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UDC 517.432
MATHEMATICS
L. A. SAKHNOVICH
NONUNITARY OPERATORS WITH ABSOLUTELY CONTINUOUS SPECTRUM ON THE UNIT CIRCLE
(Presented by Academician L. S. Pontryagin, November 3, 1967)
1°. We shall say that an operator \(T\), acting in a separable Hilbert space \(H\), belongs to the class \(\mathscr E\) if it is linearly similar to some unitary operator with absolutely continuous spectrum. We shall say that \(A \in \mathcal N\) if \((A-iE)(A+iE)^{-1} \in \mathscr E\). Let \(T\) be a bounded operator together with its inverse. We define the characteristic matrix-function \(w(\mu)\) of the operator \(T\) by the equality \((^1)\)
\[ w(0)w(\mu)=E-J\|((E-\mu T)^{-1}g_\alpha,g_\beta)\|, \]
where \(g_\alpha\) is a channel system of vectors (see \((^{2-4})\)).
Recall that the simple part of \(T\) is the operator induced by \(T\) on the subspace
\[ H_1=\overline{\sum_{k=-\infty}^{\infty} T^k D_T}, \quad \text{where } \quad D_T=\overline{(E-T^*T)H}. \]
Theorem 1. If the characteristic matrix-function of the operator \(T\) satisfies, for some \(c\), the condition
\[ \|w(\mu)\|\le c,\qquad |\mu|\ne 1, \tag{1} \]
then the simple part of \(T\) is linearly similar to a unitary operator with absolutely continuous spectrum.
In the case \(J=E\) and for a special choice of channel vectors, it was shown in the work of B. Nad’ and Ch. Foias \((^5)\) that inequality (1) is a necessary and sufficient condition for the similarity of the operator \(T\) to some unitary operator (see also \((^{6-7})\)).
It is sometimes more convenient to use Theorem 1 by writing \(w(\mu)\) in operator form (see \((^3)\)).
There exists a bounded operator \(R\), mapping some Hilbert space \(G\) into \(H\), and satisfying the conditions:
- \(E-T^*T=RJR^*\), where \(J\) acts in \(G\) and \(J=J^*\), \(J^2=E\).
- The spectrum of \(E-JR^*R\) is strictly positive.
By condition 2 there exists a \(J\)-Hermitian operator \(w(0)\) with strictly positive spectrum such that
\[ w^2(0)=E-JR^*R. \]
We define the characteristic operator-function \(w(\mu)\) of the operator \(T\) in the following way:
\[ w(0)w(\mu)=E-JR^*(E-\mu T)^{-1}R. \]
The values of the operator-function \(w(\mu)\) are operators acting in \(G\).
2°. Let the operator \(A\) have the form
\[ A=A_R+iA_I, \tag{2} \]
where \(A_R\) and \(A_I\) are self-adjoint operators acting in \(H\), with \(A_I\) bounded. There exists a bounded operator \(K\), mapping some Hilbert space \(G\) into \(H\), such that
\[ A_I=KJK^*, \]
where \(J\) acts in \(G\) and \(J^*=J,\ J^2=E\).
The operator function
\[ W(\lambda)=E-2iK^*(A-\lambda E)^{-1}KJ \]
is called the characteristic function of the operator \(A\).
For bounded dissipative operators \(A\), the definition introduced does not coincide with that given in the paper \(\left({}^{3}\right)\).
We note that the values \(W(\lambda)\) are operators in \(G\). An operator \(A\) with real spectrum will be called simple if \(T=(A-iE)\times (A-iE)^{-1}\) is simple.
Theorem 2. If the characteristic function of the operator \(A\) for some \(c\) satisfies the condition
\[ \|W(\lambda)\|\leq c,\qquad \operatorname{Im}\lambda\ne 0, \]
then the simple part of \(A\) is linearly similar to a self-adjoint operator with absolutely continuous spectrum.
For \(J=E\), a close fact is contained in the paper \(\left({}^{7}\right)\).
