On a Class of Completely Continuous Operators

V. I. Matsaev

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

In the present note we adhere to the terminology and notation of (¹). In particular, by $\mathfrak S_\infty$ is meant the Banach space of all completely continuous operators acting in a separable Hilbert space $\mathfrak H$ with the usual norm
$|A|_\infty=\sup_{f\in\mathfrak H}(|Af|/|f|)$; by $\mathfrak S_p$ $(1\le p<\infty)$, the Banach space of all operators $A\in\mathfrak S_\infty$ for which
$(|A|_p)^p=\sum s_n^p(A)=\operatorname{Sp}((A^*A)^{p/2})<\infty$, where $s_n(A)$ is the sequence, numbered in decreasing order with multiplicities taken into account, of the eigenvalues of the operator $(A^*A)^{1/2}$; and by $\mathfrak S_\omega$, the Banach space of operators $A\in\mathfrak S_\infty$ for which
$|A|_\omega=\sum(2n-1)^{-1}s_n(A)<\infty$. It is obvious that $\mathfrak S_p\subset\mathfrak S_\omega$ for any $p$ $(1\le p<\infty)$.

The set $\hat{\mathfrak S}_\omega$ of all self-adjoint operators in $\mathfrak S_\omega$ is a real subspace of the space $\mathfrak S_\omega$. In the space $\hat{\mathfrak S}_\omega$ we introduce a new norm $|H|_{\omega,K^*}$, topologically equivalent to the original one, by setting
$|H|_{\omega,K^*}=|H^+|_\omega+|H^-|_\omega$, where $H^+$ and $H^-$ are mutually orthogonal nonnegative operators whose difference is equal to $H$.

M. S. Brodskii (²) showed that every Volterra operator $A$ (i.e. every completely continuous operator $A$ with the single spectral point $\lambda=0$) admits a representation convergent in the norm of $\mathfrak S_\infty$ *

\[ A=2i\int_{\mathfrak P} PX\,dP \tag{1} \]

through its imaginary Hermitian component $X$ and a maximal eigenchain $\mathfrak P$. Conversely, if for some $X=X^*\in\mathfrak S_\infty$ the integral (1) converges in the norm of $\mathfrak S_\infty$, then $A$ is the unique Volterra operator possessing the eigenchain $\mathfrak P$ and imaginary component $X$. We shall denote the real component of the integral (1) by $\mathfrak S(\mathfrak P,X)$. A necessary condition for convergence of the integral (1) is the condition

\[ (P-Q)X(P-Q)=0, \tag{2} \]

where $(P,Q)$ is an arbitrary break of the chain $\mathfrak P$. The sufficiency of this condition for $X\in\mathfrak S_1$ was proved by M. S. Brodskii (²), and for $X\in\mathfrak S_p$ $(p>1)$ by I. Ts. Gohberg and M. G. Krein (see (¹,³)).

Theorem 1. For every operator $X\in\mathfrak S_\omega$ and every chain $\mathfrak P$ satisfying condition (2), the integral (1) converges. Moreover, the relation holds

\[ \sup |G|_\infty = \frac{2}{\pi} \sum_{j=1}^{\infty} \frac{s_j(H^+)+s_j(H^-)}{2j-1} = \frac{2}{\pi}|H|_{\omega,K^*}, \tag{3} \]

* Here we use the definition and notation for the integral of triangular truncation proposed by I. Ts. Gohberg and M. G. Krein (¹).

where the supremum on the left is taken over all Volterra operators \(A=G+iH\) with fixed imaginary component \(H=H^*\in \mathfrak S_\omega\).

On the basis of the Fan–Tzye inequalities (Ky Fan \((^4)\)) one first establishes

Lemma 1. Let \(H=H^*\in \mathfrak S_\omega\); then

\[ \sup \sum_{n=-\infty}^{\infty} (2n-1)^{-1}(H\varphi_n,\varphi_n) = |H|_{\omega,K^*}, \tag{4} \]

where the least upper bound is taken over all possible orthonormal systems \(\{\varphi_j\}_{-\infty}^{\infty}\) from \(\mathfrak H\).

Proceeding to the proof of Theorem 1, we note that it is enough to prove its first assertion for self-adjoint operators.

