UDC 513.88:519.4+517.948.35+517.948.5

MATHEMATICS

V. M. ADAMYAN, D. Z. AROV

ON SCATTERING OPERATORS AND SEMIGROUPS OF CONTRACTIONS IN A HILBERT SPACE

(Presented by Academician V. I. Smirnov on 31 III 1965)

1. Let \(\mathfrak D_+\) and \(\mathfrak D_-\) be subspaces belonging to the intersection of Hilbert spaces \(\mathfrak H\) and \(\widetilde{\mathfrak H}\). Here on \(\mathfrak D_+\) and \(\mathfrak D_-\) (but not on their linear sum) the Hilbert metrics of the spaces \(\mathfrak H\) and \(\widetilde{\mathfrak H}\) coincide. Let \(U_t\) and \(\widetilde U_t\) be groups of unitary operators in \(\mathfrak H\) and \(\widetilde{\mathfrak H}\), respectively, satisfying the conditions:
   \[ \text{1) }\quad U_{\pm t}\mathfrak D_\pm \subset \mathfrak D_\pm,\quad t>0;\qquad \text{2) }\quad U_{\pm t}f=\widetilde U_{\pm t}f,\quad f\in\mathfrak D_\pm,\quad t>0. \]
   Here discrete groups (the parameter \(t\) takes only integral values) and continuous groups \((-\infty<t<\infty)\) are considered in parallel. By \(\mathfrak H_\pm\) denote the l.c.s. \(\{U_t\mathfrak D_\pm\}_{-\infty}^{+\infty}\), and by \(\widetilde{\mathfrak H}_\pm\) the analogous subspaces for \(\widetilde U_t\). Without loss of generality, assume that
   \[ \bigcap_t U_t\mathfrak D_\pm=\bigcap_t \widetilde U_t\mathfrak D_\pm=\{0\},\qquad \mathfrak D_+\cap\mathfrak D_-=\{0\}; \tag{1} \]
   \[ \overline{\mathfrak H_+ + \mathfrak H_-}=\mathfrak H,\qquad \overline{\widetilde{\mathfrak H}_+ + \widetilde{\mathfrak H}_-}=\widetilde{\mathfrak H}. \tag{2} \]

In paper \((^1)\) the groups \(U_t\) and \(\widetilde U_t\) were studied from the point of view of scattering theory within the framework of the scheme of P. Lax and R. Phillips \((^2)\), i.e. it was assumed that the subspaces \(\mathfrak D_+\) and \(\mathfrak D_-\) are orthogonal both in \(\mathfrak H\) and in \(\widetilde{\mathfrak H}\), \(\mathfrak H_\pm=\mathfrak H\), and \(\widetilde{\mathfrak H}_\pm=\widetilde{\mathfrak H}\). Below, in particular, it is clarified how the orthogonality of the subspaces \(\mathfrak D_+\) and \(\mathfrak D_-\) is connected with the analytic properties of the scattering suboperator. The more general scheme adopted in this article makes it possible to study groups of unitary operators that are dilations of arbitrary semigroups of contractions \((^3)\).

Define the operators \(W_\pm\), acting from \(\mathfrak H_\pm\) into \(\widetilde{\mathfrak H}_\pm\), by setting, for \(f\in\mathfrak H_\pm\),
\[ W_\pm(\widetilde U,U)f=s\text{-}\lim_{T\to\pm\infty}\ s\text{-}\lim_{t\to\pm\infty}\widetilde U_{-t}U_{t-T}P_{\mathfrak D_\pm}U_Tf. \tag{3} \]
Here, as in what follows, \(P_{\mathfrak R}\) is the orthogonal projector onto the subspace \(\mathfrak R\), and \(s\text{-}\lim\) is the strong limit. The operators \(W_\pm\) isometrically map \(\mathfrak H_\pm\) onto \(\widetilde{\mathfrak H}_\pm\). We extend them to \(\mathfrak H_\pm^\perp=\mathfrak H\ominus\mathfrak H_\pm\) by the condition \(W_\pm\mathfrak H_\pm^\perp=0\) and call them wave operators. The scattering operator of the group \(\widetilde U_t\) with respect to \(U_t\) is defined as the product
\[ S(\widetilde U,U)=W_+^*(\widetilde U,U)W_-(\widetilde U,U) \bigl(=W_+(U,\widetilde U)W_-(\widetilde U,U)\bigr). \tag{4} \]
It is easy to see that
\[ S\mathfrak H\subset\mathfrak H_+,\qquad S^*\mathfrak H\subset\mathfrak H_-,\qquad U_tS=SU_t,\quad \|S\|\le 1. \tag{5} \]

Note that the operator \(S\) is unitary if and only if \(\mathfrak H_\pm=\mathfrak H\) and \(\widetilde{\mathfrak H}_\pm=\widetilde{\mathfrak H}\). Below we shall consider only the part \(S_+\) of the operator \(S\) on \(\mathfrak H_+\) and the part \(S_-\) of the operator \(S^*\) on \(\mathfrak H_-\).

