MATHEMATICS

B. M. LEVITAN

ADJOINT OPERATORS OF GENERALIZED TRANSLATION

(Presented by Academician S. L. Sobolev on 27 VI 1958)

1. Let (V_n) be a real differentiable manifold and let (T^s) be a family of generalized translation operators defined on (V_n) (see ((^1))). Let (m(E)) be a completely additive measure on (V_n). We define the family of adjoint operators (\widetilde T^s) by means of the equality

[
\int_{V_n} T_t^s f(t)\,\overline{g(t)}\,dm(t)
=
\int_{V_n} f(t)\,\overline{\widetilde T_t^s g(t)}\,dm(t).
\tag{1}
]

In ((^2)) we studied properties of adjoint operators. Some of these properties do not depend on the measure (m), but the deepest properties of the adjoint operators depend essentially on the measure.

We require that the adjoint operators satisfy the conditions:

a) in the noncommutative case

[
\widetilde T_s^r T_t^s f(t)=T_t^s \widetilde T_t^r f(t);
\tag{2}
]

b) in the commutative case

[
\widetilde T_t^s T_t^r f(t)=T_t^r \widetilde T_t^s f(t).
\tag{3}
]

It can be shown that in the commutative case, for (f(t)) from the subspace (M) (see condition (3^\circ) of generalized translation in ((^1))), conditions (2) and (3) are equivalent. Therefore, in what follows we shall confine ourselves to the study of property (2).

In this note we shall consider case II (see ((^1))). Thus, (u(s,t)=T_t^s f(t)) is the solution of the Cauchy problem

[
\widetilde N_{\alpha;s}u=N_{\alpha;t}u;
\tag{4}
]

[
u\big|_{s=0}=f(t);
\tag{5}
]

[
\left.
\frac{\partial^\lambda u}{\partial s_1^{\lambda_1}\cdots \partial s_n^{\lambda_n}}
\right|{s=0}
=
h f(t),
\tag{6}
]

where (\widetilde N_{\alpha;s}) and (N_{\alpha;t}) ((\alpha=1,2,\ldots,n)) are differential operators of the second order.

Theorem 1. Denote by (N_{\alpha;t}^{*}) the operators adjoint to the operators (N_{\alpha;t}) with respect to the measure (m).

The function (v(s,t)=\widetilde T_t^s g(t)) is the unique solution of the following Cauchy problem

[
\overline{N}{\alpha;s}v=Nv;}^{*
\tag{7}
]

[
v\big|_{s=0}=g(t);
\tag{8}
]

[
\frac{\partial^\lambda v}{\partial s_1^{\lambda_1}\ldots \partial s_n^{\lambda_n}}\bigg|{s=0}
=\overline hg(t),
\tag{9}
]

where (\overline{N}_{\alpha;s}) are operators with complex-conjugate coefficients.

Proof. Denote by (h(s)) the left-hand side (which has also become the right-hand side) of equality (1), and apply the operator (N_{\alpha;s}) to (h(s)). We obtain (using the commutation condition (N_{\alpha;s}T_t^s f(t)=T_t^s N_{\alpha;t}f(t)))

[
N_{\alpha;s}h(s)=
\int_{V_n} T_t^s N_{\alpha;t}f(t)\,\overline{g(t)}\,dm(t)
=
\int_{V_n} f(t)\,\overline{N_{\alpha;t}^{*}\widetilde T_t^s g(t)}\,dm(t)
=
]

[

\int_{V_n} f(t)\,\overline{N_{\alpha;s}\widetilde T_t^s g(t)}\,dm(t).
]

Hence, and from the arbitrariness of the function (f(t)), equation (7) follows. Setting (s=0) in (1), we obtain condition (8). Differentiating (1) (\lambda_1) times with respect to (s_1), (\lambda_2) times with respect to (s_2), etc., (\lambda_n) times with respect to (s_n), and then setting (s=0), we obtain condition (9).

If the operators (T^s) transform real functions into real ones, then the coefficients of the operators (\widetilde N_{\alpha;s}) and (N_{\alpha;t}), and the numbers (h_{\lambda_1,\ldots,\lambda_n}), are real. In this case the system (7), (8), (9) takes the form

[
N_{\alpha;s}v=N_{\alpha;t}^{*}v;
\tag{7'}
]

[
v\big|_{s=0}=g(t);
\tag{8'}
]

[
\frac{\partial^\lambda v}{\partial s_1^{\lambda_1}\ldots \partial s_n^{\lambda_n}}\bigg|{s=0}
=
hg(t).
\tag{9'}
]

In what follows we shall consider the real case.

1. Let

[
N_{\alpha;t}(f)=a_{\alpha}^{ij}(t)\frac{\partial^2 f}{\partial t_i\partial t_j}
+b_{\alpha}^{i}(t)\frac{\partial f}{\partial t_i}
+c_{\alpha}(t)f
]

with real coefficients, and suppose that the measure (m(E)) in local coordinates can be represented in the form

[
m(E)=\int_E a(t)\,dt_1\ldots dt_n,
]

where (a(t)) is a real differentiable function. It is not difficult to verify that in this case the adjoint operators (N_{\alpha;t}^{*}) have the form

[
N_{\alpha;t}^{*}(g)=
\frac{1}{a(t)}
\left[
\frac{\partial^2}{\partial t_i\partial t_j}\bigl(aa_{\alpha}^{ij}g\bigr)
-
\frac{\partial}{\partial t_i}\bigl(ab_{\alpha}^{i}g\bigr)
+
c_{\alpha}ag
\right].
\tag{10}
]

Theorem 2. In order that the adjoint operators satisfy condition (2), it is necessary and sufficient that, for an arbitrary function (f(t)), the condition

[
N_{\alpha;s}^{}T_t^s f(t)=T_t^s N_{\alpha;t}^{}f(t).
\tag{11}
]

be fulfilled.