Without loss of generality, we shall suppose that the chain \(\mathfrak P\) is continuous. First assume that \(H=H^*\) is a finite-dimensional operator. For \(X=H\), the integral (1) converges to some Volterra operator \(A=G+iH\), which belongs to \(\mathfrak S_2\). Using the integral (1), construct a Volterra operator \(B=K+iL\) with its own chain \(\mathfrak P\) and real component \(K=(\cdot,\varphi)\varphi\), where \(\varphi\) is a given unit vector from \(\mathfrak H\). The completely continuous operator \(AB\) has the continuous proper chain \(\mathfrak P\) and, consequently, is also Volterra. Since \(A,B\in\mathfrak S_2\), it follows that \(AB\in\mathfrak S_1\), and by a theorem of V. B. Lidskii \((^4)\), \(\operatorname{Sp}(AB)=0\). Taking the real components of both sides of this equality, we obtain \(\operatorname{Sp}(GK-HL)=0\), or

\[ \operatorname{Sp}(GK)=\operatorname{Sp}(HL). \tag{5} \]

By a theorem of M. S. Livshits (see, for example, \((^3)\)) the operator \(B\) is unitarily equivalent to an inessential extension of the integration operator \(I\) in \(\mathcal L^2(0,1)\):

\[ If=2\int_0^x f(t)\,dt. \]

It follows that the nonzero eigenvalues of the operator \(L\) are equal to \(2/\pi(2j-1)\) \((j=0,\pm1,\ldots)\).

Now (5) can be rewritten in the form

\[ (G\varphi,\varphi)=\frac{2}{\pi}\sum_{j=-\infty}^{\infty}(2j-1)^{-1}(H\varphi_j,\varphi_j), \tag{6} \]

where \(\{\varphi_j\}\) are the eigenvectors of the operator \(L\). Applying Lemma 1 and taking on the left in (5) the least upper bound with respect to \(\varphi\), we obtain

\[ |G|_\infty \leq \frac{2}{\pi}|H|_{\omega,K^*}. \tag{7} \]

Let now \(H\in\mathfrak S_\omega\). Take a sequence of finite-dimensional operators \(H_n=H_n^*\) such that \(|H_n-H|_\omega\to 0\). By virtue of (7) we have
\[ |\mathfrak S(\mathfrak P,H_n)-\mathfrak S(\mathfrak P,H_m)|\to 0; \]
therefore the operators \(\mathfrak S(H_n)+iH_n\) converge to some Volterra operator \(A=G+iH\), which has the proper chain \(\mathfrak P\) and, by the above-mentioned theorem of M. S. Brodskii, admits a representation in the form of an integral (1) convergent in the norm of \(\mathfrak S_\infty\). Passing in the inequality
\[ |\mathfrak S(H_n)|\leq \frac{2}{\pi}|H_n|_{\omega,K^*} \]
to the limit as \(n\to\infty\), we obtain inequality (7). Since \(A\) is the unique operator with imaginary component \(H\) and proper chain \(\mathfrak P\), this inequality may be regarded as established for any Volterra operator \(A=G+iH\) with \(H\in\mathfrak S_\omega\). In exactly the same way one can prove for \(A\) the validity of relation (6).

It remains to verify that for every \(\varepsilon>0\) there exists a Volterra operator \(A=G+iH\) with given imaginary component \(H\) \((\in\mathfrak S_\omega)\), such that

\(G \mid > \dfrac{2}{\pi}|H|_{\omega,K^*}-\varepsilon\). By Lemma 1 there exists an orthonormal basis \(\{\varphi_j\}_{-\infty}^{\infty}\) for which (4) is fulfilled. Take an operator \(B=K+iL\), unitarily equivalent to the integration operator and such that \(L\varphi_n=(2/\pi(2n-1))\varphi_n\) \((n=0,\pm1,\ldots)\). Let \(\mathfrak P\) be its proper chain. Putting \(G=\mathfrak S(\mathfrak P,H)\) and using (6), we obtain

\[ |G|\ge (G\varphi,\varphi)=\frac{2}{\pi}\sum_{-\infty}^{\infty}(2j-1)^{-1}(H\varphi_j,\varphi_j)> \frac{2}{\pi}(|H|_{\omega,K^*}-\varepsilon). \]

Theorem 1 is completely proved.

Equality (3) led the author to suppose that the belonging of the operator \(X\in\mathfrak S_\infty\) to the class \(\mathfrak S_\omega\) is not only a sufficient but also a necessary condition for the integral (1) to converge for any continuous chain \(\mathfrak P\). The validity of this supposition was proved by I. Ts. Gokhberg and M. G. Krein, who also communicated to the author the following proposition.

Let \(\{\varphi_j\}_{-\infty}^{\infty}\) be some orthonormal system. Then there exists a continuous chain \(\mathfrak P\) such that, for any completely continuous self-adjoint operator

\[ X=\sum_j \lambda_j(\,\cdot\,,\varphi_j)\varphi_j \quad \left( (2j-1)\lambda_j>0;\ j=0,\pm1,\pm2,\ldots;\ \sum_{j=-\infty}^{\infty}\frac{\lambda_j}{2j-1}=\infty \right) \]

the integral (1) diverges (even in the sense of weak convergence). Conversely, for any continuous chain \(\mathfrak P\) one can construct an orthonormal system \(\{\varphi_j\}_{-\infty}^{\infty}\) such that the integral (1) diverges for any operator \(X\) only of the indicated type.