In what follows \(U\) denotes the operator \(U_1\), if \(U_t\) is a discrete group, and the Cayley transform \(U=(A+iI)(A-iI)^{-1}\) of the infinitesimal ope-

operator \(A=s\text{-}\lim_{t\downarrow 0}\dfrac{1}{-it}(U_t-I)\), if \(U_t\) is a continuous group; \(\mathfrak N_{\pm}\) are subspaces \(\mathfrak D_{\pm}\ominus U^{\pm1}\mathfrak D_{\pm}\); \(\Lambda\) is the interval \((-\pi,\pi)\) and \((-\infty,\infty)\), respectively, in the discrete and continuous cases; \(C_{\pm}\) are the interior and exterior of the unit circle or the upper and lower half-planes of the complex plane; \(\mathcal L_2(\mathfrak N_{\pm},\Lambda)\) are Hilbert spaces of measurable vector-functions \(f(\lambda)\) with values in \(\mathfrak N_{\pm}\) and finite norm

\[ \|f\|_{\mathcal L_2}^{2}=\frac{1}{2\pi}\int_{\Lambda}\|f(\lambda)\|_{\mathfrak N_{\pm}}^{1}\,d\lambda<\infty; \]

\(\mathcal L_2^{\pm}(\mathfrak N_{\pm},\Lambda)\) are subspaces of vector-functions from \(\mathcal L_2(\mathfrak N_{\pm},\Lambda)\) that are boundary values of vector-functions, analytic in \(C_{\pm}\), of the Hardy classes \(H_2^{\pm}(\mathfrak N_{\pm})\).

From condition (1) and the definition of the spaces \(\mathfrak H_{\pm}\) there follows the existence of isometric mappings \(F_{\pm}\) of the spaces \(\mathfrak H_{\pm}\) onto \(\mathcal L_2(\mathfrak N_{\pm},\Lambda)\) such that

\[ F_{\pm}U_tF_{\pm}^{-1}f(\lambda)=e^{it\lambda}f(\lambda), \]

\[ F_{\pm}\mathfrak D_{\pm}=\mathcal L_2^{\pm}(\mathfrak N_{\pm},\Lambda). \]

By virtue of (5), there exist measurable almost everywhere nonexpanding operator-functions \(S_{\pm}(\lambda)\), acting for each \(\lambda\in\Lambda\) in \(\mathfrak N_{\pm}\), such that

\[ F_{\pm}S_{\pm}F_{\pm}^{-1}f(\lambda)=S_{\pm}(\lambda)f(\lambda),\qquad f(\lambda)\in\mathcal L_2(\mathfrak N_{\pm},\Lambda). \]

The operators \(S_{\pm}(\lambda)\) are unitary almost everywhere if and only if \(\mathfrak H_{+}=\mathfrak H_{-}\) and \(\widetilde{\mathfrak H}_{+}=\widetilde{\mathfrak H}_{-}\); in this case the groups \(U_t\) and \(\widetilde U_t\) have only Lebesgue spectrum on \(\Lambda\) of multiplicity \(r=\dim\mathfrak N_{+}=\dim\mathfrak N_{-}\). In the general case the groups \(U_t\) and \(\widetilde U_t\) have absolutely continuous, but not necessarily Lebesgue, spectrum \({}^{(3)}\); however, their parts considered respectively on \(\mathfrak H_{\pm}\) and \(\widetilde{\mathfrak H}_{\pm}\) have only Lebesgue spectrum of multiplicity \(r_{\pm}=\dim\mathfrak N_{\pm}\).

1. Of interest are the cases when the operator-functions \(S_{\pm}(\lambda)\) are boundary values of analytic functions. From the results of P. Lax \({}^{(4)}\) it follows

Theorem 1. In order that the operator-functions \(S_{\pm}(\lambda)\) be boundary values (in the strong sense) of analytic operator-functions \(S_{\pm}(\zeta)\):

a) on \(C_{+}\), it is necessary and sufficient that \(S_{+}\mathfrak D_{+}\subset \mathfrak D_{+}\);

b) on \(C_{-}\), it is necessary and sufficient that \(S_{\pm}(\mathfrak H_{\pm}\ominus\mathfrak D_{\pm})\subset \mathfrak H_{\pm}\ominus\mathfrak D_{\pm}\).

