UDC 517.432

MATHEMATICS

L. A. SAKHNOVICH

ON DISSIPATIVE OPERATORS WITH ABSOLUTELY CONTINUOUS SPECTRUM

(Presented by Academician M. V. Keldysh on 30 VI 1965)

§ 1. In the present paper we consider a non-self-adjoint operator \(A\) of class \(i\Omega\) \((^{1})\) with continuous spectrum. In addition, we shall assume that the operator \(A\) is dissipative, i.e. \((A-A^*)/i \geq 0\). As M. S. Livshits \((^{1})\) showed, the characteristic matrix-function of such an operator is represented in the form

\[ w(\lambda)=\int_0^l \exp\left[-\,\frac{i\beta^2(t)}{\alpha(t)-\lambda}\,dt\right], \]

where the function \(\alpha(t)\) increases monotonically and is bounded, while the matrix \(\beta(x)\) is nonnegative and \(\operatorname{sp}\beta^2(x)\equiv 1\).

Assume additionally that the operator \(A\) has absolutely continuous spectrum, i.e. the function \(t=\alpha(x)\) has an absolutely continuous inverse function \(x=\sigma(t)\). In this case

\[ w(\lambda)=\int_0^b \exp\left[-\,\frac{i\beta_1^2(t)}{t-\lambda}\,dt\right], \]

where \(\beta_1(t)=p(t)\beta(\sigma(t))\), \(p(t)=\sqrt{\sigma'(t)}\), \(a=\alpha(0)\), \(b=\alpha(l)\).

The triangular model of the operator \(A\), as follows from \((^{1})\), can be written in the form

\[ \vec{A}f=xf(x)+i\int_a^x f(t)\beta_1(t)\,dt\,\beta_1(x)\qquad (a\leq x\leq b). \tag{1} \]

Theorem 1. The additional component of the operator \(\vec{A}\) consists of those and only those vector-functions \(f(x)\in L_r^2[a,b]\) for which, almost everywhere, the equality

\[ f(x)\beta_1(x)\equiv 0,\qquad \text{if } \|\beta_1(x)\|\leq M. \]

§ 2. For what follows, the behavior of the multiplicative integral

\[ w(b,\lambda)=\int_a^b \exp\left[-\,\frac{i\beta_1^2(t)}{t-\lambda}\,dt\right]\qquad \left(\int_a^b \|\beta_1^2(t)\|\,dt<\infty\right) \tag{2} \]

as \(\tau\to 0\) \((\lambda=\sigma+i\tau)\) is essential.

Theorem 2. For almost all \(\sigma\in[a,b]\) there exist limiting values

\[ w^{\pm}(\sigma)=\lim_{\tau\to\pm 0} w(b,\lambda) \]

and the formulas hold

\[ w^{\pm}(\sigma)=\lim_{\varepsilon\to 0}\int_a^{\sigma-\varepsilon} \exp\left[-\frac{i\beta_1^2(t)}{t-\gamma}\,dt\right]\exp[\pm\pi\beta_1^2(\sigma)] \int_{\sigma+\varepsilon}^{b}\exp\left[-\frac{i\beta_1^2(t)}{t-\gamma}\,dt\right], \tag{3} \]

where the limits are understood in the sense of strong convergence.

The theorem was previously proved by us under the condition that \(\operatorname{sp}\beta_1^2(t)\) is bounded \((^{2,3})\).

From formula (3) it follows that

\[ w^{\pm}(\sigma)=R^{\pm1}(\sigma)u(b,\sigma), \tag{4} \]

where

\[ R^{\pm1}(\sigma)=\lim_{\varepsilon\to+0}\int_a^{\sigma-\varepsilon} \exp\left[-\frac{i\beta_1^2(t)\,dt}{t-\sigma}\right]\exp[\pm\pi\beta_1^2(\sigma)] \left[\int_a^{\sigma-\varepsilon} \exp\left[-\frac{i\beta_1^2(t)}{t-\sigma}\,dt\right]\right]^{-1} = \]

\[ =\exp[\pm\pi r^2(\sigma)], \tag{5} \]

\[ U(b,\sigma)=\lim_{\varepsilon\to+0}\int_a^{\sigma-\varepsilon} \exp\left[-\frac{i\beta_1^2(t)}{t-\sigma}\,dt\right] \int_{\sigma+\varepsilon}^{b}\exp\left[-i\frac{\beta_1^2(t)}{t-\sigma}\,dt\right]. \tag{6} \]