The possibility of divergence of the integral (1) for some \(X\in\mathfrak S_\infty\), under a special choice of the chain \(\mathfrak P\), was discovered by M. S. Brodskii \((^2)\).

It can be proved that for every operator \(X=X^*\in\mathfrak S_\infty\) there is always a continuous chain \(\mathfrak P\) for which the integral (1) converges.

1. L. A. Sakhnovich \((^6)\) showed that if, for a bounded operator with real spectrum \(A=G+iH\), the imaginary component \(H\in\mathfrak S_2\), then this operator has a sufficiently rich supply of invariant subspaces. I. Ts. Gokhberg and M. G. Krein, and independently of them the author, showed that this fact holds for \(H\in\mathfrak S_p\).

There is the following more general and complete proposition:

Theorem 2. An operator \(A=G+iH,\ H\in\mathfrak S_\omega\), having real spectrum, is an \(S\)-operator in the sense of \((^7)\).

Let us explain that an operator \(A\) with real spectrum is called an \(S\)-operator if it possesses the following properties. For every finite interval \(\Delta\) of the real axis there exists a subspace \(L(\Delta)\), invariant with respect to the operator \(A\), such that: a) on \(L(\Delta)\) the operator \(A\) is everywhere defined and bounded; b) the spectrum of the part of the operator \(A\) induced on \(L(\Delta)\) consists of the intersection of the spectrum of the operator \(A\) with the interval \(\Delta\) and, possibly, the endpoints of the interval \(\Delta\); c) every invariant subspace on which the operator \(A\) is everywhere defined, bounded, and has as spectrum a part of the segment \(\Delta\), is contained in \(L(\Delta)\); d) the system of invariant subspaces corresponding to any covering of the real axis by intervals is complete in \(\mathfrak H\).

Apparently, Theorem 2 admits the following converse: for every operator \(H=H^*\in\mathfrak S_\omega\) there is an operator \(G=G^*\) such that the operator \(A=G+iH\), on each of its invariant subspaces, has a spectrum coinciding with the spectrum of the whole operator \(A\), which in this case is real and consists of more than one point.

Theorem 2 is proved on the basis of Theorem 3 from \((^7)\) and the following lemma.

Lemma 2. The resolvent of an operator with real spectrum \(A=G+iH\), where \(G=G^*\) (in general, unbounded), and \(H=H^*\in \mathfrak S_\infty\), satisfies the following estimate:

\[ \ln |R_\lambda|=\ln |(A-\lambda I)^{-1}|\leq C\,[1+|\operatorname{Im}\lambda|^{-2} n(2|\operatorname{Im}\lambda|^{-1})], \]

where \(n(t)\) denotes the number of \(1/s_k(H)\) not exceeding \(t\).

1. Using other estimates of the resolvent, one can prove the following theorem:

Theorem 3. The system of eigenvectors and associated vectors of the operator \(A=H(I+T)\), where \(H=H^*\in\mathfrak S_\infty\), and \(T\in\mathfrak S_\omega\), is complete in its range.

This theorem can easily be given a form similar to that of the theorem of M. V. Keldysh from \({}^{(8)}\), namely:

If the equation \(y=Ay+\lambda Hy\) has a discrete spectrum, \(H=H^*\in\mathfrak S_\infty\) is a complete operator (see \({}^{(8)}\)), \(A\in\mathfrak S_\omega\), then the system of eigenvectors and associated vectors of this equation is complete in \(\mathfrak H\).

Physical-Technical Institute of Low Temperatures
of the Academy of Sciences of the Ukrainian SSR

Received
16 III 1961

REFERENCES

1. I. Ts. Gokhberg, M. G. Krein, DAN, 137, No. 5 (1961).
2. M. S. Brodskii, Uspekhi Mat. Nauk, 16, issue 1 (1961).
3. Fan Ky, Proc. Nat. Acad. Sci., 1, 35 (1949); 2, 36 (1950).
4. V. B. Lidskii, DAN, 125, No. 3 (1958).
5. I. Ts. Gokhberg, M. G. Krein, DAN, 128, No. 2 (1959).
6. L. A. Sakhnovich, Izv. Vyssh. Uchebn. Zaved., Mathematics, No. 4, (11) (1959).
7. Yu. I. Lyubich, V. I. Matsaev, DAN, 13, No. 1 (1960).
8. M. V. Keldysh, DAN, 77, No. 1 (1951).