V. M. ADAMYAN, D. Z. AROV

ON A CERTAIN CLASS OF SCATTERING OPERATORS AND CHARACTERISTIC OPERATOR-FUNCTIONS OF CONTRACTIONS

(Presented by Academician V. I. Smirnov on 20 VI 1964)

1°. In questions of scattering theory for two groups of unitary operators \(U_t\) and \(\widetilde U_t\) with Lebesgue spectrum, acting in one and the same Hilbert space \((t\) runs through a sequence of integers or the whole real axis), the \(S\)-operator of scattering is defined as the product:

\[ S(\widetilde U,U)=W_+(U,\widetilde U)W_-(\widetilde U,U) \left(=W_+^{-1}(\widetilde U,U)W_-(\widetilde U,U)\right), \tag{1} \]

where

\[ W_{\pm}(\widetilde U,U)=s-\lim_{t\to\pm\infty}\widetilde U_{-t}U_t \]

are the so-called wave operators.

Below we shall consider unitary groups satisfying the conditions of the scheme of P. Lax and R. Phillips \((^1)\). This scheme makes it possible to define the wave operators and the scattering operator for the case of unitary groups acting in different spaces.

In investigations on the theory of unitary dilations of contraction operators, B. Sz.-Nagy and C. Foias \((^3)\) made essential use of the notion of the characteristic operator-function \(W(\zeta)\) of a contraction \(T\),

\[ W(\zeta)=-T+\zeta(I-TT^*)^{1/2}(I-\zeta T^*)^{-1}(I-T^*T)^{1/2}, \tag{2} \]

introduced for the first time in this form in \((^2)\).

Thanks to the investigations \((^{1,3})\), it is possible to establish a connection between the scattering suboperator and the characteristic operator-function in one class of problems of scattering theory. In the class under consideration, the question of the possibility of reconstructing the group from the \(S\)-operator is also clarified, and the analytic properties of the scattering suboperator are studied.

2°. Let \(V_t^{\pm}(t\ge 0)\) be two semigroups of isometric operators acting in orthogonal subspaces \(\mathfrak D_{\pm}\) of the Hilbert space \(\mathfrak H=\mathfrak D_+\oplus\mathfrak D_-\), \(V_t^{\pm}\mathfrak D_{\pm}\subset \mathfrak D_{\pm}\). We shall assume that

\[ \bigcap_{t>0} V_t^{\pm}\mathfrak D_{\pm}=\{0\}. \tag{3} \]

The general case is easily reduced to this one.

A group of unitary operators \(U_t\), acting in a Hilbert space \(\mathfrak H_U\), will be called a (unitary) coupling of the semigroups \(V_t^{\pm}\) if: 1) \(\mathfrak D_{\pm}\subset \mathfrak H_U\); 2) \(U_{\pm t}f=V_t^{\pm}f\), \(f\in\mathfrak D_{\pm}\) \((t>0)\); 3) \(\bigvee U_t\mathfrak D_{\pm}=\mathfrak H_U\).

In the discrete case, when \(t\) runs through the integers, for the existence of such couplings it is necessary and sufficient that the dimensions

\[ m_{\pm}=\dim(\mathfrak D_{\pm}\ominus V_1^{\pm}\mathfrak D_{\pm}) \]

be equal.

In the continuous case this condition must be satisfied for the Cayley transforms of the infinitesimal operators of the semigroups \(V_t^{\pm}\). Indeed,

\[ B^{\pm}=s-\lim_{t\downarrow 0}\frac{1}{it}(V_t^{\pm}-I) \]

are maximal symmetric operators in the spaces \(\mathfrak D_{\pm}\) with defect indices \((0,m_{\pm})\). Therefore, for the Cayley transforms

\[ V_{\pm}(\zeta)=(B^{\pm}-\zeta I)(B^{\pm}-\bar\zeta I)^{-1} \]

of the operators \(B^{\pm}\), we have

\[ V_{\pm}(\zeta)\mathfrak D_{\pm}\subset\mathfrak D_{\pm}\quad(\operatorname{Im}\zeta>0),\qquad m_{\pm}=\dim\left(\mathfrak D_{\pm}\ominus V_{\pm}(\zeta)\mathfrak D_{\pm}\right). \]

From (3) it follows for \(V_t^{\pm}\) that

\[ \bigcap_k V_{\pm}^{k}(\zeta)\mathfrak D_{\pm}=\{0\}. \]

The infinitesimal operator \(A\) of an arbitrary coupling \(U_t\) of the semigroups \(V_t^{\pm}\) is a self-adjoint extension, with exit into the space \(\mathfrak H_U\), of the symmetric operator \(B=B^+\oplus(-B^-)\), whence it follows that the Cayley transform \(V(\zeta)\) of the operator \(A\) gives a coupling of the semigroups generated by the operators \(V_\pm(\zeta)\). Conversely, an arbitrary coupling of the semigroups \(V_\pm^k(\zeta)\) gives the Cayley transform of the infinitesimal operator of some coupling of the semigroups \(V_t^{\pm}\).