Let \(H\) and \(H_1\) denote the closures of the manifolds into which \(L_r^2[a,b]\) is mapped upon multiplication of its elements respectively by \(\beta_1(x)\) and \(R(x)-R^{-1}(x)\).

By Theorem 1, the operator \(\vec A\) induces its simple part on \(H\).

In what follows we shall consider the operator \(\vec A\) only on the space \(H\).

In \((^{4-6})\) we constructed mutually inverse operators \(B\) and \(B^{-1}\), defined on dense sets respectively in \(H_1\) and \(H\), by means of the formulas

\[ B\varphi=\frac{1}{\sqrt{2\pi}}\,\frac{d}{dx}\int_a^x \varphi(\sigma)\sqrt{R(\sigma)-R^{-1}(\sigma)}\,U(x,\sigma)\,d\sigma\,\beta_1^{-1}(x), \tag{7} \]

\[ B^{-1}f=\frac{1}{\sqrt{2\pi}}\left\{\int_a^x [f(\sigma)\beta_1^{-1}(\sigma)]'U^*(\sigma,x)\,d\sigma +f(a)\beta_1^{-1}(a)\right\}\sqrt{R(x)-R^{-1}(x)}. \tag{8} \]

Here

\[ U(x,\sigma)=\lim_{\varepsilon\to+0}\int_a^{\sigma-\varepsilon} \exp\left[-\frac{i\beta_1^2(t)}{t-\sigma}\,dt\right] \int_{\sigma+\varepsilon}^{b}\exp\left[-\frac{i\beta_1^2(t)}{t-\sigma}\,dt\right]. \]

Formulas (7)—(8) can be given meaning also in the case when the matrix \(\beta_1(x)\) has no inverse on a set of positive measure \((^{4-6})\).

Item 3. The relation holds

\[ \vec A=BQB^{-1}, \tag{9} \]

where \(Q\) is the operator of multiplication by the independent variable,

\[ Qf=xf,\qquad f\in H_1. \]

We note that, in deriving relations (7)—(9), in works \((^{4-6})\) we assumed \(\operatorname{sp}\beta_1^2(t)\) to be bounded. However, it is easy to dispense with this condition by using Theorem 2.

If $\vec A$ and $Q$ are connected by a relation of the type (9), then they are called linearly similar (${}^{4-5}$). If, moreover, $B$ and $B^{-1}$ are bounded, then the operators $\vec A$ and $Q$ are called linearly equivalent.

Theorem 3. The relations

\[ \left\| e^{-\frac{\pi}{2} r^2} B^{-1} \right\| = 1, \qquad \left\| B e^{-\frac{\pi}{2} r^2} \right\| = 1, \]

hold, where the operator $e^{-\frac{\pi}{2} r^2}$ is defined by the formula

\[ e^{-\frac{\pi}{2} r^2} f = f(x)e^{-\frac{\pi}{2} r^2(x)} . \]

Theorem 4. In order that the operator $\vec A$ be linearly equivalent to a self-adjoint operator, it is necessary and sufficient that

\[ \operatorname{vrai\,sup}\|\beta_1^2(x)\| = M < \infty . \tag{10} \]

Corollary. If condition (10) is satisfied, then the operator $\vec A$ is equivalent to $Q$, and

\[ \|B\| = e^{\frac{\pi}{2}M}, \qquad \|B^{-1}\| = e^{\frac{\pi}{2}M}. \]

In terms of the characteristic matrix-function, Theorem 4 admits the following reformulation.

