Reports of the Academy of Sciences of the USSR
1961, Volume 138, No. 3

MATHEMATICS

M. S. BRODSKII

A CRITERION FOR THE UNICELLULARITY OF VOLTERRA OPERATORS

(Presented by Academician V. I. Smirnov, January 9, 1961)

Let \(A\) be a Volterra operator acting in a separable Hilbert space \(\mathfrak H\). If \(\mathfrak H_1\) and \(\mathfrak H_2\) are invariant subspaces of it \((\mathfrak H_1 \subset \mathfrak H_2,\ \dim\{\mathfrak H_2 \ominus \mathfrak H_1\} > 1)\), then, by the theorem of Aronszajn and Smith \((^1)\), there exists an invariant subspace \(\mathfrak H_3\) such that \(\mathfrak H_1 \subset \mathfrak H_3 \subset \mathfrak H_2\). The operator \(A\) is called unicellular* \((^2)\) if, of any two invariant subspaces of it, one belongs to the other. In the present paper a necessary and sufficient condition is established for the unicellularity of an operator \(A\), which is satisfied by its characteristic operator-function.

1. Generalizing a definition of M. S. Livšic \((^3)\), represent the imaginary part of the operator \(A\) in the form
   \[ \frac{A-A^*}{2i}=RJR^*, \]
   where \(R\) is a completely continuous mapping of some Hilbert space \(\mathfrak H_W\) into \(\mathfrak H\), and \(J\) is an operator acting in \(\mathfrak H_W\) and satisfying the conditions \(J=J^*,\ J^2=E\). The operator-function
   \[ W(\lambda)=E-2iR_* (A-\lambda E)^{-1}RJ \tag{1} \]
   is called the characteristic operator-function for the operator \(A\). If \(\mathfrak H_0\) is some subspace in \(\mathfrak H\), then the operator-function
   \[ W_0(\lambda)=E-2iR_0^*(A_{P_0}-\lambda E)^{-1}R_0J \quad (A_{P_0}f=P_0Af\ (f\in\mathfrak H_0),\ R_0=P_0R), \tag{2} \]
   where \(P_0\) is the projection operator onto \(\mathfrak H_0\), is called the projection of \(W(\lambda)\) onto \(\mathfrak H_0\) and is denoted by the symbol \(\operatorname{Pr}_{\mathfrak H_0} W(\lambda)\).

From the general theory of characteristic functions \((^{4,5})\) the following assertion follows:

Lemma 1. If in the space \(\mathfrak H\) there exist two distinct subspaces \(\mathfrak H_1\) and \(\mathfrak H_2\), invariant with respect to \(A\), such that
\[ \operatorname{Pr}_{\mathfrak H_1} W(\lambda)=\operatorname{Pr}_{\mathfrak H_2} W(\lambda), \]
then the operator \(A\) commutes with some nonscalar unitary operator.

We shall also need:

Lemma 2. If the operator \(A\) commutes with some nonscalar unitary operator \(U\), then
\[ \mathfrak H=\mathfrak H^{(1)}\oplus \mathfrak H^{(2)} \quad (\mathfrak H^{(k)}\ne 0,\ k=1,2), \]
where \(\mathfrak H^{(1)}\) and \(\mathfrak H^{(2)}\) are invariant with respect to \(A\).

Proof. Denote by \(\mathfrak H_0\) the totality of all eigenvectors of the operator
\[ \frac{A-A^*}{2i}, \]
corresponding to some nonzero eigenvalue. Since \(\mathfrak H_0\) is finite-dimensional and invariant with respect to \(U\), there exists in \(\mathfrak H_0\) an eigenvector \(f\) of the operator \(U\). Putting \(Uf=tf\), denote by \(\mathfrak H^{(1)}\) the subspace consisting of all

* The operator \(A\) is called Volterra if it is completely continuous and has no nonzero points of the spectrum.

vectors \(g\) for which \(Ug = \tau g\). It is easy to see that \(\mathfrak H^{(1)}\) and \(\mathfrak H^{(2)}=\mathfrak H\ominus \mathfrak H^{(1)}\) are invariant with respect to \(A\).

1. The operator-function \(W(\lambda)\) has the following properties:

I. The function \(W(\lambda)\) expands into a norm-convergent series

\[ W(\lambda)=E+\frac{1}{\lambda}W_1+\frac{1}{\lambda^2}W_2+\cdots \quad (\lambda\ne 0), \]

where \(W_k\) \((k=1,2,\ldots)\) are completely continuous operators.