Taking into account (3) and properties 1)–3) of the coupling, we obtain that an arbitrary coupling has only Lebesgue spectrum of multiplicity \(m=m_\pm\).

For two couplings \(U_t\) and \(\widetilde U_t\) of the semigroups \(V_t^{\pm}\), with exit into the spaces \(\mathfrak H_U\) and \(\mathfrak H_{\widetilde U}\), we define the wave operators by the formulas

\[ W_\pm(U,\widetilde U)=s-\lim_{T\to\mp\infty}\left(s-\lim_{t\to\pm\infty} U_{-t}\widetilde U_t P_{\widetilde U_T\mathfrak D_\pm}\right), \tag{4} \]

where \(P_{\widetilde U_T\mathfrak D_\pm}\) are the projectors from \(\mathfrak H_{\widetilde U}\) onto \(\widetilde U_T\mathfrak D_\pm\).

It is easy to see that the operators introduced exist and
\(W_\pm f=s-\lim_{t\to\pm\infty} U_{-t}\widetilde U_t f\) for \(f\in U_T\mathfrak D_\pm\). The operators \(W_\pm(U,\widetilde U)\) isometrically map \(\mathfrak H_{\widetilde U}\) onto \(\mathfrak H_U\),

\[ U_t W_\pm(U,\widetilde U)=W_\pm(U,\widetilde U)\widetilde U_t,\qquad W_\pm(\widetilde U,U)=W_\pm^{-1}(U,\widetilde U) \]

and the multiplication theorem holds: for three couplings \(U_t^{(1)}, U_t^{(2)}\), and \(U_t^{(3)}\) of the semigroups \(V_t^{\pm}\), the equality

\[ W_\pm(U^{(1)},U^{(3)})= W_\pm(U^{(1)},U^{(2)})W_\pm(U^{(2)},U^{(3)}) \]

is valid.

The scattering operator \(S(\widetilde U,U)\) of the group \(\widetilde U_t\) with respect to the group \(U_t\) is defined, as usual, by the product (1) of the wave operators. The operator \(S(\widetilde U,U)\) is unitary on \(\mathfrak H_U\) and commutes with the group \(U_t\).

If for two couplings \(U_t\) and \(\widetilde U_t\) of the semigroups \(V_t^{\pm}\) there exists a coupling \(U_t^{(1)}\) such that \(S(U,U^{(1)})=S(\widetilde U,U^{(1)})\), then for an arbitrary coupling \(U_t^{(2)}\) the equality \(S(U,U^{(2)})=S(\widetilde U,U^{(2)})\) holds. In this case we shall say that the scattering operators of the groups \(U_t\) and \(\widetilde U_t\) coincide.

Theorem 1. For two couplings \(U_t\) and \(\widetilde U_t\), the scattering operators coincide if and only if, for all \(t\),

\[ P_U U_t P_U=P_{\widetilde U}\widetilde U_t P_{\widetilde U}, \tag{5} \]

where \(P_U\) and \(P_{\widetilde U}\) are the projectors, respectively, from \(\mathfrak H_U\) and \(\mathfrak H_{\widetilde U}\) onto \(\mathfrak H\).

In what follows the exposition will be carried out only for discrete groups. The results formulated extend to the case of continuous groups by passing to the Cayley transforms of the corresponding infinitesimal operators. In this connection the following is essential.

Lemma. For any two couplings \(U_t\) and \(\widetilde U\) \((-\infty<t<\infty)\), the equality

\[ W_\pm(U,\widetilde U)=W_\pm(V(\xi),\widetilde V(\xi)), \]

holds, where \(V(\xi)\) and \(\widetilde V(\xi)\) are the Cayley transforms of the infinitesimal operators of the groups \(U_t\) and \(\widetilde U_t\).

The assertion of the lemma under other conditions was first obtained in [4] (see also [5]).

Since below we shall be speaking only about discrete groups, instead of saying that the group \(U_t\) is a coupling of the semigroups \(V_t^{\pm}\), we shall say that the operator \(U=U_1\) is a coupling of the operators \(V_\pm=V_1^{\pm}\).