Let us now consider the operator
\[ A=A_0+B, \tag{3} \]
where \(A_0\) is a self-adjoint operator, \(B\) is a bounded operator. Represent \(B\) in the form
\[ B=B_1^*B_2, \tag{4} \]
where \(B_1\) and \(B_2\) are bounded operators.
An operator of the form (3) admits representations (2).
Put \(R_0(\lambda)=(A_0-\lambda E)^{-1},\ Q(\lambda)=B_2R_0(\lambda)B_1^*\).
Theorem 3. Let the operators \(Q(\lambda)\), \((E+Q(\lambda))^{-1}\), \(K^*R_0(\lambda)K\), \(K^*R_0(\lambda)B_1^*\), \(B_2R_0(\lambda)K\) be uniformly bounded off the real axis. Then the simple part of \(A\) is linearly similar to a self-adjoint operator with absolutely continuous spectrum.
It is useful to compare this result with a theorem of T. Kato \(\left({}^{8}\right)\).
\(3^\circ\). Consider in the space \(L^2(E_m)\) the Schrödinger operator
\[ A=A_0+q, \]
where
\[ A_0u=-\sum_{i=1}^{m}\frac{\partial^2}{\partial x_i^2}u,\qquad qu=q(x)u,\qquad x\in E_m,\quad m\geq 3. \]
Introduce the notation
\[ B_1f=|q(x)|^{1/2}f(x),\qquad B_2f=\frac{\overline{q(x)}}{|q(x)|^{1/2}}f(x),\qquad Q(\lambda)=B_2R_0(\lambda)B_1^*. \]
Theorem 4. Let the function \(q(x)\) be bounded and \(q(x)\in L^p(E_m)\) \((1\leq p<m/2)\). If the operator function \([E+Q(\lambda)]^{-1}\) is uniformly bounded for \(\operatorname{Im}\lambda\ne 0\), then the operator \(A\) is linearly similar to a self-adjoint operator with absolutely continuous spectrum, i.e. \(A\in\mathcal N\).
Corollary. If in Theorem 4 \(p=1\), then the operators \(A\) and \(A_0\) are linearly similar and comparable.
Recall that the operators \(A\) and \(A_0\) are called comparable \(\left({}^{1}\right)\) if there exist wave operators, bounded together with their inverses,
\[ W_{\pm}(A,A_0)=s\text{-}\lim_{t\to\pm\infty} e^{iAt}e^{-iA_0t}, \]
and, moreover,
\[ AW_{\pm}(A,A_0)=W_{\pm}(A,A_0)A_0. \]
Under somewhat weaker requirements on \(q(x)\), in the work of T. Kato \((^8)\) the linear similarity and comparability of the operators \(A\) and \(A_0\) were proved. In that work, however, it was additionally assumed that
\[ \|Q(\lambda)\|\leq \rho <1. \]
\(4^\circ.\) Consider a differential expression of the form
\[ ly=-y''+yp(r), \]
where \(y(r)=[y_1(r),\,y_2(r),\ldots,y_m(r)]\) \((m<\infty)\) is a vector function, \(p(r)\) is a matrix of order \(m\), and
\[ \sigma=\int_0^\infty \|p(r)\|\,dr<\infty. \]
The expression \(l(y)\) and the condition
\[ y'(0)-y(0)\theta=0 \qquad (\theta\text{ is a matrix}) \]
generate the differential operator \(L_\theta\). The non-self-adjoint operator \(L_\theta\) was studied in the work of M. A. Naimark \((^9)\) for \(m=1\) and
\[ \int_0^\infty (1+r^2)\|\nu(r)\|\,dr<\infty. \]
Introduce the matrix \(y(r,s)\), which satisfies the equation
\[ l(y)-s^2y=0,\qquad \operatorname{Im}s\geq 0, \]
and the condition
\[ \lim_{r\to\infty} y(r,s)e^{-isr}=E. \]
Denote by \(D(s)=y'(0,s)-y(0,s)\theta\). We shall compare the operator \(L_\theta\) with the simplest operator \(\mathscr L_0\):
\[ \mathscr L_0 f=-d^2 f/dr^2,\qquad f(0)=0,\qquad f\in L_m^2(0,\infty). \]
Theorem 5. If there exists a \(c\) such that
\[ \|D^{-1}(s)\|\leq c,\qquad \operatorname{Im}s\geq 0,\qquad \text{and}\quad \int_0^\infty (1+r^{3/2})\|p(r)\|\,dr<\infty, \]
then the operators \(L_\theta\) and \(\mathscr L_0\) are linearly similar and comparable.
