UDC 513.88+517.948.35

MATHEMATICS

M. S. BRODSKII, L. E. ISAEV

TRIANGULAR REPRESENTATIONS OF DISSIPATIVE OPERATORS WITH RESOLVENT OF EXPONENTIAL TYPE

(Presented by Academician V. I. Smirnov on 3 III 1969)

A bounded linear operator \(A\), acting in a separable Hilbert space \(\mathfrak H\), will be assigned to the class \(\Lambda^{(\mathrm{exp})}\) if:
1) the spectrum of the operator \(A\) is concentrated at zero;
2) \(A_I=(A-A^*)/2i \ge 0\);
3) \((I-\lambda A)^{-1}\) is a function of exponential type. The type of growth of the resolvent \((I-\lambda A)^{-1}\) we shall agree to call the type of the operator \(A\) and denote by \(\sigma(A)\). If \(\sigma(A)=0\), then \(A=0\) \((^1)\).

It will be shown below that for every operator \(A\in\Lambda^{(\mathrm{exp})}\) there exists an orthogonal resolution of the identity \(P(x)\) \((0\le x\le \sigma(A))\) such that, in the sense of uniform convergence,

\[ A=2i\int_{0}^{\sigma(A)} P(x)A_I\,dP(x). \tag{1} \]

Representations of the form (1) are well known \((^2,^3)\) for the class of all completely continuous operators satisfying condition 1). Operators of the class \(\Lambda^{(\mathrm{exp})}\), as is easy to see, are not all completely continuous.

1. We shall say that a function of the complex variable \(W(\lambda)\), whose values are bounded linear operators acting in a separable Hilbert space \(\mathfrak G\), belongs to the class \(\Omega_{\mathfrak G}^{(\mathrm{exp})}\) if: I) \(W(\lambda)\) is an entire function of exponential type; II) \(W^*(\lambda)W(\lambda)-I=0\) \((\operatorname{Im}\lambda=0)\); III) \(W^*(\lambda)W(\lambda)-I\ge 0\) \((\operatorname{Im}\lambda<0)\); IV) \(W(0)=I\). The type of the function \(W(\lambda)\) will be denoted by \(\sigma(W)\).

For a given operator \(A\in\Lambda^{(\mathrm{exp})}\) one can, obviously, construct a bounded linear operator \(K\), acting from some separable Hilbert space \(\mathfrak G\) into \(\mathfrak H\), so that the equality \(KK^*=A_I\) holds. The collection

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

is called an exponential node, and the operator-function

\[ W_\theta(\lambda)=I+2i\lambda K^*(I-\lambda A)^{-1}K \tag{2} \]

the characteristic function of the node \(\theta\). Direct verification shows that \(W_\theta(\lambda)\in\Omega_{\mathfrak G}^{(\mathrm{exp})}\). Conversely, every function \(W(\lambda)\in\Omega_{\mathfrak G}^{(\mathrm{exp})}\) is characteristic for some exponential node \((^4)\). If \(W(\lambda)\) is the characteristic function of the exponential node

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

then \(\sigma(W)=\sigma(A)\) \((^4)\).

Let

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

be an exponential node and \(\mathfrak H=\mathfrak H_1\oplus\mathfrak H_2\). Define in \(\mathfrak H_j\) \((j=1,2)\) the operator \(A_j\), putting \(A_jh=P_jAh\) \((h\in\mathfrak H_j)\), where \(P_j\) is the orthoprojector onto \(\mathfrak H_j\). If the subspace \(\mathfrak H_1\) is invariant with respect to \(A\), then

\[ \theta_j=\begin{pmatrix} A_j & P_jK \\ \mathfrak H_j & \mathfrak G \end{pmatrix} \quad (j=1,2) \]

are exponential nodes and the formula \(W_\theta(\lambda)=W_{\theta_1}(\lambda)W_{\theta_2}(\lambda)\) holds \((^4)\). The node \(\theta_j\) is called the projection of the node \(\theta\) onto the subspace \(\mathfrak H_j\).