3°. Let the unitary operator \(U\) be a coupling, with exit into the space \(\mathfrak H_U\), of the isometric operators \(V_\pm\). Then, if we denote by \(\mathfrak R_\pm=\mathfrak D_\pm\oplus V_\pm\mathfrak D_\pm\), the decompositions

\[ \mathfrak D_\pm=\sum_0^\infty \oplus V_\pm^k\mathfrak R_\pm,\qquad \mathfrak H_U=\sum_{-\infty}^\infty \oplus U^k\mathfrak R_\pm . \tag{6} \]

hold. Let \(\mathcal L_2(-\pi,\pi;\mathfrak R_\pm)\) be the Hilbert spaces of vector-functions \(f(\lambda)\) with values in \(\mathfrak R_\pm\) and norm
\[ \|f\|^2=\int_{-\pi}^{\pi}\|f(\lambda)\|_{\mathfrak R_\pm}^2\,d\lambda, \]
and let \(\mathcal L_2^\pm(-\pi,\pi;\mathfrak R_\pm)\) be the subspaces of vector-functions from \(\mathcal L_2(-\pi,\pi;\mathfrak R_\pm)\) expandable in a Fourier series respectively in nonnegative and negative powers of \(e^{i\lambda}\). Consider the isometric mappings \(F_\pm\) of the space \(\mathfrak H_U\) onto \(\mathcal L_2(-\pi,\pi;\mathfrak R_\pm)\):
\[ F_+f=\sum_{-\infty}^{\infty} e^{ik\lambda}P_{\mathfrak R_+}U^{-k}f,\qquad F_-f=\sum_{-\infty}^{\infty} e^{i(k-1)\lambda}P_{\mathfrak R_-}U^{-k}f,\quad f\in\mathfrak H_U, \]
where \(P_{\mathfrak R_\pm}\) are the projectors from \(\mathfrak H_U\) onto \(\mathfrak R_\pm\).

It is easy to see that
\[ F_\pm Uf=e^{i\lambda}F_\pm f,\qquad F_\pm\mathfrak D_\pm=\mathcal L_2^\pm(-\pi,\pi;\mathfrak R_\pm). \]

In [1] an operator of the form
\[ \hat S=QF_-F_+^{-1}, \]
was introduced, where \(Q\) is an arbitrary isometric mapping of \(\mathfrak R_-\) onto \(\mathfrak R_+\), and was called the scattering operator. It was also shown there that

\[ (\hat Sf)(\lambda)=S(\lambda)f(\lambda),\qquad f\in\mathcal L_2(-\pi,\pi;\mathfrak R_+), \tag{7} \]
where \(S(\lambda)\) is the boundary value of an “inner” operator-function \(S(\xi)\) on \(\mathfrak R_+\): 1) \(S(\xi)\) is analytic inside the disk \(|\xi|<1\); 2) the boundary value (in the strong sense) \(S(\lambda)\) is unitary for almost all \(\lambda\).

One can show that, for an arbitrary “inner” operator-function on \(\mathfrak R_+\), the operator
\[ S=F_+^{-1}\hat SF_+=F_+^{-1}QF_-, \tag{8} \]
where \(\hat S\) is given by equality (7), is indeed the scattering operator \(S(U_0,U)\) for the coupling \(U\) under consideration and for some coupling \(U_0\) without exit from the space \(\mathfrak H=\mathfrak D_+\oplus\mathfrak D_-\). Here the operator \(U_0\) is determined uniquely by the \(S\)-operator, and \(U_0f=Qf,\ f\in\mathfrak R_-\). Conversely, if \(U_0\) is an arbitrary coupling without exit from \(\mathfrak H\), then
\[ S(U_0,U)=F_+^{-1}U_0F_-. \]

4°. Let now \(\widetilde U\) be an arbitrary coupling, generally speaking with exit into the space \(\mathfrak H_{\widetilde U}\). Then, since \(S(\widetilde U,U)\) commutes with the operator \(U\) and is unitary, for the operator \(\hat S=F_+SF_+^{-1}\) we obtain
\[ (\hat Sf)(\lambda)=S(\lambda)f(\lambda),\qquad f\in\mathcal L_2(-\pi,\pi;\mathfrak R_+), \]
where \(S(\lambda)\) is a unitary operator-function on \(\mathfrak R_+\), called the scattering suboperator. It follows from item 3° that when \(\mathfrak H_{\widetilde U}=\mathfrak H\), the suboperator \(S(\lambda)\) is the boundary value of an “inner” operator-function on \(\mathfrak R_+\).

In the general case the following holds.

Theorem 2. For arbitrary couplings \(U\) and \(\widetilde U\) of the operators \(V_\pm\), the suboperator \(S(\lambda)\) is representable in the form
\[ S(\lambda)=S_1^*(\lambda)S_2(\lambda), \tag{9} \]
where \(S_1(\lambda)\) and \(S_2(\lambda)\) are boundary values of “inner” operator-functions on \(\mathfrak R_+\).

