MATHEMATICS

M. S. BRODSKII

ON THE UNICELLULARITY OF REAL VOLTERRA OPERATORS

(Presented by Academician V. I. Smirnov, 1 VI 1962)

Let an involution \(S\) be given in a separable Hilbert space \(\mathfrak H\). A subspace \(\mathfrak H_0 \subset \mathfrak H\) invariant with respect to \(S\) is called \(S\)-real. If \(A\) is a certain bounded linear operator in \(\mathfrak H\) and \(AS = SA\), then the operator \(A\) is called \(S\)-real.

Let the \(S\)-real operator \(A\) be a Volterra operator, i.e. completely continuous and having no spectral points distinct from zero. It is called unicellular \((^1)\) if the set of all its invariant subspaces is ordered by inclusion, and \(S\)-unicellular if the set of its \(S\)-real invariant subspaces is ordered by inclusion. In the finite-dimensional case the notions of unicellularity and \(S\)-unicellularity are equivalent.

In the present article an analytic criterion for \(S\)-unicellularity is established for \(S\)-real Volterra operators. From it, in particular, it follows that in an infinite-dimensional space there exist non-unicellular \(S\)-real Volterra operators which, however, are \(S\)-unicellular.

1. Let a bounded linear operator \(A\) be given in the space \(\mathfrak H\), with completely continuous imaginary component

\[ \frac{A-A^*}{2i}. \]

There exists a completely continuous linear mapping \(R\) of some Hilbert space \(\mathfrak H_W\) into the space \(\mathfrak H\), and an operator \(J\) \((J=J^*, J^2=E)\) acting in \(\mathfrak H_W\), such that

\[ \frac{A-A^*}{2i}=RJR^* \]

and the set of vectors of the form \(A^nRf\) \((n=0,1,\ldots;\ f\in\mathfrak H_W)\) is complete in \(\mathfrak H\). The operator-function

\[ W(\lambda)=E-2iR^*(A-\lambda E)^{-1}RJ \tag{1} \]

is called the characteristic \((^{2-4})\) function of the operator \(A\). The characteristic function (1) will be called \(T\)-real if in the space \(\mathfrak H_W\) there exists an involution \(T\) for which \(TW(\bar\lambda)T=W(\lambda)\) and \(TJT=-J\). It is easy to show that every \(S\)-real operator \(A\) has such a \(T\)-real characteristic function \(W(\lambda)\), and that the involutions \(S\) and \(T\) are connected by the relation \(SR=RT\). The converse assertion is also true:

Theorem 1. If the operator \(A\) has a \(T\)-real characteristic function (1), then in the space \(\mathfrak H\) there exists an involution \(S\) such that \(AS=SA\) and \(SR=RT\).

Proof. From the definition of the characteristic function it follows that

\[ W^*(\mu)JW(\lambda)-J = 4\,\frac{\lambda-\bar\mu}{2}\, JR^*(A^*-\bar\mu E)^{-1}(A-\lambda E)^{-1}RJ. \]

Consequently,

\[ TR^*(A^*-\bar\mu E)^{-1}(A-\lambda E)^{-1}RT = R^*(A^*-\bar\mu E)^{-1}(A-\bar\lambda E)^{-1}R, \]

\[ \bigl(A^nRTf, A^mRTg\bigr) = \overline{\bigl(A^nRf, A^mRg\bigr)} \quad (n,m=0,1,\ldots;\ f,g\in\mathfrak H_W). \]

Define the required involution \(S\) on the dense set of vectors in \(\mathfrak H\) by setting

\[ S\left(\sum_{k=0}^{n} A^k R g_k\right) = \sum_{k=0}^{n} A^k R T g_k \qquad (n=0,1,\ldots;\ g_k\in \mathfrak H_W). \]

1. Consider an \(S\)-real operator \(A\) and its \(T\)-real characteristic function (1), satisfying the condition \(SR=RT\). To each subspace \(\mathfrak H_0\subset \mathfrak H\) invariant with respect to \(A\) there corresponds the operator-function

\[ W_0(\lambda)=E-2iR_0^{*}(A_0-\lambda E)^{-1}R_0J \qquad (R_0=P_0R), \tag{2} \]

where \(P_0\) is the projection operator onto \(\mathfrak H_0\) and \(A_0f=P_0Af\) \((f\in\mathfrak H_0)\). The function \(W_0(\lambda)\) is characteristic for the operator \(A_0\) and is called the projection of the function \(W(\lambda)\) onto the subspace \(\mathfrak H_0\).

