UDC 513.88+517.948.35

MATHEMATICS

L. E. ISAEV

ON A CLASS OF OPERATORS WITH SPECTRUM CONCENTRATED AT ZERO

(Presented by Academician I. M. Vinogradov, April 7, 1967)

Let \(A\) be a bounded linear operator acting in a separable Hilbert space \(\mathfrak H\). The operator \(A\) is called completely non-selfadjoint if there is no subspace invariant with respect to \(A\) and \(A^*\) in which it induces a selfadjoint operator.

We shall assign a completely non-selfadjoint operator \(A\) to the class \(\Lambda^{(\exp)}\) if it satisfies the following conditions: 1) \(A\) has no spectral points distinct from zero; 2) \(\left(A-\dfrac{1}{\mu}E\right)^{-1}\) is a function of exponential type; 3) \(A\) is dissipative, i.e. \(A_I=(A-A^*)/2i\geqslant0\). If, moreover, \(\operatorname{sp} A_I<\infty\) \((\operatorname{sp} A_I=\infty)\), then we shall assign \(A\) to the class \(\Lambda^{[\exp]}\) \((\Lambda_\infty^{(\exp)})\).

The class \(\Lambda^{[\exp]}\) was studied by M. S. Livshits, M. S. Brodskii, M. G. Krein, I. Ts. Gokhberg, and G. E. Kisilevskii in papers \((^{1-8})\). The present article is devoted to extending some results of M. S. Brodskii and G. E. Kisilevskii to the class \(\Lambda^{(\exp)}\).

We note that all operators of the class \(\Lambda^{[\exp]}\) are completely continuous, which cannot be said, in general, of operators of the class \(\Lambda^{(\exp)}\).

An operator \(A\in\Lambda^{(\exp)}\) can be included in the node
\[ \theta=\begin{pmatrix}A&K\\ \mathfrak H&\mathfrak G\end{pmatrix}, \]
where \(\mathfrak G\) is some separable space and \(K\) is a bounded linear operator acting from \(\mathfrak G\) into \(\mathfrak H\), such that \(KK^*=A_I\). The function
\[ W(\lambda)=E-2iK^*(A-\lambda E)^{-1}K \]
is called the characteristic function of the node \(\theta\). It has the following properties: 1) its values are bounded linear operators acting in \(\mathfrak G\); 2) \(W(1/\mu)\) is an entire function of exponential type; 3) \(\lim_{\lambda\to\infty}\|W(\lambda)-E\|=0\); 4) \(W^*(\lambda)W(\lambda)-E\geqslant0\) \((\operatorname{Im}\lambda>0)\); 5) \(W^*(\lambda)W(\lambda)-E=0\) \((\operatorname{Im}\lambda=0,\lambda\neq0)\).

We shall denote the types of growth of the functions \(\left(A-\dfrac{1}{\mu}E\right)^{-1}\) and \(W\left(\dfrac{1}{\mu}\right)\), respectively, by \(\sigma[A]\) and \(\sigma[W]\). Obviously,
\[ \sigma[W]\leqslant\sigma[A]. \tag{1} \]

By \(\Omega_{\mathfrak G}^{(\exp)}\) we denote the class of all functions possessing properties 1)—5).

Consider the Hilbert space \(\widetilde{\mathfrak L}_2(0,l)\) \((l<\infty)\) of matrices of the form
\[ f(x)=\|f_1(x)\ f_2(x)\ldots\|, \tag{2} \]
where \(f_j(x)\) \((j=1,2,\ldots)\) are functions measurable on \((0,l)\) satisfying the condition
\[ \sum_{j=1}^{\infty}\int_0^l |f_j(x)|^2\,dx<\infty. \]
The scalar product in \(\widetilde{\mathfrak L}_2(0,l)\) is defi-

we define by the formula

\[ (f,g)=\sum_{j=1}^{\infty}\int_{0}^{l} f_j(x)\overline{g_j(x)}\,dx \qquad \left(g(x)=\|g_1(x)\ g_2(x)\ldots\|\in \widetilde{\mathscr L}_2(0,l)\right). \]

Define in \(\mathscr L_2(0,l)\) the operator \(\widetilde I_l\), which assigns to the matrix (2) the matrix

\[ \widetilde I_l f(x)=\|I_l f_1(x)\ I_l f_2(x)\ldots\| \qquad \left(I_l f_j(x)=2i\int_x^l f_j(t)\,dt\right). \]

Theorem 1*. If the operator \(T\) is induced by the operator \(\widetilde I_l\) in one of its invariant subspaces, then \(T\in\Lambda^{(\exp)}\) and \(\sigma[T]\le 2l\). Conversely, if \(A\in\Lambda^{(\exp)}\) and \(\sigma[A]\le 2l\), then there exists an invariant subspace of the operator \(\widetilde I_l\) in which an operator \(T\), unitarily equivalent to \(A\), is induced.

Proof. The first assertion of the theorem is trivial. To prove the second, embed \(A\) in the simple dissipative node

\[ \theta=\begin{pmatrix} A & K \\ \mathfrak H & \mathfrak G \end{pmatrix}, \]

and let \(W(\lambda)\) be the characteristic operator-function of this node. For all \(\lambda\ne 0\) there exists the operator \(W^{-1}(\lambda)\), and the estimate

\[ \|W(\lambda)\|\le e^{2l|\operatorname{Im}\frac{1}{\lambda}|} \]

holds, from which it follows that \(W^{-1}(\lambda)e^{2il/\lambda}\in\Omega_{\mathfrak H}^{(\exp)}\). To complete the proof it remains to refer to Theorem 2 of paper \((^2)\).

