MATHEMATICS

I. Ts. Gokhberg and M. G. Krein

On Volterra Operators with an Imaginary Component of One Class or Another

(Presented by Academician A. N. Kolmogorov, 16 III 1961)

1. In the present note we shall adhere to the terminology and notation of the preceding note (¹). In particular, by \(\mathfrak S_p\) \((1 \le p < \infty)\) we shall denote the Banach space of all operators \(A \in \mathfrak S_\infty\) (\(\mathfrak S_\infty\) is the set of all completely continuous operators acting in the Hilbert space \(\mathfrak H\)) for which

\[ (|A|_p)^p=\sum_n s_n^p(A)=\operatorname{Sp}\bigl((A^*A)^{p/2}\bigr)<\infty . \]

L. A. Sakhnovich (²) (see also (³)) showed that, if the imaginary component
\(A_J\,(=(A-A^*)/2i)\) of a Volterra operator belongs to the class \(\mathfrak S_2\), then the real component \(A_R\,(=(A+A^*)/2)\) belongs to the same class, and
\(|A_R|_2=|A_J|_2\). Subsequently, the authors (¹, ³, ⁴) obtained various characteristics of the component \(A_R\) of a Volterra operator \(A\) with \(A_J\in \mathfrak S_1\). In the autumn of 1959, at the Fifth All-Union Conference on Functional Analysis in Baku, while discussing with S. G. Krein a number of results reported by the authors, it became clear that the following holds:

Theorem 1. If one of the components of a Volterra operator \(A\) has finite order \(\ge 1\), then the other component has the same order.

Let us explain that the order \(\rho(A)\) \((0\le \rho(A)<\infty)\) of an operator \(A\) is the lower bound of the numbers \(p\) for which \(A\in\mathfrak S_p\).

For the case \(1<\rho(A_J)<2\), this theorem is derived from the theorem of L. A. Sakhnovich (²) and Theorem 2 of (³) with the aid of the following lemma.

Lemma 1. Let \(T_z\) be an operator-function with values in \(\mathfrak S_\infty\), holomorphic in the strip \(a_1\le \operatorname{Re} z\le a_2\). If

\[ |T_{a_j+iy}| \le C_j \quad (-\infty<y<\infty;\ j=1,2;\ 1\le p_1<p_2\le \infty), \]

then for every intermediate \(x\) \((a_1<x<a_2)\) one has

\[ |T_{x+iy}|_{p_x}\le C_1^{\,t_x} C_2^{\,1-t_x}, \]

where

\[ p_x^{-1}=t_x p_1^{-1}+(1-t_x)p_2^{-1}, \qquad t_x=(x-a_1)/(a_2-a_1). \]

This lemma, which is a peculiar development of the well-known three-lines theorem in the theory of functions, was obtained by the authors jointly with S. G. Krein, to whom the authors are also indebted for assistance in clarifying the role of this lemma in the questions of interest to them.

The validity of Theorem 1 for the case \(2<\rho(A_J)<\infty\) follows already from the preceding on the basis of the general Theorem 2 of (¹).

1. Recently V. I. Matsaev (⁵) developed a new method for investigating the dependence between the behavior of the Hermitian components of a Volterra operator,

based on subtle theorems of function theory already established by him (⁶). In particular, this method enabled him to obtain the following refinement of Theorem 1:

Theorem 2. Let \(A\) be a Volterra operator. If \(A_J \in \mathfrak S_p\), \((1<p<\infty)\), then \(A_R \in \mathfrak S_p\) and

\[ |A_R|_p \leqslant C_p |A_J|_p, \tag{1} \]

where \(C_p\) is a certain constant depending only on \(p\).

Theorem 2 from (¹) also makes it possible to assert that

\[ C_p=C_{p/(p-1)} \quad (1<p<\infty), \]

if by \(C_p\) one understands the sharp constant in (1).

After the result of V. I. Matsaev, the authors discovered that only a little needs to be added to their methods in order to obtain Theorem 2, moreover with the following characteristics of the sharp constant \(C_p\):

1) the constant \(C_p\) is a logarithmically convex (and therefore continuous) function of the argument \(\xi=p^{-1}\) on the interval \((0<\xi<1)\);

2) as a function of the argument \(p\), the constant \(C_p\) is monotone and increases without bound on the interval \((2\leq p<\infty)\);

3) the growth of the constant \(C_p\) is bounded by the inequality \(C_{2p}\leqslant C_p+\sqrt{1+C_p^2}\) \((1<p<\infty)\), and also by the inequalities \(\alpha p\leqslant C_p\leqslant \beta p\) \((2\leq p<\infty)\), where \(\alpha\) and \(\beta\) are certain constants (in particular, one may take \(\alpha=1/2\pi\) and \(\beta=(e^{2/3}\ln 2)^{-1}(<1)\)).