Proof. Since

[
N_{\alpha;t}(f)=\left.\frac{\partial^2 T_t^s f(t)}{\partial s_\alpha^2}\right|{s=0},
\qquad
N,}^{*}(f)=\left.\frac{\partial^2 \widetilde T_t^s f(t)}{\partial s_\alpha^2}\right|_{s=0
]

(11) follows from (2).

To prove that (2) follows from (11), introduce the notation

[
\Phi(s,r,t)=\widetilde T_s^r T_t^s f(t), \qquad
\Psi(s,r,t)=T_t^s \widetilde T_t^r f(t).
]

The function (\Phi(s,r,t)), as a function of the variables (s) and (r), is the solution of the following Cauchy problem:

[
N_{\alpha;r}\Phi=N_{\alpha;s}^{*}\Phi;
\tag{12}
]

[
\Phi\big|_{r=0}=T_t^s f(t);
\tag{13}
]

[
\left.
\frac{\partial^\lambda \Phi}
{\partial r_1^{\lambda_1}\ldots \partial r_n^{\lambda_n}}
\right|{r=0}
=
hT_t^s f(t).
\tag{14}
]

We shall show that (\Psi) also satisfies these conditions.

Conditions (13) and (14) are verified directly. Let us verify equations (12). We have, using (7′) and (11):

[
N_{\alpha;r}\Psi
=
T_t^s N_{\alpha;r}\widetilde T_t^r f(t)
=
T_t^s N_{\alpha;t}^{}\widetilde T_t^r f(t)
=
N_{\alpha;s}^{}\Psi .
]

Remark. Differentiating equality (11) twice with respect to (t_\beta) and then putting (t=0), we obtain

[
\widetilde N_{\beta;s}N_{\alpha;s}^{}(f)
=
N_{\alpha;s}^{}\widetilde N_{\beta;s}(f).
\tag{15}
]

Thus condition (15) is a consequence of condition (11). It can be shown that, for the equivalence of conditions (11) and (15) (and consequently also of conditions (2) and (15)), it is necessary and sufficient that, for any function (f(t)), the following hold:

[
\begin{aligned}
1)\quad&
N_{\alpha;s}^{}T_t^s f(t)\big|{s=0}
=
N^{}f(t);
\[4pt]
2)\quad&
\left.
\frac{\partial^\lambda}
{\partial s_1^{\lambda_1}\ldots \partial s_n^{\lambda_n}}
\,N_{\alpha;s}^{}T_t^s f(t)
\right|{s=0}
=
h^{}N_{\alpha;t}f(t).
\end{aligned}
\tag{16}
]

3. The situation is greatly simplified in the case when (N_{\alpha;t}) are first-order operators, which corresponds to a shift on a group.

Thus, let

[
N_{\alpha;t}(f)=b_\alpha^{\,i}(t)\frac{\partial f}{\partial t_i},
\qquad
\widetilde N_{\alpha;s}(f)=\widetilde b_\alpha^{\,i}(s)\frac{\partial f}{\partial s_i},
]

where

[
b_\alpha^{\,i}(0)=\widetilde b_\alpha^{\,i}(0)=\delta_\alpha^i .
]

In this case the adjoint operators have the form

[
N_\alpha^{*}(g)
=
-\,N_\alpha(g)
-\frac{1}{a}N_\alpha(a)\,g
-B_\alpha g,
\tag{17}
]

where (B_\alpha(t)=\partial b_\alpha^{\,i}(t)/\partial t_i) (summation over (i)).

Further, if (u(s,t)=T_t^s f(t)), then

[
u\big|{s=0}=f(t),
\qquad
N(u)\big|{s=0}=N(f).
]

Therefore from condition (16) it follows that

[
\frac{1}{a}N'{\alpha}(a)+B\alpha=\lambda,
\tag{18}
]

where (\lambda_{\alpha}) are constant numbers. Each solution of system (17) gives a measure (a(t)) for which the corresponding adjoint operators satisfy condition (3). From (17) and (18) it follows that the adjoint operators have the form

[
N^{*}{\alpha}(g)=-Ng .}(g)-\lambda_{\alpha
]

For (\lambda_{\alpha}=0), (a(t)) satisfies the system

[
\frac{1}{a}N_{\alpha}(a)+B\alpha=0
]

and is the density of the Haar measure.

I express my gratitude to I. M. Gel'fand for his interest in the work and for discussion of some results.

Received
25 VI 1958

REFERENCES

¹ B. M. Levitan, DAN, 123, No. 1 (1958). ² B. M. Levitan, Uspekhi matem. nauk, 6, issue 1, 3 (1949).