II. \(W^*(\lambda)JW(\lambda)-J\ge 0,\ \operatorname{Im}\lambda>0;\quad W^*(\lambda)JW(\lambda)-J=0,\ \operatorname{Im}\lambda=0,\ \lambda\ne 0.\)

We shall denote by \((\Omega_J)\) the totality of all operator-functions satisfying conditions I and II. It can be shown that each function of this class is characteristic for some Volterra operator.

Let \(W_1(\lambda)\) and \(W_2(\lambda)\) belong to the class \((\Omega_J)\). The notation \(W_2(\lambda)<W_1(\lambda)\) will mean that there exists a function \(W_3(\lambda)\in(\Omega_J)\) such that \(W_1(\lambda)=W_2(\lambda)W_3(\lambda)\). In this case we shall say that \(W_2(\lambda)\) is a divisor of the function \(W_1(\lambda)\). If the subspace \(\mathfrak H_0\) is invariant with respect to \(A\), then

\[ W(\lambda)=\operatorname{Pr}_{\mathfrak H_0}W(\lambda)\operatorname{Pr}_{\mathfrak H\ominus\mathfrak H_0}W(\lambda) \tag{6} \]

where the functions \(\operatorname{Pr}_{\mathfrak H_0}W(\lambda)\) and \(\operatorname{Pr}_{\mathfrak H\ominus\mathfrak H_0}W(\lambda)\) are characteristic, respectively, for the Volterra operators \(P_0Af\) \((f\in\mathfrak H_0)\) and \((E-P_0)Af\) \((f\in\mathfrak H\ominus\mathfrak H_0)\), and therefore belong to the class \((\Omega_J)\). Thus to every invariant subspace of the operator \(A\) there corresponds a divisor of its characteristic operator-function. It is easy to verify that the converse assertion is false.

An operator-function of the form

\[ T(\lambda)=E+\frac{2i\sigma}{\lambda}PJ \quad (\sigma>0,\ PJP=0), \]

where \(P\) is the operator of projection onto a one-dimensional subspace in \(\mathfrak H_W\), will be called elementary. Clearly, all elementary functions belong to the class \((\Omega_J)\). Two elementary functions

\[ T_1=E+\frac{2i\sigma_1}{\lambda}P_1J,\qquad T_2=E+\frac{2i\sigma_2}{\lambda}P_2J \]

we shall agree to call similar if \(P_1=P_2\).

Let \(W(\lambda)=W_1(\lambda)W_2(\lambda)\) \((W_k(\lambda)\in(\Omega_J))\). We shall say that \(W_1(\lambda)\) is a proper divisor of the function \(W(\lambda)\) if \(W_1(\lambda)\) and \(W_2(\lambda)\) cannot be represented in the form

\[ W_1(\lambda)=W_{11}(\lambda)W_{22}(\lambda),\qquad W_2(\lambda)=W_{21}(\lambda)W_{22}(\lambda)\quad (W_{ij}(\lambda)\in(\Omega_J)), \]

where \(W_{12}(\lambda)\) and \(W_{21}(\lambda)\) are similar elementary functions.

Lemma 3. If the set of vectors of the form \(A^nRf\) \((f\in\mathfrak H_W,\ n=0,1,2,\ldots)\) is complete in \(\mathfrak H\), then a divisor \(W_1(\lambda)\) of the function \(W(\lambda)\) coincides with the projection of this function onto some invariant subspace of the operator \(A\) if and only if it is proper.

Lemma 4. If \(W_1(\lambda)\) is an improper divisor of the function \(W(\lambda)\), then there exist proper divisors \(W_2(\lambda)\) and \(W_3(\lambda)\) such that \(W_2(\lambda)<W_1(\lambda)<W_3(\lambda)\) and \(W_3(\lambda)=W_2(\lambda)W_4(\lambda)\), where \(W_4(\lambda)\) is an elementary function.

1. A characteristic operator-function \(W(\lambda)\) will be called ordered if, for any two of its divisors, one is a divisor of the other.

Theorem. A simple Volterra operator \(A\) if and only if

* A Volterra operator \(A\) is called simple if in \(\mathfrak H\) there is no subspace annihilated by the operators \(A\) and \(A^*\). If \(A\) is a simple Volterra operator, and \(P_0\) is the projector onto its invariant subspace \(\mathfrak H_0\), then the set of vectors of the form \(A^nRf\) \((f\in\mathfrak H_W,\ n=0,1,2,\ldots)\) is complete in \(\mathfrak H\), and the set of vectors of the form \(A^nP_0Rf\) \((f\in\mathfrak H_W,\ n=0,1,2,\ldots)\) is complete in \(\mathfrak H_0\).

case it is unicellular when its characteristic operator-function \(W(\lambda)\) is ordered.