Theorem 2. For an invariant subspace \(\mathfrak H_0\) with respect to \(A\) to be \(S\)-real, it is necessary and sufficient that the projection (2) satisfy the condition

\[ TW_0(\bar\lambda)T=W_0(\lambda). \]

1. The characteristic function \(W(\lambda)\) of a Volterra operator has the following properties:
   1) \(W(\lambda)-E\) is an entire function of \(\dfrac{1}{\lambda}\), taking completely continuous values and satisfying the condition \(W(\infty)=E\);
   2)

\[ W^{*}(\lambda)JW(\lambda)-J\ge 0\quad (\operatorname{Im}\lambda>0), \qquad W^{*}(\lambda)JW(\lambda)-J=0\quad (\operatorname{Im}\lambda=0). \]

Denote by \(\Omega_J\) the set of all operator-functions possessing properties 1) and 2). If

\[ W_1(\lambda)=W_2(\lambda)W_3(\lambda) \qquad (W_k(\lambda)\in\Omega_J), \]

then we shall say that \(W_2(\lambda)\) is a left divisor of the function \(W_1(\lambda)\), and that the function \(W_2(\lambda)\) precedes the function \(W_1(\lambda)\).

Consider an \(S\)-real Volterra operator \(A\) and its \(T\)-real characteristic function \(W(\lambda)\). If \(SR=RT\), then every projection of the function \(W(\lambda)\) onto an \(S\)-real invariant subspace of the operator \(A\) is a \(T\)-real left divisor of the function \(W(\lambda)\). The converse assertion is false in general.

Theorem 3. Let \(A\) be a Volterra \(S\)-real operator, let \(W(\lambda)\) be its \(T\)-real characteristic operator-function, and let \(SR=RT\). For the operator \(A\) to be \(S\)-unicellular, it is necessary and sufficient that the set of all left \(T\)-real divisors of the function \(W(\lambda)\) be ordered.

1. Consider an \(S\)-real simple (1) Volterra operator \(A\) with a finite-dimensional imaginary component. Since the spectrum of this component is symmetric with respect to the origin, in the space

\[ \frac{A-A^{*}}{2i}\,\mathfrak H \]

there exists an orthonormal basis

\[ \{e_1^{+},\ldots,e_n^{+},e_1^{-},\ldots,e_n^{-}\} \]

such that

\[ Se_\alpha^{+}=e_\alpha^{-}, \qquad \frac{A-A^{*}}{2i}\,e_\alpha^{+}=\omega_\alpha e_\alpha^{+}\ (\omega_\alpha>0), \qquad \frac{A-A^{*}}{2i}\,e_\alpha^{-}=-\omega_\alpha e_\alpha^{-}. \]

In an arbitrary \(2n\)-dimensional unitary space \(\mathfrak H_W\), choose an orthonormal basis

\[ \{h_1^{(1)},\ldots,h_n^{(1)},h_1^{(2)},\ldots,h_n^{(2)}\} \]

and define linear operators \(J\) and \(R\) by setting

\[ Jh_\alpha^{(1)}=h_\alpha^{(1)},\qquad Jh_\alpha^{(2)}=-h_\alpha^{(2)},\qquad Rh_\alpha^{(1)}=\omega_\alpha^{1/2}e_\alpha^{+},\qquad Rh_\alpha^{(2)}=\omega_\alpha^{1/2}e_\alpha^{-}. \]

In addition, define an involution \(T\) in the space \(\mathfrak H_W\), specifying it by the equalities \(Th_\alpha^{(1)}=h_\alpha^{(2)}\). Then \(SR=RT\), and

\[ W(\lambda)=E-2iR^{*}(A-\lambda E)^{-1}RJ \]

is a \(T\)-real characteristic operator-function of the operator \(A\). Consider the matrix

\[ V(\lambda)= \left\| \begin{matrix} v_{11}(\lambda) & v_{12}(\lambda)\\ v_{21}(\lambda) & v_{22}(\lambda) \end{matrix} \right\| \qquad \left( v_{ij}(\lambda)= \left\| \bigl(W(\lambda)h_\alpha^{(i)},h_\beta^{(j)}\bigr) \right\|_{\alpha,\beta=1}^{n} \right). \tag{3} \]

The condition \(T W(\bar\lambda)T = W(\lambda)\) is equivalent to the condition

\[ v_{11}(\lambda)=\overline{v_{22}(\bar\lambda)},\qquad v_{12}(\lambda)=\overline{v_{21}(\bar\lambda)}. \tag{4} \]

By virtue of Theorem 3, the operator \(A\) is then and only then \(S\)-unicellular when the set of all right divisors of the matrix function \(V(\lambda)\) satisfying conditions of the form (4) is ordered.