Proof of Theorem 2 and of properties 1), 2), 3) of the constant \(C_p\). Denote by \(\mathfrak R\) the totality of all self-adjoint finite-dimensional operators. Let \(\mathfrak P\) be an arbitrary continuous chain (see (¹)).

If \(H\in\mathfrak R\), then the integral

\[ \mathfrak S(H)=\mathfrak S(H;\mathfrak P)=i\int_{\mathfrak P}(PH\,dP-dP\,HP) \]

converges, and \(\mathfrak S(H)\in\mathfrak S_p\) \((p>1)\), while \(\mathfrak S(H)+iH\) is a Volterra operator with its own chain \(\mathfrak P\).

Put

\[ \gamma_p=\sup_{H\in\mathfrak R}\bigl(|\mathfrak S(H)|_p/|H|_p\bigr) \quad (1<p<\infty;\ \gamma_p\leqslant\infty). \]

Together with the operator \(\mathfrak S(H)+iH\), the operator \(B=(\mathfrak S(H)+iH)^2\) is also Volterra and has the chain \(\mathfrak P\).

Writing that \(B_R=\mathfrak S(B_J)\) gives the identity *

\[ [\mathfrak S(H)]^2 = H^2+\mathfrak S\bigl(H\mathfrak S(H)+\mathfrak S(H)H\bigr). \tag{2} \]

Since for any \(X,Y\in\mathfrak S_{2p}\), \(|XY|_p\leqslant |X|_{2p}|Y|_{2p}\), it follows that

\[ |[\mathfrak S(H)]^2|_p=(|\mathfrak S(H)|_{2p})^2 \leqslant (|H|_{2p})^2+2\gamma_p |H|_{2p}|\mathfrak S(H)|_{2p}. \]

Hence we find

\[ |\mathfrak S(H)|_{2p}/|H|_{2p}\leqslant \gamma_p+\sqrt{1+\gamma_p^2}, \]

which gives

\[ \gamma_{2p}\leqslant \gamma_p+\sqrt{1+\gamma_p^2}. \]

From what has been proved and from the quoted theorem of L. A. Sakhnovich it follows that \(\gamma_{2^n}<\infty\) \((n=1,2,\ldots)\).

* We point out that the idea of establishing and using the identity (2) arose with the authors in connection with the remarkably simple proof, proposed by M. Kotlyar (⁷), of M. Riesz’s theorem on the Hilbert transform.

Starting from any operator \(H\in\mathfrak K\), we form, by means of its polar representation \(H=UH_1\) (\(U\) unitary, and \(H_1\) a nonnegative operator), the holomorphic operator-valued function \(T_z=\mathfrak S(UH_1^z)\). Application to this operator-valued function of Lemma 1 with \(a_1=p/p_1,\ a=p/p_2\) \((1<p_1<p<p_2<\infty)\) shows that, since \(\gamma_{p_1}<\infty\) and \(\gamma_{p_2}<\infty\), we also have \(\gamma_p<\infty\), and, moreover, \(\gamma_p\) is a logarithmically convex function of \(p^{-1}\) \((p_1\le p\le p_2)\). Since \(\gamma_2=1\) and \(\gamma_{2^n}<\infty\) \((n=2,3,\ldots)\), it follows that \(\gamma_p<\infty\) for \(2\le p<\infty\).

From the indicated properties of the constant \(\gamma_p\) it is already easy to derive that it has properties 1), 2), 3), formulated for the constant \(C_p\).

Let us show that Theorem 2 is valid for \(C_p=\sup\gamma_p\), where the supremum is taken over all continuous chains \(\mathfrak P\). Let \(A\) be an arbitrary Volterra operator. Without loss of generality one may suppose that the proper chain \(\mathfrak P\) of the operator \(A\) is continuous. Obviously,
\[ A_R=\mathfrak S(A_J;\mathfrak P). \]

Let \(\{H_n\}_1^\infty\) be a sequence of operators from \(\mathfrak K\) such that
\[ |H_n-A_J|_p\to 0. \]
Using (1) and the closedness of the operator \(\mathfrak S=\mathfrak S(\,\cdot\,;\mathfrak P)\), we obtain
\[ |\mathfrak S(H_n)-A_R|\to 0. \]
Passing to the limit as \(n\to\infty\) in the relation
\[ |\mathfrak S(H_n)|_p\le C_p|H_n|_p \]
we obtain (1) in the general case.

Properties 1), 2), 3) of the constant \(C_p\), obviously, follow from the corresponding properties of the constant \(\gamma_p\).