1. We shall need the following properties of operators of the class \(\Lambda^{(\exp)}\).

a) If \(A\in\Lambda^{(\exp)}\) and \(0<\sigma_0<\sigma(A)\), then there exists a subspace \(\mathfrak H_0\) invariant with respect to \(A\) such that the operator \(A_0\) induced in it satisfies the condition \(\sigma(A_0)=\sigma_0\).

Indeed, there exists a subspace \(\mathfrak H_1\) invariant with respect to \(A\), in which a unicellular operator \(A_1\) is induced, whose type coincides with the type of the operator \(A\) \((^1)\). From the unicellularity of the operator \(A_1\) it follows, as was shown in \((^4)\), that it is completely continuous and has a nuclear imaginary component. By virtue of the criterion of unicellularity \((^5)\),
\[ \sigma(A)=2\,\operatorname{sp}(A_1)_I. \]
It remains to note that the traces of the imaginary components of operators induced in invariant subspaces of the operator \(A_1\) take all values from \(0\) to \(\operatorname{sp}(A_1)_I\) \((^6)\).

b) If \(A\in\Lambda^{(\exp)}\), then \(\|A\|\leqslant \sigma(A)\).

Let us suppose first that the operator \(A\) is unicellular. Then it is completely continuous \((^4)\) and is representable in the form (1). Let
\[ 0=x_0<x_1<\cdots<x_n=\sigma(A) \]
be some partition of the segment \([0,\sigma(A)]\). Consider an orthonormal basis \(\{e_\alpha\}_1^\infty\), consisting of eigenvectors of the operator \(A_I\), and denote by \(\omega_\alpha\) the eigenvalues corresponding to the vectors \(e_\alpha\). Since
\[ \left|\left(\sum_{j=1}^{n} P(x_j)A_I\Delta P_j f,g\right)\right| \leqslant \sum_{j=1}^{n}\sum_{\alpha=1}^{\infty} \omega_\alpha\left|(\Delta P_j f,e_\alpha)\right| \left|(P(x_j)e_\alpha,g)\right| \leqslant \]
\[ \leqslant \|g\|\sum_{\alpha=1}^{\infty}\omega_\alpha \sum_{j=1}^{n}\left|(\Delta P_j f,e_\alpha)\right| \leqslant \|f\|\,\|g\|\,\operatorname{sp} A_I \qquad \bigl(\Delta P_j=P(x_j)-P(x_{j-1})\bigr), \]
then
\[ \|A\|\leqslant 2\,\operatorname{sp} A_I\;(=\sigma(A)). \]

In the case when \(A\) is not unicellular, for an arbitrary vector \(f\in\mathfrak H\) we construct the subspace \(\mathfrak H_f\), which is the closure of the linear span of the vectors \(A^n f\) \((n=0,1,\ldots)\). The operator \(A_f\), induced in the subspace \(\mathfrak H_f\), is unicellular \((^7)\). Consequently, \(\|A_f\|\leqslant \sigma(A_f)\), and since \(\sigma(A_f)\leqslant\sigma(A)\), then
\[ \|Af\|=\|A_f f\|\leqslant \sigma(A)\|f\| \quad (f\in\mathfrak H), \]
and hence \(\|A\|\leqslant\sigma(A)\).

1. Let \(A\in\Lambda^{(\exp)}\) and let \(\mathfrak H_\gamma\) \((\gamma\in\Gamma)\) be all possible subspaces invariant with respect to \(A\), in which operators are induced whose types do not exceed some fixed value \(x\in(0,\sigma(A)]\).

Embed the operator \(A\) in the node
\[ \theta=\begin{pmatrix} A & K\\ \mathfrak H & \mathfrak G \end{pmatrix} \]
and denote by \(\theta_\gamma\) and \(\theta_x\) the projections of the node \(\theta\) onto \(\mathfrak H_\gamma\) and onto the closure \(\mathfrak H_x\) of the linear span of the subspaces \(\mathfrak H_\gamma\) \((\gamma\in\Gamma)\). According to one of the theorems of \((^8)\),
\[ \sigma(W_{\theta_x})=\sup_{\gamma\in\Gamma}\sigma(W_{\theta_\gamma}). \]
Denoting by \(A_x\) the operator induced in \(\mathfrak H_x\), we obtain
\[ \sigma(A_x)=\sigma(W_{\theta_x})=\sup_{\gamma\in\Gamma}\sigma(A_\gamma)=x. \]

