D. L. BERMAN

ON THE THEORY OF SEMIGROUPS OF LINEAR OPERATORS

(Presented by Academician S. N. Bernstein on 16 V 1963)

\(1^\circ\). Denote by \(\widetilde C\) the space of all continuous \(2\pi\)-periodic functions \(f(x)\) with norm
\[ \|f\|=\max_{0\le x<2\pi}|f(x)|. \]

Let \(U_n\) be a linear operator from \(\widetilde C\) into \(\widetilde C\), taking every \(f\in \widetilde C\) into a trigonometric polynomial of order \(\le n\). Introduce the operator
\[ \widetilde U_n(f)=\frac{1}{2\pi}\int_0^{2\pi}(U_n(f_t))_{-t}\,dt, \]
where \(f_t(x)=f(x+t)\). In the paper \((^1)\) it was proved that for any \(f\in \widetilde C\) the equality
\[ \widetilde U_n(f,x)=\int_0^{2\pi} f(x+t)\Phi_n(t)\,dt, \tag{1} \]
holds, where \(\Phi_n(t)=\widetilde U_n(D_n,-t)\) and \(D_n\) is the Dirichlet kernel of order \(n\). Hence it follows that
\[ \|U_n\|\ge \int_0^{2\pi}\bigl|\widetilde U_n(D_n,t)\bigr|\,dt. \tag{2} \]

We shall now abandon the requirement that, for every \(f\in\widetilde C\), \(U_n(f)\) be a polynomial of order \(n\), and consider an arbitrary semigroup
\[ \Omega=\{U_\xi,\ \xi>0\} \]
of linear bounded operators, measurable with respect to \(\xi\), and mapping the space \(\widetilde C\) into itself.

It turns out that for the operators \(\Omega\) there are analogues of equality (1) and inequality (2).

\(2^\circ\). Define the operator
\[ \widetilde U_\xi(f)=\frac{1}{2\pi}\int_0^{2\pi}(U_\xi(f_t))_{-t}\,dt. \tag{3} \]

Theorem 1. Let \(\Omega\) be an arbitrary semigroup of linear bounded operators from \(\widetilde C\) into \(\widetilde C\), measurable with respect to \(\xi\). Then the operators
\[ \Omega_1=\{\widetilde U_\xi,\ \xi>0\}, \]
where the operator \(\widetilde U_\xi\) is defined by formula (3), also form a semigroup of linear bounded operators measurable with respect to \(\xi\); moreover, the operators \(\Omega_1\) have the following properties:

1) The operators \(\Omega_1\) map \(\widetilde C\) into \(\widetilde C\).

2) The operators \(\Omega_1\), being constructed for arbitrary operators \(\Omega\), commute with the group of real translations, i.e.
\[ \widetilde U_\xi(f_t)=(\widetilde U_\xi(f))_t,\qquad f\in\widetilde C,\qquad -\infty<t<\infty. \]

3) If the operators \(\Omega\) commute with the group of real translations, then \(\Omega_1\equiv\Omega\).

4) \(\|\widetilde U_\xi\|\leqslant \|U_\xi\|\).

Proof. Property 1) is obvious.

Let us prove property 2). To this end we compute \((\widetilde U_\xi(f_t))_{-t}\). According to equality (3) we have

\[ (\widetilde U_\xi(f_t))_{-t} = \frac{1}{2\pi}\int_0^{2\pi} (U_\xi(f_{t+t_1}))_{-(t+t_1)}\,dt_1 . \tag{4} \]

Since \((U_\xi(f_t))_{-t}\) is a \(2\pi\)-periodic function of \(t\), it follows from equality (4) that

\[ (\widetilde U_\xi(f_t))_{-t} = \frac{1}{2\pi}\int_0^{2\pi} (U_\xi(f_z))_{-z}\,dz . \]

Consequently, \((\widetilde U_\xi(f_t))_{-t}=\widetilde U_\xi(f)\), or \(\widetilde U_\xi(f_t)=(\widetilde U_\xi(f))_t\). Thus property 2) is proved.

Property 3) is obvious.

Let us prove property 4). For any \(-\infty<t<\infty\), \((U_\xi(f_t))_{-t}\in\widetilde C\). Therefore the integral on the right-hand side of equality (3) may be regarded as a Bochner integral \((^2)\). It is known \((^2)\) that for the Bochner integral the inequality

\[ \left\|\int f\,dt\right\|\leqslant \int \|f\|\,dt \]

holds. Hence it follows from equality (3) that

\[ \|\widetilde U_\xi(f)\| \leqslant \frac{1}{2\pi}\int_0^{2\pi} \|(U_\xi(f_t))_{-t}\|\,dt . \tag{5} \]

Since \(U_\xi\) is a linear operator and \(\|f_t\|=\|f\|\), it follows from inequality (5) that \(\|\widetilde U_\xi(f)\|\leqslant \|U_\xi\|\|f\|\).