According to (1), the subspaces \(\mathfrak D_{+}\) and \(\mathfrak D_{-}\) intersect only in zero. Therefore one can construct the space \(\mathfrak H_0=\mathfrak D_{+}\oplus\mathfrak D_{-}\). For definiteness in what follows, we shall assume that \(r_{+}\ge r_{-}\). Then there exists a subspace \(\mathfrak D_{+}'\subset\mathfrak D_{+}\), for which \(U_t\mathfrak D_{+}'\subset\mathfrak D_{+}'\) and \(r_{+}'=\dim\mathfrak N_{+}'=r_{-}\) \((\mathfrak N_{+}'=\mathfrak D_{+}'\ominus U\mathfrak D_{+}')\). In the role of the group \(\widetilde U_t\) henceforth will appear the group \(U_t'\) in the space \(\mathfrak H_0'=\mathfrak D_{+}'\oplus\mathfrak D_{-}\). We note that the group \(U_t'\) is determined uniquely up to an isometric mapping of \(\mathfrak N_{-}\) onto \(\mathfrak N_{+}'\). For the corresponding operator-functions \(S_{\pm}(\lambda)\) the following is valid

Theorem 2. In order that the operator-functions \(S_{\pm}(\lambda)\) be boundary values of operator-functions analytic in \(C_{\pm}\), it is necessary and sufficient that the subspaces \(\mathfrak D_{+}'\) and \(\mathfrak D_{-}\) be orthogonal in \(\mathfrak H\). *

When the condition of Theorem 2 is fulfilled, in the case of discrete groups

\[ \begin{aligned} S_{+}(\xi)&=U^{0}P_{\mathfrak N_{-}}(U-\xi I)^{-1}P_{\mathfrak N'_{+}},\qquad &&|\xi|<1;\\ S_{-}(\xi)&=U^{0-1}P_{\mathfrak N'_{+}}(U^{-1}-\xi^{-1}I)^{-1}P_{\mathfrak N_{-}},\qquad &&|\xi|>1. \end{aligned} \tag{6} \]

* In the general case, if \(\mathfrak D_{+}\) and \(\mathfrak D_{-}\) are orthogonal both in \(\mathfrak H\) and in \(\widetilde{\mathfrak H}\), for the suboperators \(S_{\pm}(\lambda)\) of the operators \(S_{\pm}=S_{\pm}(\widetilde U,U)\) there remains valid a factorization theorem analogous to that formulated in \({}^{(4)}\).

For continuous groups, from (6) we obtain

\[ \begin{aligned} S_+(\zeta) &= -\frac{(\zeta-i)^2}{2i}\, U_0 P_{\mathfrak N_-}(A-\zeta I)^{-1}P_{\mathfrak N'_+}, \qquad \operatorname{Im}\zeta<0;\\ S_-(\zeta) &= \frac{(\zeta+i)^2}{2i}\, U_0^{-1}P_{\mathfrak N'_+}(A-\zeta I)^{-1}P_{\mathfrak N_-}, \qquad \operatorname{Im}\zeta>0 . \end{aligned} \tag{7} \]

3. Suppose that \(\mathfrak H_0\subset \mathfrak H\), and consider in \(\mathfrak K=\mathfrak H\ominus\mathfrak H_0\) the one-parameter family of operators \(T_t\) defined by the formula

\[ T_t=P_{\mathfrak K}U_tP_{\mathfrak K},\qquad t>0 . \tag{8} \]

The family \(T_t\) forms a nonexpanding semigroup \((^2)\). It is easy to see that the group \(U_t\) is a unitary dilation in the sense of Nagy \((^3)\) of the semigroup \(T_t\), where property (1) is necessary and sufficient for the minimality of the dilation (l.c.s. \(\{U_t\mathfrak K\}_{-\infty}^{+\infty}=\mathfrak H\)), and property (2) is necessary and sufficient for the complete nonunitarity of the semigroup \(T_t\) (absence of a unitary part).

The condition \(\mathfrak H_+=\mathfrak H\) (\(\mathfrak H_-=\mathfrak H\)) is equivalent to the requirement \(s\text{-}\lim_{t\to\infty}T_t=0\) \((s\text{-}\lim_{t\to\infty}T_t^*=0)\). Let us note that if \(Z\) is the Cayley transform of the infinitesimal operator of the semigroup \(T_t\), then the condition \(s\text{-}\lim_{t\to\infty}T_t=0\) is equivalent to the condition \(s\text{-}\lim_{n\to\infty}Z^n=0\). This is a consequence of the equality

\[ s\text{-}\lim_{t\to\infty}T_t^*T_t = s\text{-}\lim_{n\to\infty}Z^{*n}Z^n . \tag{9} \]

Let now a nonexpanding semigroup \(T_t\) be defined in some Hilbert space \(\mathfrak K\), and let \(U_t\) be some unitary dilation of it with range in the space \(\mathfrak H\).

