Full Text
E. R. Tsekanovskii
On Model Elements of Non-Self-Adjoint Operators
(Presented by Academician S. L. Sobolev on 16 X 1961)
1°. In the papers \((^1,^2)\) the question of reducing operators of class \((\Omega)\) to triangular form was studied. It is not difficult to see that an operator \(A\) of class \((\Omega)\) can be represented as the coupling of two operators \(A_1\) and \(A_2\), where \(A_1\) is a complete* operator, while \(A_2\) is an operator whose spectrum consists only of real points. To reduce the operator \(A_1\) to triangular form it is sufficient to construct an orthonormal basis \(\psi_1,\psi_2,\ldots\) by successive orthogonalization of eigen- and associated elements.
The triangular model of the operator \(A_2\) is the continual analogue of a triangular matrix. In this latter case, in a Hilbert space one cannot indicate a basis analogous to the basis \(\psi_1,\psi_2,\ldots\).
In the present paper generalized elements of a Hilbert space are introduced. In doing so, to each operator of the class under consideration there is put in correspondence a basis consisting of ordinary or generalized elements, with respect to which the operator has the form of a triangular model. From what was said above it is clear that it is sufficient to study operators with purely real spectrum.
2°. A bounded linear operator \(A\), acting in a Hilbert space \(H\), will be assigned to the class \(\Omega\) if it has the following properties:
1) the entire spectrum of the operator lies on the real axis;
2) the rank of non-Hermiticity of the operator is equal to \(r\) \((r<\infty)\);
3) \(A\) is a simple operator, i.e. \(H\) coincides with the closure of the linear span of all vectors of the form
\[
A^n f\left(n=0,1,\ldots,\ f\in \frac{A-A^*}{i}H\right).
\]
Let us extend \(H\) to some Hilbert space \(\widetilde H=H\oplus H_0\). An operator \(\widetilde A\), acting in \(\widetilde H\), is called a simple extension of the operator \(A\) \((^3)\) if the subspaces \(H\) and \(H_0\) are invariant for it and if \(\widetilde A\) generates in \(H\) the operator \(A\), and in \(H_0\) some Hermitian operator. Every operator satisfying conditions 1) and 2), but not satisfying condition 3), is a simple extension of an operator of class \(\Omega\). Let \(A\in\Omega\). Consider the Hilbert space \(L_2^{(r)}\), whose elements are matrices \(f(x)=\|f_1(x), f_2(x),\ldots,f_r(x)\|\), defined on \([0,l]\), where \(l\) is the sum of the moduli of the eigenvalues of the operator \(A\). We define the scalar product by the formula
\[
(f(x),g(x))=\int_0^l f(x)g^*(x)\,dx.
\]
Then, as is known \((^1)\), some simple extension \(\widetilde A\) of the operator \(A\) is unitarily equivalent to the operator acting in \(L_2^{(r)}\)
\[
\widetilde Bf(x)=\alpha(x)f(x)+i\int_x^l f(t)\pi(t)\,dt\,J\pi^*(x),
\]
* An operator is called complete if it possesses a complete system of eigen- and associated elements.
where \(\alpha(x)\) is a bounded nondecreasing function; \(\pi(x)=\|\pi_{ij}(x)\|\) is a square matrix of order \(r\); \(J=\|\delta_{ij}\|\) is a diagonal matrix, each element of whose main diagonal is equal either to \(+1\) or to \(-1\), i.e.
\[
\widetilde A=U^{-1}\widetilde B U.
\]
It is also known \({}^{(3)}\) that if \(A\in\Omega\), then there exists such a simple extension \(\widetilde A\) of it that
\[
\widetilde A=\int_0^l \alpha(\lambda)\,dE_\lambda
+i\int_0^l E_\lambda\,\frac{\widetilde A-\widetilde A^*}{i}\,dE_\lambda,
\tag{1}
\]
where \(E_\lambda\) is a continuous orthogonal resolution of the identity. Let us also note that if \(F_\lambda\) is the projection operator in \(L_2^{(r)}\) which assigns to the vector \(h(t)\) the vector
\[
F_\lambda h(t)=
\begin{cases}
h(t), & 0\leq t\leq \lambda,\\
0, & \lambda<t\leq l,
\end{cases}
\]
then, as shown in \({}^{(3)}\),
\[
E_\lambda=U^{-1}F_\lambda U.