1. Let \(\Pi=\{\pi_n\}_1^\infty\) be an arbitrary nonincreasing sequence of positive numbers \(\pi_n\) such that
   \[ \lim \pi_n=0,\qquad \sum \pi_n=\infty. \]
   By \(\mathfrak S_\Pi\) we shall denote the Banach space of all operators \(A\in\mathfrak S_\infty\) for which
   \[ |A|_\Pi=\sup\left(\left(\sum_1^n s_k(A)\right)\Big/\sum_1^n \pi_k\right)<\infty, \]
   and by \(\mathfrak S_\Pi^{(0)}\) the subspace of \(\mathfrak S_\Pi\) which is the closure in \(\mathfrak S_\Pi\) of the set \(\mathfrak K\) of all finite-dimensional operators. It turns out that the space \(\mathfrak S_\Pi^{(0)}\) consists of all operators \(A\in\mathfrak S_\infty\) for which
   \[ \lim\left(\sum_1^n s_k(A)\Big/\sum_1^n \pi_k\right)=0. \]
   Denote by \(\mathfrak S_\pi\) the Banach space of all operators \(A\) \((\in\mathfrak S_\infty)\) for which
   \[ |A|_\pi=\sum_1^\infty \pi_k s_k(A)<\infty. \]
   By Theorem 1, from (1) it follows that the space \(\mathfrak S_\pi\) is conjugate to \(\mathfrak S_\Pi^{(0)}\), while the space \(\mathfrak S_\Pi\) is conjugate to \(\mathfrak S_\pi\).

If the sequence \(\{\pi_n\}\) is regular, i.e.
\[ \sum_1^n \pi_k=O(n\pi_n), \]
then the operators \(A\in\mathfrak S_\Pi\) are characterized by the fact that
\[ s_n(A)=O(\pi_n), \]
and the operators \(A\in\mathfrak S_\Pi^{(0)}\) by the fact that
\[ s_n(A)=o(\pi_n)\qquad (n\to\infty). \]
In what follows we shall suppose that
\[ \pi_n=n^{-\chi}L(n)\qquad (n=1,2,\ldots;\ 0<\chi<1), \]
where \(L(\nu)\) \((1\le \nu<\infty)\) is a positive continuously differentiable slowly varying function. The last term means that
\[ \nu L'(\nu)/L(\nu)\to 0,\qquad \nu\to\infty. \]
It is easily verified that such a sequence \(\{\pi_n\}\) is regular.

V. I. Matsaev \((^5)\) showed that if, for a Volterra operator \(A\),
\[ s_n(A_J)=O(\pi_n)\qquad (1/2<\chi<1), \]
then also
\[ s_n(A_R)=O(\pi_n), \]
and this assertion remains valid if in it \(O\) is replaced by \(o\).

It turns out that the assertion of V. I. Matsaev holds for all \(\chi\) \((0<\chi<1)\). Moreover, the following holds:

Theorem 3. Let \(A\) be a Volterra operator with \(A_J\in\mathfrak S\), where \(\mathfrak S\) is one of the spaces \(\mathfrak S_\Pi^{(0)},\mathfrak S_\pi,\mathfrak S_\Pi\). Then \(A_R\in\mathfrak S\), and
\[ |A_R|_{\mathfrak S}\le M|A_J|_{\mathfrak S}, \]
where \(M\) is a constant depending only on \(\chi\) \((0<\chi<1)\) and the function \(L(\nu)\).

Corollary. If, for a Volterra operator \(A\),
\[ s_n(A_J)=O(\pi_n), \]
then
\[ s_n(A_R)=O(\pi_n), \]
and
\[ M'\sup\bigl(s_n(A_J)/\pi_n\bigr)\le \sup\bigl(s_n(A_R)/\pi_n\bigr)\le \]

\[ \leq M'' \sup \bigl(s_n(A_J)/\pi_n\bigr), \]
where \(M'\), \(M''\) are constants depending only on \(x\) and the function \(L(v)\). If \(s_n(A_J)=o(\pi_n)\), then also \(s_n(A_R)=o(\pi_n)\).

The proof of Theorem 3 has much in common with the proof of Theorem 2; however, it is considerably more complicated. It uses the theorem of V. I. Matsaev cited above, identity (2), a theorem of the type of Theorem 2 from [1], and a lemma analogous to Lemma 1 for a special system of spaces.

Moldavian Branch
of the Academy of Sciences of the USSR

Odessa Civil Engineering Institute

Received
20 II 1961

References

1. I. Ts. Gokhberg, M. G. Krein, DAN, 137, No. 5 (1961)*.
2. L. A. Sakhnovich, Izv. Vysshikh uchebn. zaved., Mathematics, No. 4 (11) (1959).
3. I. Ts. Gokhberg, M. G. Krein, DAN, 128, No. 2 (1959).
4. M. G. Krein, DAN, 130, No. 2 (1960).
5. V. I. Matsaev, DAN, 139, No. 3 (1961).
6. V. I. Matsaev, DAN, 132, No. 32 (1960).
7. M. Cotlar, Revista Matem. Cuyana, 1, No. 2 (1955).

* In our note “On the theory of triangular representations of non-self-adjoint operators” (1), the following misprints occurred: