UDC 517.43

MATHEMATICS

V. B. KOROTKOV

ON INTEGRAL OPERATORS WITH CARLEMAN KERNELS

(Presented by Academician S. L. Sobolev on 17 IV 1965)

Let \(\Omega\) be a measurable subset of \(n\)-dimensional Euclidean space \(R_n\). A Carleman kernel is a complex-valued function \(K(s,t)\) defined on \(\Omega \times \Omega\) and satisfying three conditions:

A. \(K(s,t)\) is measurable.

B. For almost all \(s\) in \(\Omega\),
\[ \int_{\Omega} |K(s,t)|^2\,dt < \infty. \]

C. For almost all \((s,t)\),
\[ K(s,t)=\overline{K(t,s)}. \]

A linear (in general, unbounded) operator \(T\), defined on an everywhere dense linear manifold \(D_T\) in \(L_2(\Omega)\), is called a Carleman operator if it is representable in the form
\[ (Tf)(s)=\int_{\Omega} K(s,t)f(t)\,dt,\qquad f \in D_T, \tag{1} \]
where \(K(s,t)\) is a Carleman kernel.

Operators of this kind were first considered by T. Carleman \((^1)\). The problem of describing the specific properties of Carleman operators was posed by J. von Neumann in the paper \((^2)\), where there are two distinct formulations of the problem. In the first, the question concerns the characteristic properties of the spectrum of Carleman operators; in the second, the question concerns finding conditions for representability of an operator in the form (1) with a Carleman kernel. In his paper J. von Neumann considered the problem in the first formulation and, for self-adjoint operators, obtained a definitive result.

In the second formulation the problem was studied by N. I. Akhiezer \((^3)\). In this work N. I. Akhiezer singled out from the totality of all Carleman kernels a certain class of kernels and found a criterion for representability of a symmetric operator in the form (1) with a kernel from this class.

In the present paper a solution of the problem is given for the entire totality of Carleman kernels. The criterion is formulated in Theorem 2. Throughout the paper the linear operator \(T\) is assumed to be defined on an everywhere dense linear manifold \(D_T\) in \(L_2(\Omega)\).

1. Operators of Carleman type. We shall call a linear operator \(T\) an integral operator of Carleman type if it is representable in the form (1) with a kernel satisfying only conditions A and B.

Definition. We shall say that a linear operator \(T\) has a majorant if there exists a measurable, almost everywhere finite, nonnegative function \(\Lambda(s)\) such that for any function \(f \in D_T\), \(\|f\|=1\), almost everywhere
\[ |(Tf)(s)| \leq \Lambda(s). \tag{2} \]
The function \(\Lambda(s)\) will be called a majorant.

Lemma 1. Let \(\varphi(s)\), \(s \in \Omega\), be a measurable abstract function taking values in \(L_2(\Omega)\). Then there exists a kernel \(l(s,t)\), satisfying conditions A and B, such that for almost all \(s \in \Omega\), \(l(s,t)\), as a function of \(t\), coincides with \(\varphi(s)\).

Theorem 1. A linear operator \(T\) is an integral operator of Carleman type if and only if \(T\) has a majorant.

Necessity. As a majorant one may choose the function
\[ L(s)=\left(\int_{\Omega}|K(s,t)|^2dt\right)^{1/2}. \]

Sufficiency. Let \(\Lambda(s)\) be a majorant of the operator \(T\). Put
\[ E_k=E(s:\ \Lambda(s)\leq k),\qquad V_k=E(s:\ s\in R_n,\ \|s\|\leq k) \]
and
\[ \Omega_k=\Omega\cap E_k\cap V_k,\qquad k=1,2,\ldots. \]
Then \(\Omega_1\subseteq\Omega_2\subseteq\Omega_3\subseteq\ldots\) and
\[ m\left(\Omega\setminus\bigcup_{k=1}^{\infty}\Omega_k\right) = m\left(\Omega\setminus\bigcup_{k=1}^{\infty}E_k\right)=0. \tag{3} \]

Define, on the linear set \(D_T\), everywhere dense in \(L_2(\Omega)\), the functionals
\[ F_E^{(k)}(f)=\int_E (Tf)(s)\,ds,\qquad E\subseteq\Omega_k,\qquad k=1,2,\ldots. \tag{4} \]

By virtue of (2), for any \(f\in D_T,\ \|f\|=1\),
\[ |F_E^{(k)}(f)|\leq \int_{E\cap E_k}|(Tf)(s)|\,ds\leq kmE,\qquad E\subseteq\Omega_k. \]

Consequently, the linear and bounded functionals \(F_E^{(k)}\) on \(D_T\) can be extended by continuity to the whole space \(L_2(\Omega)\), and moreover
\[ \|F_E^{(k)}\|\leq kmE,\qquad E\subseteq\Omega_k. \]
But then, by the theorem of N. Dunford and B. Pettis ((4), p. 584), there exists, and moreover uniquely (up to equivalence), a measurable abstract function \(\varphi_k(s)\), \(s\in\Omega_k\), such that
\[ F_E^{(k)}=\int_E \varphi_k(s)\,ds,\qquad E\subseteq\Omega_k. \tag{5} \]

Extend \(\varphi_k(s)\) by zero for \(s\in\Omega\setminus\Omega_k\). By (4) and by the uniqueness, \(\varphi_k(s)=\varphi_{k+1}(s)\) for almost all \(s\in\Omega_k\), \(k=1,2,\ldots\). Hence, for almost all \(s\in\Omega\), \(\lim_{k\to\infty}\varphi_k(s)\) exists; moreover the limiting function \(\varphi(s)\) is also measurable and \(\varphi(s)=\varphi_k(s)\) for almost all \(s\in\Omega_k\), \(k=1,2,\ldots\). Let \(f\in D_T\). Since for every measurable set \(E\) from \(\Omega_k\), \(k=1,2,\ldots\),
\[ \int_E (Tf)(s)\,ds = F_E^{(k)}(f) = \left\langle f,\int_E \varphi_k(s)\,ds\right\rangle = \int_E (f,\varphi_k(s))\,ds = \int_E (f,\varphi(s))\,ds, \]
it follows, by (3), that for almost all \(s\in\Omega\),
\[ (Tf)(s)=(f,\varphi(s)). \]
The proof is completed by applying Lemma 1.

Lemma 2. Let \(T\) be an integral operator of Carleman type with kernel \(K(s,t)\). Then every majorant of it, for almost all \(s\in\Omega\), satisfies the inequality
\[ \left(\int_{\Omega}|K(s,t)|^2\,dt\right)^{1/2}\leq \Lambda(s). \tag{6} \]

Proof. Let \(F,\ F\subset D_T\), be a countable set lying on the unit sphere of \(L_2(\Omega)\) and everywhere dense in it. Using the definition of a majorant and the countability of \(F\), we find in \(\Omega\) a set \(N_0\) of measure zero such that for all \(s\in\Omega\setminus N_0\) and \(f\in F\),
\[ \left|\int_{\Omega}K(s,t)f(t)\,dt\right|\leq \Lambda(s). \]
From this (6) follows immediately.

II. Carleman Operators

Let \(\xi(s)\ge 0,\ s\in\Omega\), be a measurable function finite almost everywhere. It is known that the linear manifold

\[ [L_2(\Omega)]_\xi=(f:\ f\in L_2(\Omega),\ \int_\Omega \xi(s)|f(s)|\,ds<\infty) \]

is everywhere dense in \(L_2(\Omega)\) \((^3)\).

Theorem 2. In order that a linear operator \(T\) be a Carleman integral operator, it is necessary and sufficient that:

1) the operator \(T\) have a majorant \(\Lambda(s)\);
2) \(D_{T^*}\supseteq [L_2(\Omega)]_\Lambda\);
3) for any \(f,g\) from \([L_2]_\Lambda\), \((T^*f,g)=(f,T^*g)\).

Necessity. Put

\[ K(s)=\left(\int_\Omega |K(s,t)|^2dt\right)^{1/2}. \]

Following (3), consider the linear manifold \(D_B\) of all functions \(f\in L_2(\Omega)\) for which

\[ \int_\Omega K(s,t)f(t)\,dt=\varphi_f(s)\in L_2(\Omega) \]

and define on the linear manifolds \(D_B\) and \([L_2]_K\subseteq D_B\) linear operators \(B\) and \(A\):

\[ Bf=\varphi_f,\quad f\in D_B,\qquad Ag=\varphi_g,\quad g\in [L_2]_K. \]

\(A\) is a symmetric operator and \(A^*=B\) \((^3)\). Let \(T\) be a Carleman integral operator with kernel \(K(s,t)\). By Theorem 1, \(T\) has a majorant \(\Lambda(s)\). Since \(T\subseteq B\), we have \(T^*\supseteq B^*=A^{**}\supseteq A\). Consequently,

\[ D_{T^*}\supseteq D_A=[L_2]_K\supseteq \text{(see (6))}\supseteq [L_2]_\Lambda. \]

The third assertion follows from the inclusions \(T^*\supseteq A\), \(D_A\supseteq [L_2]_\Lambda\) and the symmetry of the operator \(A\).

Sufficiency. Since \(T\) has a majorant \(\Lambda(s)\), by Theorem 1, \(T\) is representable in the form (1) with a kernel \(K(s,t)\) satisfying conditions A, B. It remains to show that the kernel \(K(s,t)\) is Hermitian. As above, construct from the kernel \(K(s,t)\) an operator \(B\). It is clear that \(T\subseteq B\). Since for any \(f\in [L_2]_\Lambda\) and \(g\in L_2(\Omega)\),

\[ \int_\Omega\int_\Omega |K(s,t)|\,|f(s)|\,|g(t)|\,dt\,ds \le \|g\|\int_\Omega K(s)|f(s)|\,ds \le \|g\|\int_\Omega \Lambda(s)|f(s)|\,ds, \tag{7} \]

we have

\[ \int_\Omega \overline{K(s,t)}\,f(s)\,ds\in L_2(\Omega). \]

This makes it possible to define on \([L_2]_\Lambda\) the linear operator \(A\):

\[ (Af)(t)=\int_\Omega \overline{K(s,t)}\,f(s)\,ds,\qquad f\in [L_2(\Omega)]_\Lambda. \]

In view of (7) and Fubini’s theorem, for any \(g\in D_B\) and \(f\in D_A=[L_2]_\Lambda\),

\[ \int_\Omega\left(\int_\Omega \overline{K(s,t)}\,f(s)\,ds\right)\overline{g(t)}\,dt = \int_\Omega f(s)\left(\int_\Omega K(s,t)\overline{g(t)}\,dt\right)ds. \]

Consequently, \(B\subseteq A^*\). But \(T\subseteq B\). Therefore \(A\subseteq A^{**}\subseteq T^*\). Hence, by condition 3, for any \(f\) and \(g\) from \(D_A=[L_2]_\Lambda\),

\[ \int_\Omega\left(\int_\Omega \overline{K(s,t)}\,f(s)\,ds\right)\overline{g(t)}\,dt = \int_\Omega f(s)\overline{\left(\int_\Omega \overline{K(t,s)}\,g(t)\,dt\right)}ds. \]

Interchanging (on the basis of (7)) the order of integration on the left and using the fact that \(f\) and \(g\) range over the everywhere dense set \([L_2(\Omega)]_\Lambda\) in \(L_2(\Omega)\), we obtain that \(\overline{K(s,t)}=\overline{K(t,s)}\) for almost all \(t\in\Omega\) for almost every \(s\in\Omega\). The theorem is proved.

III. Operators of Special Form

Let the linear operator \(T\) have the form

\[ Tf=\sum_{k=1}^{\infty}\lambda_k(f,\varphi_k)\psi_k,\qquad f\in D_T,\ \varphi_k\in D_T, \tag{8} \]

where \(\lambda_k\) is a sequence of numbers, \(\varphi_k\) is an orthonormal system, and the series in (8) converges to \(Tf\) in the norm of \(L_2(\Omega)\).