\tag{2}
\]
\(3^\circ\). Denote by \(\{\Psi\}\) the set of vector-functions from \(L_2^{(r)}\) all of whose elements are infinitely differentiable functions on the interval \([0,l]\). Introduce in the set \(\{\Psi\}\) the system of norms
\[
\|f(x)\|_p=\max_{i,\ x\in[0,l]}
\{\,|f_i(x)|,\ |f_i'(x)|,\ldots,\ |f_i^{(p)}(x)|\,\}
\]
\[
(i=1,\ldots,r;\ p=1,2,\ldots).
\]
It is easy to see that the norms introduced are comparable and compatible \({}^{(4)}\).
The set \(\{\Psi\}\), with the norms specified in it, is a complete countably normed space \({}^{(4)}\). Let \(U\) be an isometric mapping of the space \(\widetilde H\) onto the Hilbert space \(L_2^{(r)}\), under which the operator \(\widetilde A\) passes into its model \(\widetilde B\).
Consider
\[
\{\Phi\}=U^{-1}\{\Psi\}.
\]
Introduce in the linear space \(\Phi\) a system of norms, setting
\[
\|f\|_p=\|f(x)\|_p\qquad (f(x)=Uf,\ p=1,2,\ldots).
\]
It is not difficult to see that \(\Phi\) is a complete countably normed perfect \({}^{(4)}\) space and that
\[
\Phi\subset\widetilde H\subset\Phi',
\]
where \(\Phi'\) is the space of linear continuous functionals defined on \(\Phi\). We shall call the space \(\Phi\) the space of basic elements, and its elements basic. We shall say that every linear continuous functional \(x(\varphi)\in\Phi'\) is generated by a generalized element \(x\), and we shall denote the value of the functional on the basic element \(\varphi\in\Phi\) by \((\varphi,x)\). Let \(x_1\) and \(x_2\) be generalized elements. By \(a_1x_1+a_2x_2\) we denote the generalized element generating the functional
\[
\overline{a_1}(\varphi,x_1)+\overline{a_2}(\varphi,x_2).
\]
The linear space of generalized elements obtained in this way will be identified with \(\Phi'\).
In what follows, by \(x_\lambda\) \((\lambda\in[0,l])\) we shall mean a generalized vector of order \(r\), all components of which are generalized elements generated by linear continuous functionals. Using the notion of generalized elements introduced, we shall show that a certain simple extension \(\widetilde A\) of the operator \(A\in\Omega\) has a system of generalized vectors \(x_\lambda\) \((\lambda\in[0,l])\) satisfying the conditions:
1) for every \(f\in\widetilde H\)
\[
f=\int_0^l f(\lambda)\,x_\lambda\,d\lambda;
\]
2) if
\[ \widetilde A f=\int_0^l g(\lambda)\,x_\lambda\,d\lambda\qquad (g=\widetilde A f), \]
then the coefficients \(f(\lambda)\) and \(g(\lambda)\) are connected with each other by the triangular model, i.e.
\[ g(\lambda)=\alpha(\lambda)f(\lambda)+i\int_\lambda^l f(t)\pi(t)\,dt\,J\pi^*(\lambda). \]
The generalized vectors \(x_\lambda\) \((\lambda\in[0,l])\) satisfying the conditions listed above will be called model vectors.
Theorem. Some simple extension \(\widetilde A\) of an operator \(A\in\Omega\) possesses a system of generalized model vectors \(x_\lambda\) \((\lambda\in[0,l])\), orthogonal and complete in the sense that for every element \(f\in\widetilde H\) the equalities
\[ f=\int_0^l f(\lambda)x_\lambda\,d\lambda,\qquad \|f\|^2=\int_0^l f(\lambda)f^*(\lambda)\,d\lambda \tag{3} \]
hold. The integral sums of the integral (3) converge to \(f\) strongly.
