tau=1}^{q}\sum_{s=1}^{\tau}\overline b_{jsl}\overline\lambda_{j\tau s}\overline\gamma_{\mu\nu j\tau} \]
\[ +\sum_{s=1}^{q}\overline b_{jsl}\bigl[(h_{js}^{*},h_{\mu\nu}^{0})+(\pi_{js}^{*},\pi_{\mu\nu}^{0})\bigr]+ \]
\[ +\sum_{\tau=1}^{q}b_{\mu\tau\nu}\bigl[(h_{jl}^{0},h_{\mu\tau})+(\pi_{jl}^{0},\pi_{\mu\tau})\bigr] +(A_1h_{jl}^{0},h_{\mu\nu}^{0})+(A_2\pi_{jl}^{0},\pi_{\mu\nu}^{0}) \tag{18} \]

carry out a one-to-one mapping of \(\Xi\) onto \(\widetilde\Xi\).

Condition 1) means that the space \(\mathfrak R'\), spanned by \(\{y_{jl}'\}\), is zero.

1. Suppose now that \(\Pi_k,\widetilde\Pi_k\) are separable, that the algebras \(R,\widetilde R\) are separable with respect to the operator norm and are given in the form of canonical models described in Sec. 5 of \((^{1})\). Let
   \[ \mathfrak H=\int_T \mathfrak H(t)\,d\sigma,\quad \widetilde{\mathfrak H}=\int_{\widetilde T}\widetilde{\mathfrak H}(\widetilde t)\,d\widetilde\sigma \]
   be the corresponding realizations of the spaces \(\widetilde{\mathfrak H},\mathfrak H\) in these canonical models. Applying to the second formula in (15) the lemma from Appendix IV in \((^{4})\), the argument on p. 223 in \((^{4})\), and then using formula (16), we obtain:

Theorem 2. Let \(R,\widetilde R\) be two equivalent canonical models, specified by means of the spaces

\[ \int_T \mathfrak H(t)\,d\sigma,\qquad \int_{\widetilde T}\widetilde{\mathfrak H}(\widetilde t)\,d\widetilde\sigma, \]

the algebras \(R_1,\widetilde R_1,R_2,\widetilde R_2\), the vector-functions \(\xi_{jl}(t), \widetilde \xi_{jl}(\widetilde t)\), and the defining manifolds \(\Xi\) and \(\widetilde \Xi\). Then there exist: a) a homeomorphism \(s\) of the space \(T\) onto \(\widetilde T\), mapping one-to-one the set \(\{t_j,\ j=1,\ldots,m\}\) of all singular points of the algebra \(R\) onto the set \(\{\widetilde t_j,\ j=1,\ldots,m\}\) of all singular points of the algebra \(\widetilde R\); b) a \(\sigma\)-measurable operator function \(V_1(t)\), defined \(\sigma\)-almost everywhere on \(T\), whose value for \(\sigma\)-almost every \(t\in T\) is an isometric operator \(V_1(t)\) mapping \(\mathfrak H(t)\) onto \(\widetilde{\mathfrak H}(st)\); c) vector-functions \(h_{j\nu}^{0}(t)\in\mathfrak H,\ \nu=1,\ldots,r_j;\ j=1,\ldots,q\), such that: 1) the measures \(\widetilde\sigma(s\Delta)\) and \(\sigma(\Delta)\), \(\Delta\subset T\), are equivalent; 2) the operator \(V\) in (15) is given by the formula \(V\{h(t)\}=\{\widetilde h(\widetilde t)\}\), where \(\widetilde h(st)=\rho(t)V_1(t)h(t)\), \(\rho(t)=d\widetilde\sigma(st)/d\sigma(t)\); 3) \(\sigma\)-almost everywhere on \(T_j\)

\[ \widetilde \xi_{j\nu}(st)=\rho(t)V_1(t)\left[h_{j\nu}^{0}(t)+\sum_{l=1}^{r_j}\overline b_{jl\nu}\xi_{jl}(t)\right], \]

and if \(t_j\) is a singular point,

\[ \widetilde k_{j\nu}=V_{1j}\left(\sum_{l=1}^{r_j}\overline b_{jl\nu}k_{jl}-\sum_{l=\nu+1}^{r_j}\overline \lambda_{jl\nu}k_{jl}\right), \]

where \(V_{1j}\) is the restriction of the operator \(V_1\) to the singular space \(K_j\).