5°. Denote \(\mathfrak K=\mathfrak H_U\ominus\mathfrak H\); let \(P_{\mathfrak K}\) be the projector from \(\mathfrak H_U\) onto \(\mathfrak K\), and introduce the semigroup of contractions on \(\mathfrak K\)

\[ T^n=P_{\mathfrak K}U^nP_{\mathfrak K}. \tag{10} \]

We shall consider such a coupling \(U\) for which \(U\mathfrak N_-\cap U\mathfrak N_+=\{0\}\). The general case is easily reduced to this one. Then \(\overline{(U-T)\mathfrak K}=\mathfrak N_+\), \(\overline{(U^*-T^*)\mathfrak K}=\mathfrak N_-\), and there exist isometric mappings \(Q_+\) and \(Q_-\) of the spaces \(\overline{(I-T^*T)^{1/2}\mathfrak K}\) and \(\overline{(I-TT^*)^{1/2}\mathfrak K}\) onto the spaces \(\mathfrak N_+\) and \(\mathfrak N_-\), such that

\[ (U-T)f=Q_+(I-T^*T)^{1/2}f,\qquad (U^*-T^*)f=Q_-(I-TT^*)^{1/2}f,\qquad f\in\mathfrak K. \]

Observe that in the case under consideration \(U\) is the minimal unitary dilation of the operator \(T\), and \(T^n f\to0\), \(T^{*n}f\to0\) \((n\to\infty)\), \(f\in\mathfrak K\) \((^3)\).

Let \(W(\zeta)\) \((|\zeta|<1)\) be the characteristic operator-function (2) of the contraction \(T\). If \(U_0\) is an arbitrary coupling of the operators \(V_\pm\) without exit from the space \(\mathfrak H\), then, according to §§ 3°, 4°, the scattering suboperator of \(U_0\) relative to \(U\) is the boundary value of a certain “inner” operator-function \(S(\zeta)\) on \(\mathfrak N_+\).

Theorem 3. The relation holds

\[ S(\zeta)=QQ_-W(\zeta)Q_+^{-1}, \tag{11} \]

where \(Q\) is the isometric mapping of \(\mathfrak N_-\) onto \(\mathfrak N_+\), induced by the operator \(U_0\).

Conversely, the boundary value of an arbitrary inner function of the form (11) is the scattering suboperator of some coupling without exit relative to the operator \(U\).

6°. If \(W(\zeta)\) is the characteristic operator-function of an arbitrary contraction \(T\) and \(\zeta_0\) is a regular point of the operator \(T\), \(|\zeta_0|<1\), then the operator \(W(\zeta_0)\) is continuously invertible and

\[ W^{-1}(\zeta_0)=-T^*+(I-T^*T)^{1/2}(\zeta_0 I-T)^{-1}(I-TT^*)^{1/2}. \]

For the class of contraction operators \(T\) considered by us, when \(W(\zeta)\) is an inner function, Theorem 3 and the results of work \((^1)\) imply the converse assertion (compare with \((^6)\)).

Moreover, the following theorem is valid, which is more conveniently formulated for the operator-function \(S(\zeta)\) connected with \(W(\zeta)\) by relation (11).

Theorem 4. For any \(\zeta\), \(|\zeta|<1\), the dimensions of the annihilating subspaces of the operators \(T-\zeta I\) and \(S(\zeta)\) are equal and

\[ \dim[\mathfrak K\ominus\overline{(T-\zeta I)\mathfrak K}] = \dim[\mathfrak N_+\ominus\overline{S(\zeta)\mathfrak N_+}]. \]

A point \(\zeta\), \(|\zeta|<1\), is a point of regular type of the operator \(T\) if and only if \(0\) is a point of regular type of the operator \(S(\zeta)\).

Couplings of semigroups of isometric operators with exit into different spaces arise in the consideration of the wave equation with different boundary conditions \((^1)\).

In conclusion the authors express their gratitude to Prof. M. G. Krein for posing the problems and for his constant attention, and also to the participants of the seminar under his direction for useful discussions.

Received
18 VI 1964

CITED LITERATURE

\(^1\) P. Lax, R. Phillips, Materials of the Joint Soviet-American Symposium, Novosibirsk, 1963.
\(^2\) Yu. L. Shmul’yan, DAN, 93, No. 6 (1963).
\(^3\) Bela Sz.-Nagy, C. Foias, C. R., 256, 323 (1963).
\(^4\) M. Sh. Birman, M. G. Krein, DAN, 144, No. 3 (1962).
\(^5\) M. Sh. Birman, Izv. AN SSSR, Ser. Mat., 27, No. 4 (1963).
\(^6\) M. S. Livshits, V. P. Potapov, DAN, 72, No. 4 (1950).