Theorem 3. The space \(\mathfrak H\) is representable in the form

\[ \mathfrak H=\mathfrak D_+\oplus\mathfrak K\oplus\mathfrak D_-, \tag{10} \]

where \(\mathfrak D_\pm\) are subspaces of the space \(\mathfrak H\) possessing the property \(U_{\pm t}\mathfrak D_\pm\subset \mathfrak D_\pm,\ t>0\). In the case of minimality of the dilation \(U_t\) (and only then) the representation (10) is unique.

The construction of the representation (10) for the minimal dilation of a discrete group is in fact contained in \((^3)\), where

\[ \mathfrak D_\pm=\text{l.c.s. }\{(I-P_{\mathfrak K})\cdot U_{\pm t}\mathfrak K\}_{t>0}. \]

4. For a semigroup of nonexpanding operators \(T_t\) in a Hilbert space \(\mathfrak K\), denote by \(T\) the operator \(T_1\), if \(T_t\) is a discrete semigroup, and the Cayley transform \(T=(B+iI)(B-iI)^{-1}\) of the infinitesimal operator

\[ B=s\text{-}\lim_{t\downarrow0}\frac{1}{-it}(T_t-I), \]

if \(T_t\) is a continuous semigroup; by \(\mathfrak D_T\) and \(\mathfrak D_{T^*}\) denote the subspaces

\[ \mathfrak D_T=\overline{(I-T^*T)^{1/2}\mathfrak K},\qquad \mathfrak D_{T^*}=\overline{(I-TT^*)^{1/2}\mathfrak K}. \]

The characteristic function of a nonexpanding operator \(T\) (of a discrete semigroup of nonexpanding operators \(T_t\)) is the function whose values are operators acting from \(\mathfrak D_T\) to \(\mathfrak D_{T^*}\) according to the formula

\[ W_T(z)=-T+z(I-TT^*)^{1/2}(I-zT^*)^{-1}(I-T^*T)^{1/2}. \tag{11} \]

The characteristic function of a continuous semigroup \(T_t\) (of a dissipative operator \(B\)) is obtained from (11) by replacing the argument \(z=(\zeta+i)/(\zeta-i)\). When the operator \(B\) is unbounded, this definition of the characteristic function of a dissipative operator coincides with the generally accepted one \((^5)\).

Let \(U_t\) be a minimal unitary dilation of the semigroup \(T_t\) with exit into the space \(\mathfrak H=\mathfrak D_+\oplus\mathfrak K\oplus\mathfrak D_-\) and \(\mathfrak N_\pm=\mathfrak D_\pm\ominus U^{\pm1}\mathfrak D_\pm\). Then there exist isometric mappings \(V_\pm\) of the spaces \(\mathfrak D_T\) and \(\mathfrak D_{T^*}\) onto \(\mathfrak N_+\) and \(\mathfrak N_-\), respectively, \((3)\), such that for \(h\in\mathfrak K\)

\[ V_+(I-T^*T)^{1/2}h=(U-T)h,\qquad V_-(I-TT^*)^{1/2}h=(U^*-T^*)h. \]

Therefore, in particular, \(r_+=\dim\mathfrak N_+=\dim\mathfrak D_T\), \(r_-=\dim\mathfrak N_-=\dim\mathfrak D_{T^*}\). By Theorem 2, the scattering suboperators \(S_\pm(\lambda)\) for \(S_\pm=S_\pm(U^0,U)\) are the boundary values of operator functions \(S_\pm(\zeta)\) analytic in \(C_\pm\).

The following theorem was established by the authors already in \((1)\) under the particular assumption that \(T_t\) and \(T_t^*\) converge strongly to zero as \(t\to\infty\).

Theorem 4. The operator functions \(S_\pm(\zeta)\) are related to the characteristic functions \(W_T(\zeta)\) and \(W_{T^*}(\zeta)\) by the relations

\[ S_+(\zeta)=U^0V_-W_T(\zeta)V_+^{-1},\qquad \zeta\in C_+; \tag{12} \]

\[ S_-(\zeta)=U^{0-1}P_{\mathfrak N_+'}\,V_+W_{T^*}(\zeta)V_-^{-1},\qquad \zeta\in C_-. \]

The authors express their gratitude to Prof. M. G. Krein for his attention to the work.

Received
29 III 1965

References Cited

1. V. M. Adamyan, D. Z. Arov, DAN, 160, No. 1 (1965).
2. P. Lax, R. Phillips, Materials for the joint Soviet-American symposium (rotaprint), Novosibirsk, 1963.
3. B. Sz.-Nagy, C. Foiaș, Acta Sci. Math., 23, 106 (1962).
4. K. Hoffman, Banach Spaces of Analytic Functions, Ch. 7, IL, 1963.
5. A. V. Shtraus, Izv. AN SSSR, ser. matem., 24, No. 1, 43 (1960).