Lemma. If \(P_\Delta\) is the orthoprojector onto the subspace
\[ \mathfrak H_\Delta=\mathfrak H_{x_2}\ominus\mathfrak H_{x_1}, \]
\((0<x_1<x_2\leqslant\sigma(A))\) and \(A_\Delta f=P_\Delta Af\) \((f\in\mathfrak H_\Delta)\), then \(\sigma(A_\Delta)=x_2-x_1\).

Proof. Denoting by \(\theta_\Delta\) the projection of the node \(\theta_{x_2}\) onto \(\mathfrak H_\Delta\), we obtain
\[ W_{\theta_2}(\lambda)=W_{\theta_1}(\lambda)W_{\theta_\Delta}(\lambda). \]
Consequently,
\[ \sigma(W_{\theta_2})\leqslant\sigma(W_{\theta_1})+\sigma(W_{\theta_\Delta}), \]
so that
\[ \sigma(A_\Delta)\geqslant x_2-x_1. \]

Fix a vector \(f\in\mathfrak H_\Delta\) and consider the operator \(A_f\) induced in \(\mathfrak H_f\). Obviously,
\[ x_1<\sigma(A_f)\leqslant x_2. \]
Let \(P_f\) be the orthoprojector onto the subspace
\[ \mathfrak G_f=\mathfrak H_f\ominus(\mathfrak H_f\cap\mathfrak H_{x_1}), \]
and
\[ B_f h=P_f A_f h\qquad (h\in\mathfrak G_f). \]
Since the operator induced in \(\mathfrak H_f\cap\mathfrak H_{x_1}\) has type \(x_1\), it follows that
\[ \operatorname{sp}(B_f)_I\leqslant \tfrac12(x_2-x_1). \]
Introduce the operators \(T\) and \(C_f\), of which the first assigns to each vector of \(\mathfrak H_f\) its orthogonal projection onto \(\mathfrak H_\Delta\), while the second is induced by the operator \(A_\Delta\) in the subspace
\[ \mathfrak K_f=T\mathfrak H_f \]
invariant for it. The operator
\[ A_\Delta T(=TA_f) \]
is completely continuous. Consequently, there exist orthonormi-

orthonormal sequences \(\{\varphi_\alpha\}_1^\nu\) \((\nu \leq \infty)\) and \(\{\psi_\alpha\}_1^\nu\), belonging respectively to the subspaces \(\mathfrak H_f\) and \(\mathfrak H_\Delta\), such that \(A_\Delta T h=\sum_{\alpha=1}^{\nu}(h,\varphi_\alpha)\omega_\alpha\psi_\alpha\) \((h\in\mathfrak H_f,\ \omega_\alpha>0)\). It is not hard to see that \(\{\varphi_\alpha\}_1^\nu\) and \(\{\psi_\alpha\}_1^\nu\) are bases of the subspaces \(\mathfrak H_f\) and \(\mathfrak R_f\). From the equalities

\[ (A_f\varphi_\alpha,\varphi_\alpha) = \frac{1}{\omega_\alpha} \left( \sum_{\beta=1}^{\nu}(A_f\varphi_\alpha,\varphi_\beta)\omega_\beta,\psi_\alpha \right) = \frac{1}{\omega_\alpha}(A_\Delta T A_f\varphi_\alpha,\psi_\alpha) = (A_\Delta\psi_\alpha,\psi_\alpha) \]

it follows that

\[ \operatorname{sp}(C_f)_I = \sum_{\alpha=1}^{\nu}((C_f)_I\psi_\alpha,\psi_\alpha) = \sum_{\alpha=1}^{\nu}((A_\Delta)_I\psi_\alpha,\psi_\alpha) = \sum_{\alpha=1}^{\nu}((A_f)_I\varphi_\alpha,\varphi_\alpha) = \]

\[ = \sum_{\alpha=1}^{\nu}((B_f)_I\varphi_\alpha,\varphi_\alpha) \leq \frac12(x_2-x_1). \]

For every operator \(A\in\Lambda^{(\exp)}\) the inequality \(\sigma(A)\leq 2\operatorname{sp}A_I\) holds. Consequently, \(\sigma(C_f)\leq x_2-x_1\). Since \(\mathfrak H_\Delta\) is the closure of the linear span of all subspaces \(\mathfrak R_f\) \((f\in\mathfrak H_\Delta)\), it follows that \(\sigma(A_\Delta)\leq x_2-x_1\).

