Abstract
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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,\qquad
y'{jl}=U^{-1}\widetilde y,\qquad
U^{-1}\widetilde{\mathfrak N}=\mathfrak N'.
\tag{2}
]
Then ({x'{jl}}) is a basis in (\mathfrak N'), ({y''), biorthogonal to ({x'}}) is a basis in (\widehat{\mathfrak N{jl}}); (\mathfrak N,\widehat{\mathfrak N}') are skew-related. Moreover, ({x}) for fixed (j), form a basis in (\mathfrak N_j); consequently,}}), as well as ({x'_{jl
[
x'{jl}=\sum,}^{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'+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}^{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). From (8) we conclude that ((h,y'})), (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{jl})=(h,h^0)), 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
+\sum_{\mu=l}^{r_j}\lambda^{j\mu l}y
+h^_{jl}+\pi^_{jl},
]
(h,h_{jl},h^{jl}\in\mathfrak H;\ \pi,\pi,\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
+\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}^{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}^{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.
- 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,}(\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,\qquad l=1,\ldots}^{q}\sum_{\nu=1}^{r_\mu}(h_{jl}^{0},h_{\mu\nu}^{0})x_{\mu\nu}+y_{jl}+h_{jl}^{0
]
[
\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'},{y'}}) 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{jl}}) in (\mathfrak N',\mathfrak N'), respectively, we conclude that (\xi(A)=0) for (j=m+1,\ldots,q).}(st)=\rho(t)V_1(t)[h_{j\nu}^{0}(t)+\xi_{j\nu}(t)]=0), and therefore (h_{jl
Steklov Mathematical Institute
Academy of Sciences of the USSR
Received
8 IX 1964
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