Proof. Consider the functional
\[ (\varphi,f_\lambda^{(i)})=(\varphi,E_\lambda e_i) \qquad (\varphi\in\Phi;\ i=1,2,\ldots,r), \]
where \(E_\lambda\) is the continuous orthogonal resolution of the identity corresponding to the operator \(A\); \(e_i(t)=Ue_i\); \(e_i(t)=\|0,\ldots,0,1,0,\ldots,0\|\), \(e_i\in\widetilde H\); \(i=1,2,\ldots,r\). Using equality (2), one can show that the functional \((\varphi,f_\lambda^{(i)})\) \((i=1,\ldots,r;\ \lambda\in[0,l])\) has strongly bounded variation \({}^{(4)}\), and therefore, by a known theorem \({}^{(4)}\), it is weakly differentiable with respect to the parameter \(\lambda\), i.e.
\[ (\varphi,x_\lambda^{(i)})=\frac{d}{d\lambda}(\varphi,E_\lambda e_i) \qquad (\varphi\in\Phi;\ i=1,\ldots,r;\ \lambda\in[0,l]). \]
Let us show that the functional \((\varphi,x_\lambda^{(i)})\) belongs to the space \(\Phi'\). Indeed,
\[ (\varphi,x_\lambda^{(i)}) =\frac{d}{d\lambda}(\varphi,E_\lambda e_i) =\frac{d}{d\lambda}(U^{-1}\varphi(t),\,U^{-1}F_\lambda Ue_i) = \]
\[ =\frac{d}{d\lambda}(\varphi(t),F_\lambda e_i(t)) =\frac{d}{d\lambda}\int_0^\lambda \varphi_i(t)\,dt =\varphi_i(\lambda). \tag{4} \]
Equality (4) proves our assertion.
Now let \(\varphi\in\Phi,\ f\in\widetilde H\). Then
\[ (\varphi,f)=\sum_{i=1}^r\int_0^l \varphi_i(\lambda)\,\overline{f_i(\lambda)}\,d\lambda. \]
In view of (4), the last equality may be rewritten in the form
\[ (\varphi,f) =\sum_{i=1}^r\int_0^l \overline{f_i(\lambda)}\,(\varphi,x_\lambda^{(i)})\,d\lambda =\sum_{i=0}^r\int_0^l(\varphi,f_i(\lambda)x_\lambda^{(i)})\,d\lambda = \]
\[ =\int_0^l\left(\varphi,\sum_{i=1}^r f_i(\lambda)x_\lambda^{(i)}\right)\,d\lambda =\int_0^l(\varphi,f(\lambda)x_\lambda)\,d\lambda. \tag{5} \]
By virtue of the perfection of the space \(\Phi\) (strong and weak convergence coincide), the integral sums converge strongly. Equalities (3) follow
now from (5). Suppose further that
\[ \widetilde{A}f=\int_0^l g(\lambda)x_\lambda\,d\lambda \qquad (g=\widetilde{A}f,\; g(\lambda)=Ug). \]
Then
\[ g(\lambda)=\frac{d}{d\lambda}(\widetilde{A}f,E_\lambda e), \]
where
\[ e=\begin{pmatrix} e_1\\ \vdots\\ e_r \end{pmatrix}. \]
Since \(\widetilde{A}=U^{-1}\widetilde{B}U\), we have
\[ \begin{aligned} g(\lambda) &=\frac{d}{d\lambda}(U^{-1}\widetilde{B}Uf,\;U^{-1}E_\lambda Ue) =\frac{d}{d\lambda}(\widetilde{B}f(t),E_\lambda e(t)) \\ &=\frac{d}{d\lambda}\left\{\int_0^\lambda a(t)f(t)\,dt +i\int_0^\lambda\left[\int_t^l f(\xi)\pi(\xi)\,d\xi\; J\pi^*(t)\right]dt\right\} \\ &=a(\lambda)f(\lambda)+i\int_\lambda^l f(t)\pi(t)\,dt\; J\pi^*(\lambda), \end{aligned} \]
where
\[ e(t)=Ue= \left\| \begin{array}{ccccc} 1&0&.&.&0\\ 0&1&.&.&0\\ .&.&.&.&.\\ 0&0&.&.&1 \end{array} \right\|. \]
The theorem is proved.
Kharkov Mining
Institute
Received
27 VII 1961
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