The conditions of Theorem 5 can be weakened if dissipative operators are considered.
Theorem 6. Let the operator \(L_\theta\) satisfy the following conditions:
-
\[ \frac{\theta-\theta^*}{i}\geq 0,\qquad \frac{p(r)-p^*(r)}{i}\geq 0. \]
-
\[ \int_0^\infty \|p(r)\|\,dr<\infty. \]
-
There exists a \(c\) such that \(\|D(s)\|\leq c,\ \|D^{-1}(s)\|\leq c\).
Then the operators \(L_\theta\) and \(\mathscr L_0\) are linearly similar and comparable.
\(5^\circ.\) Let the operator \(A\) act in \(L_m^2(a,b)\) \((-\infty\leq a<b\leq\infty)\) and be defined by the formula
\[ A=A_0+iB, \]
where \(A_0f=xf\), and \(B\) is a self-adjoint nuclear operator.
The operator \(B\) can be written in the form
\[ Bf=\frac12\sum_{\alpha,\beta}^n (f,g_\alpha)\, j_{\alpha\beta}g_\beta,\qquad \sum_{\alpha=1}^n \|g_\alpha\|^2<\infty, \]
where \(n\leq\infty\), \(J=\|j_{\alpha\beta}\|\) is a certain Hermitian matrix such that \(J^2=E\).
Introduce the matrices
\[ G(x)=\|g_\alpha(x)g_\beta^*(x)\|_{\alpha,\beta=1}^n, \qquad V(\lambda)=\frac12\int_a^b \frac{G(x)}{x-\lambda}\,dx . \]
Theorem 7. Let the matrix \(G(x)\) satisfy the conditions:
- The function \(G(x)\) is bounded on the segment \([a,b]\), and
\[ \int_a^b \|G(x)\|\,dx<\infty . \]
- There exists an \(M\) such that, for all \(t\) and for some \(c>0\),
\[ \int_{t-c}^{t+c}\left\|\frac{G(x)-G(t)}{x-t}\right\|\,dx \le M . \]
-
\(\overline{\lim}\,\|G(x)\|\ln(x-a)<\infty,\quad x\to a.\)
-
\(\overline{\lim}\,\|G(x)\|\ln(b-x)<\infty,\quad x\to b.\)
(If \(a=-\infty\) or \(b=\infty\), then respectively condition 3 or 4 is omitted.)
If the matrix-function \([E+iV(\lambda)J]^{-1}\) is uniformly bounded for \(\operatorname{Im}\lambda\ne0\), then the operators \(A\) and \(A_0\) are linearly similar and comparable.
6°. Consider the operator
\[ Af=xf+i\int_a^x f(t)\beta(t)J\,dt\,\beta(x) \qquad (-\infty<a<b<\infty), \]
where \(\beta(t)\) is a nonnegative matrix of order \(m\) (\(m\le\infty\)), and \(J=J^*\) and \(J^2=E\). A broad class of operators (2) is reduced to such triangular form.
Theorem 8. Let the following conditions be fulfilled:
- There exists a matrix \(U(t)\) such that
\[ \beta^2(t)J=U^*(t)H(t)U^{-1}(t), \]
where \(H(t)\) is a self-adjoint matrix.
-
The matrices \(U(t)\), \(U^{-1}(t)\), and \(H(t)\) are uniformly bounded on the segment \([a,b]\).
-
The inequality holds
\[ \|U(t_2)-U(t_1)\|\le K|t_2-t_1|, \qquad t_1,t_2\in[a,b], \]
where \(K\) is some constant.
Then the simple part of \(A\) is linearly similar to some self-adjoint operator with absolutely continuous spectrum.
We note that for \(J=E\) one may put \(U(t)=E\) (see (6)).
Odessa Electrotechnical Institute of Communications
Received
21 X 1967
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