Theorem 4′*. In order that a dissipative operator $A$ of class $i\Omega$ with absolutely continuous spectrum be linearly equivalent to a self-adjoint operator, it is necessary and sufficient that

\[ \operatorname{vrai\,sup}\|w^{+}(\sigma)\| < \infty . \]

Theorem 3 makes it possible to study the operator $A$ also in the case when condition (10) is not satisfied.

Theorem 5. The multiplicity of the spectrum of the operator $A$ is equal to

\[ N = \operatorname{vrai\,sup}\operatorname{rang}\beta_1^2(x) = \operatorname{vrai\,sup}\operatorname{rang}\,[w^{+}(x)-w^{-}(x)] . \]

Corollary. The operator $A$ decomposes into the sum of $N$ induced operators of first multiplicity.

Theorem 6. There exist invariant subspaces $H^{(n)}$ and $H_1^{(n)}$ of the operators $A$ and $Q$, respectively, such that the operators $A^{(n)}$ and $Q^{(n)}$ induced on them are linearly equivalent. Moreover, the projection operators onto these subspaces satisfy

\[ P^{(n)} \xrightarrow[n\to\infty]{} I, \qquad P_1^{(n)} \xrightarrow[n\to\infty]{} I . \]

All the results remain valid also for operators not belonging to the class $i\Omega$, whose characteristic matrix-function admits the representation (2).

Let us note that the theorems formulated here can be carried over to the case when one or both endpoints of the segment $[a,b]$ are infinite. In this case the operators $A$ and $Q$ are unbounded.

4. Let us study the behavior of $e^{iAt}$ as $t\to\pm\infty$. We shall compare it with the behavior of $e^{iA_1t}$ as $t\to\pm\infty$, where $A_1$ is the real component of $A$ ($A_1=(A+A^*)/2$). Let $G$ be the subspace corresponding to the absolutely continuous part of the spectrum of $A_1$, and let $P_G$ be the projection operator onto $G$.

* Note added in proof. A more general result has been obtained by B. S. Naim and K. Hoffman (${}^{10}$).

Theorem 7. Let the prime operator \(A\) satisfy the conditions of Theorem 4. Then the strong limits exist

\[ W_{\pm}(A,A_1)=\lim_{t\to\pm\infty} e^{iAt}e^{-iA_1t}P_G, \]

\[ W_{\pm}(A_1,A)=\lim_{t\to\pm\infty} e^{iA_1t}e^{-iAt}. \]

Moreover, the operators \(W_{\pm}(A,A_1)\) and \(W_{\pm}(A_1,A)\) are bounded together with their inverses, and the relations hold

\[ A=W_{\pm}(A,A_1)A_1W_{\pm}(A,A_1)^{-1},\qquad A_1=W_{\pm}(A_1,A)AW_{\pm}(A_1,A)^{-1}, \]

\[ W_{\pm}(A,A_1)=W_{\pm}^{-1}(A_1,A), \]

where \(A_1\) is considered only on \(G\).

Thus, Theorem 7 extends to non-self-adjoint operators the well-known Rosenblum–Kato theorem \((^{7,8})\). We note that some results in this direction were obtained earlier \((^9)\).

Introduce the scattering operator \(S\) by the formula

\[ S=W_-^{-1}(A,A_1)W_+(A,A_1). \]

The operator \(S\) is defined in \(G\) and commutes in \(G\) with \(A\), \(\|S\|\leqslant 1\).

Theorem 8. The operator \(S\) is unitarily equivalent to multiplication by the matrix

\[ S(x)=I-\sqrt{R(x)-R^{-1}(x)}\,R^{1/2}(x)U(b,x)(I+R(x)U(b,x))^{-1}\times \]

\[ \times R^{-1/2}(x)\sqrt{R(x)-R^{-1}(x)} \]

in the space \(H_1\). Here \(R\) and \(U\) are related to the characteristic function \(w\) of the operator \(A\) by relations (4)—(6).

Odessa Electrotechnical
Institute of Communications

Received
21 VI 1965

REFERENCES

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