Theorem 3. In order that the linear operator (8) be an integral operator of Carleman type, it is necessary and sufficient that

for almost all \(s \in \Omega\)

\[ \lambda^{2}(s)=\sum_{k=1}^{\infty}|\lambda_k|^{2}|\psi_k(s)|^{2}<\infty . \tag{9} \]

Necessity. By virtue of (8) and Bessel’s inequality, for almost all \(s \in \Omega\)

\[ K^{2}(s)=\int_{\Omega}|K(s,t)|^{2}\,dt \geq \sum_{k=1}^{\infty}|(K(s,t)\overline{\varphi_k(t)})_t|^{2} = \sum_{k=1}^{\infty}|(T\varphi_k)(s)|^{2} = \lambda^{2}(s). \tag{10} \]

Sufficiency. Let \(f \in D_T\), \(\|f\|=1\). By virtue of (8) and (9), the series
\[ \sum_{k=1}^{\infty}\lambda_k(f,\varphi_k)\psi_k(s) \]
converges absolutely for almost all \(s \in \Omega\) to \((Tf)(s)\). But then

\[ |(Tf)(s)|^{2} = \left|\sum_{k=1}^{\infty}\lambda_k(f,\varphi_k)\psi_k(s)\right|^{2} \leq \sum_{k=1}^{\infty}|\lambda_k|^{2}|\psi_k(s)|^{2} = \lambda^{2}(s). \]

Consequently, \(\lambda(s)\) is a majorant for \(T\). The theorem is proved.

Like any majorant, \(\lambda(s)\) satisfies inequality (6). Hence, together with (10), it follows that for almost all \(s \in \Omega\)

\[ \lambda(s)=K(s). \tag{11} \]

Taking into account (11), Theorem 2 and Theorem 3, we arrive at the following theorem:

Theorem 4. In order that a linear operator \(T\) of the form (8) be a Carleman integral operator, it is necessary and sufficient that:

1) the function \(\lambda(s)\) be finite almost everywhere;
2) \([L_2(\Omega)]_{\lambda} \subseteq D_{T^*}\);
3) for any \(f\) and \(g\) from \([L_2(\Omega)]_{\lambda}\),
\[ (T^*f,g)=(f,T^*g). \]

Remark 1. In the case of a self-adjoint operator the second and third conditions of Theorem 4 may be omitted.

Remark 2. Theorems 1–4 are also valid for operators acting in the real Hilbert space \(L_2(\Omega)\).

In conclusion let us consider an example. Let \(\Omega=[0,1]\), \(\chi_n^{(k)}\), \(n=0,1,2,\ldots,\ k=1,2,\ldots,2^n\), be the Haar functions, and let
\[ T\chi_n^{(k)}=\alpha_n^{(k)}\chi_n^{(k)}, \]
where \(\alpha_n^{(k)}\) are real numbers. Let
\[ a_n=\min_{1\leq k\leq 2^n}|\alpha_n^{(k)}|,\qquad A_n=\max_{1\leq k\leq 2^n}|\alpha_n^{(k)}|. \]
The self-adjoint operator \(T\) is a Carleman operator if the series
\[ \sum_{n=0}^{\infty}2^n A_n^2 \]
converges, and is not one if the series
\[ \sum_{n=0}^{\infty}2^n a_n^2 \]
diverges. Indeed,

\[ \sum_{n=0}^{\infty}2^n a_n^2 \leq \lambda^2(s) \leq \sum_{n=0}^{\infty}2^n A_n^2. \]

Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR

Received
30 III 1965

References

1. T. Carleman, Sur les équations integrales singulières a noyau réel et symmetrique, Uppsala, 1923.
2. J. Neumann, Actualités Sci. et Ind., 229 (1935).
3. N. I. Akhiezer, UMN, 2, no. 5 (21), 93 (1947).
4. N. Dunford, J. T. Schwartz, Linear Operators. General Theory, IL, 1962.