1. In the space \(\mathfrak L_2^{(2)}[0,1]\), whose elements are vector functions
   \(f(x)=\|f_1(x)\ f_2(x)\|\) \((0\le x\le 1)\) and in which the scalar product is defined by the equality
   \[ (f(x),g(x))=\int_0^1 f(t)g^*(t)\,dt, \]
   let us define a simple Volterra operator*
   \[ A\bigl(\|f_1(x)\ f_2(x)\|\bigr) = \int_x^1 \|f_1(t)\ f_2(t)\| \left\| \begin{matrix} 0&-1\\ 1&0 \end{matrix} \right\|\,dt. \tag{5} \]

It is obvious that the operator \(A\) commutes with the involution
\[ S\bigl(\|f_1(x)\ f_2(x)\|\bigr) = \|\overline{f_1(x)}\ \overline{f_2(x)}\|. \]
It is not unicellular, since the sets of vectors of the form
\[ \|f_1(x)\ i f_1(x)\| \quad \text{and} \quad \|f_1(x)\ - i f_1(x)\| \]
are invariant, mutually orthogonal subspaces with respect to \(A\). A direct calculation shows that the imaginary component of the operator \(A\) is two-dimensional and that the matrix (3) has the form
\[ V(\lambda)= \left\| \begin{matrix} e^{i/\lambda}&0\\ 0&e^{-i/\lambda} \end{matrix} \right\|. \]
Let
\[ V(\lambda)=V_1(\lambda)V_2(\lambda), \]
where
\[ V_2(\lambda)= \left\| \begin{matrix} a(\lambda)&b(\lambda)\\ \overline{b(\bar\lambda)}&\overline{a(\bar\lambda)} \end{matrix} \right\| \]
is a right divisor of the matrix \(V(\lambda)\) satisfying conditions (4). From the equalities
\[ V_k(\lambda)JV_k^*(\lambda)-J\ge 0 \qquad \left( J= \left\| \begin{matrix} 1&0\\ 0&-1 \end{matrix} \right\|,\ \operatorname{Im}\lambda\ge 0,\ k=1,2 \right) \]
it follows easily that
\[ a(\lambda)\overline{a(\bar\lambda)} - b(\lambda)\overline{b(\bar\lambda)} \equiv 1, \qquad a(\lambda)=1+\frac{i\mu}{\lambda}+\cdots\quad(\mu>0) \tag{6} \]
and that the function \(a(\lambda)\) has no zeros. Consequently,
\[ a(\lambda)=e^{g(\lambda)}, \]
where \(g(\lambda)\) is an entire function of \(1/\lambda\). Moreover, by virtue of the multiplicative representation
\[ V_2(\lambda)=\int_0^1 e^{\frac{t}{\lambda}\,dE(t)J}\quad (2) \]
there is the estimate
\[ |e^{g(\lambda)}|\le e^{\sigma/\lambda}\qquad(\sigma>0), \]
and therefore
\[ g(\lambda)=\alpha+\beta/\lambda. \]
Using formulas (6), we obtain:
\[ \alpha=0,\qquad \beta=i\mu,\qquad a(\lambda)=e^{i\mu/\lambda},\qquad b(\lambda)=0. \]
Thus,
\[ V_2(\lambda)= \left\| \begin{matrix} e^{i\mu/\lambda}&0\\ 0&e^{-i\mu/\lambda} \end{matrix} \right\| \qquad(0<\mu<1). \]
This proves the orderedness of the set of all right divisors of the matrix function \(V(\lambda)\) satisfying conditions (4), i.e., the \(S\)-unicellularity of the operator (5) is proved.

Odessa State Pedagogical Institute
named after K. D. Ushinsky

Received
29 V 1962

REFERENCES

1. M. S. Brodskii, DAN, 138, No. 3, 512 (1961).
2. M. S. Livshits, Matem. sborn., 34 (76), 1, 145 (1954).
3. M. S. Brodskii, Matem. sborn., 39(81), 2, 179 (1956).
4. M. S. Brodskii, M. S. Livshits, UMN, 13, issue 1 (79), 3 (1958).

* Several years ago M. G. Krein expressed the hypothesis of the \(S\)-unicellularity of all \(S\)-real simple Volterra operators with two-dimensional imaginary component. Having learned of the criterion of \(S\)-unicellularity formulated in this article, M. G. Krein recommended that the author apply it to example (5).