Theorem 2. Let \(U_\xi\in\Omega_1\). Then there exist a set \(N\) of integers and a set \(\Lambda=\{\lambda_j\}_{j\in N}\) of complex numbers such that the function*

\[ K(t,\xi)=\sum_{j\in N}\frac{e^{jti}-1}{2\pi ji}\,e^{\lambda_j\xi} \tag{6} \]

is of bounded variation on \([-\pi,\pi]\) for every \(\xi>0\), and for every \(f\in\widetilde C\) the equality

\[ \widetilde U_\xi(f,x)=\int_{-\pi}^{\pi} f(x-t)\,d_tK(t,\xi) \tag{7} \]

holds.

This theorem follows directly from Theorem 1 and the theorem of Hille \((^2)\), according to which, for a semigroup of operators from \(\widetilde C\) into \(\widetilde C\) commuting with the group of real translations, representation (7) holds.

Theorem 3. Let \(U_\xi\in\Omega\). Then the inequality

\[ \|U_\xi\|\geqslant \operatorname*{Var}_{t\in[-\pi,\pi]} K(t,\xi), \tag{8} \]

holds, where \(K(t,\xi)\) is determined from Theorem 2.

Proof. According to Theorem 1,

\[ \|\widetilde U_\xi\|\leqslant \|U_\xi\|. \tag{9} \]

* If \(j=0\in N\), the corresponding term on the right-hand side of (6) is taken to be equal to \(t\).

On the other hand, it is known that the norm of the operator

\[ \sigma(f,x)=\int_{-\pi}^{\pi} f(x-t)\,d_tK(t,\xi), \]

which maps \(\widetilde C\) into \(\widetilde C\), is equal to \(\operatorname{Var}_t K\). Therefore, according to Theorem 2,

\[ \|\widetilde U_\xi\|=\operatorname{Var}_{t\in[-\pi,\pi]} K(t,\xi). \tag{10} \]

From (9) and (10), (8) follows.

Corollary. Let the operators \(\Omega\) be such that \(\overline{\lim}_{\xi\to\infty}\operatorname{Var} K(x,\xi)=\infty\). Then the relation

\[ U_\xi(f,x)\to f(x),\qquad \xi\to\infty \]

cannot hold uniformly for all \(f\in\widetilde C\).

Theorems 1–3 also remain valid for operators \(\Omega\) that map \(\widetilde L\) into \(\widetilde L\), where \(\widetilde L\) is the space of all summable \(2\pi\)-periodic functions with norm

\[ \|f\|=\int_{0}^{2\pi}|f(t)|\,dt. \]

In this case the proofs do not differ essentially from the proofs for the case of the space \(\widetilde C\).

\(3^\circ\). Denote by \(L\) the space of all functions \(f(x)\) summable on the entire real axis with norm

\[ \|f\|=\int_{-\infty}^{\infty}|f(x)|\,dx. \]

Let \(U\) be a bounded linear operator from \(L\) into \(L\). Put

\[ \widetilde U(f)=\lim_{\substack{\tau_1\to-\infty\\ \tau_2\to+\infty}} \frac{1}{\tau_2-\tau_1}\int_{\tau_1}^{\tau_2}(U(f_t))_{-t}\,dt. \tag{11} \]

It is obvious that \(\widetilde U\) is also a bounded linear operator from \(L\) into \(L\).

Theorem 4. The operator \(\widetilde U\) has the properties:

1) For any \(f\in L\) and any \(t\), \(\widetilde U(f_t)=(\widetilde U(f))_t\).

2) \(\|\widetilde U\|\leq \|U\|\).

Theorem 5. Let \(\widetilde U\) be an arbitrary bounded linear operator from \(L\) into \(L\). Then there exists a function \(\Phi(t)\) of bounded variation on \((-\infty,\infty)\) such that, for any \(f\in L\), the equality

\[ \widetilde U(f,x)=\int_{-\infty}^{\infty} f(x-t)\,d\Phi(t) \]

holds, where the operator \(\widetilde U\) is defined according to formula (11).

Theorem 6. Let \(U\) be an arbitrary bounded linear operator from \(L\) into \(L\). Then

\[ \|U\|\geq \int_{-\infty}^{\infty}|d\Phi(t)|, \]

where \(\Phi(t)\) is defined from Theorem 5.

Received
16 V 1963

CITED LITERATURE

1. D. L. Berman, DAN, 144, No. 3 (1962).
2. E. Hille, R. Phillips, Functional Analysis and Semigroups, IL, 1962.