Conversely, if all these conditions are satisfied, then: \(\alpha)\) the operator \(V_1\) maps \(\mathfrak H\) onto \(\widetilde{\mathfrak H}\) in such a way that \(\overline R_1\) is mapped onto \(\widetilde R_1\); \(\beta)\) for the vector-functions

\[ h_{jl}(t)=(A(t)-\lambda_j)\xi_{jl}(t)-\sum_{\mu=l+1}^{r_j}\lambda_{j\mu l}\xi_{j\mu}(t)\quad\text{for }t\ne t_j,\qquad h_{jl}(t_j)=k_{jl}, \]

\[ \widetilde h_{jl}(\widetilde t)=(\widetilde A(\widetilde t)-\widetilde\lambda_j)\widetilde\xi_{jl}(\widetilde t)-\sum_{\mu=l+1}^{r_j}\overline{\widetilde\lambda}_{j\mu l}\widetilde\xi_{j\mu}(\widetilde t)\quad\text{for }\widetilde t\ne\widetilde t_j,\qquad \widetilde h_{jl}(\widetilde t_j)=\widetilde k_{jl}, \]

relation (16) is satisfied.

Corollary. Let \(R\) be a separable c.s.a. with only real c.f. in the separable space \(\Pi_k\). If \(\lambda_{m+1},\ldots,\lambda_q\) are all regular c.f. of the algebra \(R\) with eigenvectors on the principal null subspace \(\mathfrak N\), then the skew-orthogonal subspace \(\mathfrak N'\) can be chosen so that \(h_{jl}(A)=0\) for \(j=m+1,\ldots,q\).

Indeed, from the regularity of \(\lambda_j\) it easily follows that \(\{\xi_{jl}(t)\}\in\mathfrak H\), and therefore one may set \(h_{jl}^{0}(t)=-\xi_{jl}(t)\) for \(j=m+1,\ldots,q\); \(h_{jl}(t)=0\) for \(j=1,\ldots,q\).

Put further

\[ y'_{jl}=-\frac12\sum_{\mu=1}^{q}\sum_{\nu=1}^{r_\mu}(h_{jl}^{0},h_{\mu\nu}^{0})x_{\mu\nu}+y_{jl}+h_{jl}^{0},\qquad l=1,\ldots \]

\[ \ldots,r_j;\quad j=1,\ldots,q, \]

and denote by \(\mathfrak N'\) the subspace spanned by \(\{y'_{jl}\}\). Then \(\mathfrak N'\) is a null subspace skew-orthogonal to \(\mathfrak N\), \(\{y'_{jl}\}\) is a basis in \(\mathfrak N'\), biorthogonal to \(\{x_{jl}\}\); put \(\mathfrak H'=\mathfrak M\cap\mathfrak N'^\perp\), \(\Pi'=\Omega\cap\mathfrak N'^\perp\). Applying Theorems 1 and 2 to the algebras \(R\) and \(\widetilde R=R\), realized by means of the decompositions \(\Pi_k=(\mathfrak N\dotplus\mathfrak N')\oplus\mathfrak H\oplus\Pi\), \(\Pi_k=(\mathfrak N\dotplus\mathfrak N')\oplus\mathfrak H'\oplus\Pi'\), the bases \(\{x_{jl}\}\) in \(\mathfrak N\), \(\{y_{jl}\},\{y'_{jl}\}\) in \(\mathfrak N',\mathfrak N'\), respectively, we conclude that \(\xi_{j\nu}(st)=\rho(t)V_1(t)[h_{j\nu}^{0}(t)+\xi_{j\nu}(t)]=0\), and therefore \(h_{jl}(A)=0\) for \(j=m+1,\ldots,q\).

Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
8 IX 1964

REFERENCES

1. M. A. Naimark, DAN, 161, No. 1 (1965).
2. M. A. Naimark, DAN, 156, 734 (1964).
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4. J. Dixmier, Les algèbres d’opérateurs dans l’espace hilbertien, Paris, 1957.