The lemma is proved. Denote by \(P(x)\) \((0<x\leq\sigma(A))\) the orthogonal projector onto \(\mathfrak H_x\), and put \(P(0)=0\). We shall call the function \(P(x)\) \((0\leq x\leq\sigma(A))\) the extremal spectral function of the operator \(A\). It is easy to show that \(P(x)\) is continuous on the interval \((0,\sigma(A)]\). If \(A\) is a completely non-self-adjoint operator \((^4)\), then it is also continuous at the point \(0\).

Theorem 1. An operator \(A\in\Lambda^{(\exp)}\) admits a norm-convergent triangular representation (1), where \(P(x)\) is its extremal spectral function.

Proof. It is enough to show \((^2)\) that for every \(\varepsilon>0\) there exists a \(\delta>0\) such that, if the partition \(0=x_0<x_1<\cdots<x_n=\sigma(A)\) satisfies the condition \(x_j-x_{j-1}<\delta\), then

\[ \left\|\sum_{j=1}^{n}\Delta P_j A \Delta P_j\right\|<\varepsilon \]

\((\Delta P_j=P(x_j)-P(x_{j-1}))\). Put \(\delta=\varepsilon\). Using assertion b) and the lemma, we obtain:

\[ \left\|\sum_{j=1}^{n}\Delta P_j A \Delta P_j\right\| = \max_j\|\Delta P_j A \Delta P_j\| \leq \max_j\sigma(\Delta P_j A \Delta P_j) < \varepsilon. \]

Theorem 2. If \(W(\lambda)\in\Omega_{\mathfrak S}^{(\exp)}\), then in the sense of uniform convergence

\[ W(\lambda)=\int_{0}^{\sigma(W)} e^{i\lambda x}\,dF(x), \]

where \(F(x)\) is a strictly increasing operator-function satisfying the condition
\[ \|F(x')-F(x'')\|\leq |x'-x''|. \]

Proof follows from formula (1) with the aid of arguments analogous to those given in work \((^9)\).

Theorem 2 is a special case of a more general assertion obtained by Yu. P. Ginzburg \((^{10})\) by a purely analytic method.

Odessa State Pedagogical Institute
named after K. D. Ushinsky

Received
26 II 1969

CITED LITERATURE

1. M. S. Brodskii, Triangular and Jordan Representations of Linear Operators, “Nauka,” 1969.
2. M. S. Brodskii, Uspekhi Mat. Nauk, 16, no. 1 (97), 135 (1961).
3. I. Ts. Gokhberg, M. G. Krein, Theory of Volterra Operators in Hilbert Space and Its Applications, “Nauka,” 1967.
4. L. E. Isaev, Dokl. Akad. Nauk SSSR, 178, no. 4, 783 (1968).
5. M. S. Brodskii, G. E. Kisilevskii, Izv. Akad. Nauk SSSR, Ser. Mat., 30, no. 6, 1213 (1966).
6. M. S. Brodskii, Scientific Notes of the Odessa State Pedagogical Institute, 24, issue 1, 3 (1959).
7. G. E. Kisilevskii, Dokl. Akad. Nauk SSSR, 173, no. 5, 1006 (1967).
8. M. S. Brodskii, Mathematical Investigations, Kishinev, 3, issue 1 (7), 3 (1968).
9. M. S. Brodskii, Dokl. Akad. Nauk SSSR, 138, no. 4, 751 (1961).
10. Yu. P. Ginzburg, Dokl. Akad. Nauk SSSR, 170, no. 1, 23 (1966).