Theorem 2. Every function \(W(\lambda)\in\Omega_{\mathfrak H}^{(\exp)}\) is characteristic for some node

\[ \theta=\begin{pmatrix} A & K \\ \mathfrak H & \mathfrak G \end{pmatrix}, \]

where \(A\in\Lambda^{(\exp)}\) and \(\sigma[A]=\sigma[W]\).

It is known that every completely noncontinuous operator has a nontrivial invariant subspace \((^{10})\) (see also \((^{11})\), § 65). It turns out that operators of the class \(\Lambda_\infty^{(\exp)}\) also possess this property. Moreover, the following holds.

Theorem 3. For every operator \(A\in\Lambda_\infty^{(\exp)}\) there exists a continuous strictly increasing function \(P(x)\) \((0\le x\le \infty)\) satisfying the following conditions: 1) the values of the function \(P(x)\) are orthoprojectors in \(\mathfrak H\), with \(P(0)=0\), \(P(\infty)=E\); 2) all subspaces \(P(x)\mathfrak H\) are invariant with respect to \(A\);

\[ \text{3) }\quad \operatorname{sp}\bigl(P(x)A_I P(x)\bigr)=x \qquad (0\le x\le \infty). \tag{3} \]

Proof. Consider the operator \(B=-A^*\). By Theorem 1, the generality of the argument will not be violated if we assume that \(B\) is induced by the operator \(\widetilde I_l\) in some invariant subspace \(\mathfrak H\subset\widetilde{\mathscr L}_2(0,l)\) \((l=\tfrac12\sigma[A])\). In \(\widetilde{\mathscr L}_2(0,l)\) there exists a subspace \(\mathfrak H_0\) possessing the following properties: 1) \(\mathfrak H\) and \(\mathfrak H_0^\perp=\widetilde{\mathscr L}_2(0,l)\ominus\mathfrak H_0\) are invariant with respect to \(\widetilde I_l\); 2) in \(\mathfrak H_0\) the operator \(\widetilde I_l\) induces a Volterra operator with a nuclear imaginary component; 3) \(\mathfrak H\cap\mathfrak H_0^\perp\ne\mathfrak H\). From Lemma 4 of paper \((^4)\) it follows that \(\mathfrak H\cap\mathfrak H_0^\perp\ne 0\) and that \(A\) induces in its invariant subspace \(\mathfrak H\ominus(\mathfrak H\cap\mathfrak H_0^\perp)\) a Volterra operator with nuclear imaginary component. The subsequent arguments differ only insignificantly from those given in Sec. 1 of paper \((^3)\).

An operator is called unicellular if the set of all its invariant subspaces is ordered by inclusion.

Theorem 4. The class \(\Lambda_\infty^{(\exp)}\) contains no unicellular operators. In other words, every unicellular operator of the class \(\Lambda^{(\exp)}\) is Volterra and has a nuclear imaginary component.

The proof follows from Theorem 3 and the criterion for unicellularity of Volterra operators with nuclear imaginary components, established by M. S. Brodskii and G. E. Kisilevskii in the work \((^4)\).

\[ \text{* For the case } A\in\Lambda^{[\exp]} \text{ Theorem 1 was proved by G. E. Kisilevskii.} \]

Theorem 5. Let \(A\) be a completely non-self-adjoint operator whose spectrum contains no points different from zero. In order that \(A\) belong to the class \(\Lambda^{(\exp)}\), it is necessary and sufficient that there exist a sequence of subspaces invariant with respect to \(A\),
\(\mathfrak H_1 \subset \mathfrak H_2 \subset \ldots\), satisfying the following conditions:
1) \(\displaystyle \bigcup_{j=1}^{\infty}\mathfrak H_j=\mathfrak H\);
2) the operators \(A_j\), induced in \(\mathfrak H_j\), belong to the class \(\Lambda^{[\exp]}\);
3) the set of numbers \(\sigma[A_j]\) is bounded.

Proof. Necessity follows from Theorem 3. To prove sufficiency, include \(A\) and \(A_j\) in simple dissipative nodes

\[ =\begin{pmatrix} A & K\\ \mathfrak H & \mathfrak G \end{pmatrix} \quad\text{and}\quad \theta_j=\begin{pmatrix} A_j & P_jK\\ \mathfrak H_j & \mathfrak G \end{pmatrix}, \]

where \(P_j\) is the orthoprojector onto \(\mathfrak H_j\). Let \(W(\lambda)\) and \(W_j(\lambda)\) be the characteristic functions of these nodes. Since
\(\lim_{j\to\infty} W_j(\lambda)f=W(\lambda)f\) \((f\in\mathfrak G)\) and the set of numbers \(\sigma[A_j]\) is bounded, \(W(\lambda)\) has finite type of growth. The required result follows from Theorem 2.

The function \(P(x)\) \((0\le x\le\infty)\), introduced in Theorem 3, makes it possible to obtain the following generalization of the known theorem on the triangular representation of Volterra operators \((^3)\) (see also \((^{11})\), §115).

Theorem 6. Every operator \(A\in\Lambda^{(\exp)}\) can be represented in the form

\[ A=2i\lim_{x\to\infty}\int_0^x P(t)A_I\,dP(t), \]

where all the integrals exist in the sense of the operator norm, and the passage to the limit is carried out in the sense of strong convergence.

The author expresses deep gratitude to M. S. Brodskii for his guidance of this work.

Odessa State Pedagogical Institute named after K. D. Ushinsky

Received
5 IV 1967

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