Proof. Let \(W(\lambda)\) be an ordered function. If the operator \(A\) is not unicellular, then there exist subspaces \(\mathfrak H_1\) and \(\mathfrak H_2\), invariant with respect to \(A\), such that neither of them belongs to the other. The projections \(W_1(\lambda)\) and \(W'_1(\lambda)\) of the function \(W(\lambda)\) onto these subspaces are comparable with one another. Suppose, for example, \(W_1(\lambda) < W'_1(\lambda)\). From Lemma 3 and the relations

\[ W'_1(\lambda)=W_1(\lambda)W_2(\lambda),\quad W(\lambda)=W_1(\lambda)W_2(\lambda)W_3(\lambda)\quad \bigl(W_2(\lambda),W_3(\lambda)\in(\Omega_J)\bigr), \]

it follows that \(W_1(\lambda)\) is a proper divisor of the function \(W'_1(\lambda)\). Applying Lemma 3 again, we find an \(A\)-invariant subspace \(\mathfrak H_2\subset \mathfrak H_1\) such that the projection of \(W(\lambda)\) onto \(\mathfrak H_2\) will be equal to \(W_1(\lambda)\). Since \(\mathfrak H_2\ne\mathfrak H_1\), by Lemmas 1 and 2 the space \(\mathfrak H\) is representable in the form \(\mathfrak H=\mathfrak H^{(1)}\oplus\mathfrak H^{(2)}\) \((\mathfrak H^{(k)}\ne0,\ k=1,2)\), where \(\mathfrak H^{(1)}\) and \(\mathfrak H^{(2)}\) are invariant with respect to \(A\).

Introduce the projection operator \(P^{(k)}\) \((k=1,2)\) onto the subspace \(\mathfrak H^{(k)}\), and consider the projections

\[ W^{(k)}(\lambda)=E-2iR^*P^{(k)}(A_{P^{(k)}}-\lambda E)^{-1}P^{(k)}RJ \quad (k=1,2). \]

One of these projections, for instance \(W^{(1)}(\lambda)\), must be a divisor of the other,

\[ W^{(2)}(\lambda)=W^{(1)}(\lambda)W^{(3)}(\lambda) \quad \bigl(W^{(3)}(\lambda)\in(\Omega_J)\bigr). \]

Expanding both sides of the last equality in a series in a neighborhood of the point at infinity and comparing the coefficients of \(1/\lambda\), we obtain

\[ P^*P^{(2)}R=R^*P^{(1)}R+H\quad (H\ge 0). \tag{3} \]

Neither of the subspaces \(\mathfrak H^{(1)}\) and \(\mathfrak H^{(2)}\) can be annihilated by the operator \(\dfrac{A-A^*}{2i}\), since otherwise the operator \(A\) would not be simple. Consequently, in the subspace \(\mathfrak H^{(1)}\) there is an eigenvector \(e\) \((\|e\|=1)\) of the operator \(\dfrac{A-A^*}{2i}\), corresponding to a nonzero eigenvalue \(\omega\). Thus

\[ (RJHJR^*e,e)= \left(\frac{A-A^*}{2i}(P^{(2)}-P^{(1)})\frac{A-A^*}{2i}e,e\right) =-\omega^2, \]

which contradicts equality (3).

Now let \(A\) be a unicellular operator. Then the comparability of two proper divisors of the function \(W(\lambda)\) is obvious. The comparability of divisors in the remaining cases follows easily from Lemma 4.

Odessa State Pedagogical Institute
named after K. D. Ushinsky

Received
9 I 1961

CITED LITERATURE

1. N. Aronszajn, K. T. Smith, Collected Translations: Mathematics, 2, No. 1, 97 (1958).
2. M. S. Brodskii, DAN, 111, No. 5, 926 (1956).
3. M. S. Livshits, Mat. sborn., 34 (76), 1, 145 (1954).
4. M. S. Brodskii, Mat. sborn., 39 (81), 2, 179 (1956).
5. M. S. Brodskii, M. S. Livshits, UMN, 13, issue 1 (79), 3 (1958).
6. M. S. Brodskii, DAN, 97, No. 5, 761 (1954).