Doklady of the Academy of Sciences of the USSR
1967. Volume 173, No. 5

UDC 513.88:517.948.32

MATHEMATICS

G. E. KISELEVSKII

ON CYCLIC SUBSPACES OF DISSIPATIVE OPERATORS

(Presented by Academician I. N. Vekua, 10 VI 1966)

1. Let \(\mathfrak H\) be a separable Hilbert space and let \(A\) be a bounded linear operator acting in \(\mathfrak H\). If the operator \(A\) is unicellular \((^{1-5})\), then there exists for it a vector \(g \in \mathfrak H\) such that the linear span of the sequence \(g, Ag, A^2g,\ldots\) is dense in \(\mathfrak H\). In the present article, for a certain class of operators, the converse assertion is established.

2. We shall agree to assign an operator \(A\) to the class \(\Omega_0\) when the following conditions are fulfilled: 1) the entire spectrum of the operator \(A\) is concentrated at zero; 2) the operator \(A\) is dissipative, i.e. its imaginary component
   \[ A_I=\frac{1}{2i}(A-A^*) \]
   is nonnegative;

\[ \text{3) }\quad \sigma(A)=\overline{\lim_{\lambda\to0}\left(|\lambda|\,|\ln|\lambda||\,\|(A-\lambda E)^{-1}\|\right)}<\infty . \tag{1} \]

The class \(\Omega_0\) contains, in particular, all Volterra dissipative operators whose imaginary component is nuclear \((^{3,6})\).

If the subspace \(\mathfrak H_0\) is invariant with respect to an operator \(A \in \Omega_0\) and \(A_0=A|_{\mathfrak H_0}\) is the restriction of the operator \(A\) to the subspace \(\mathfrak H_0\), then \(A_0\) also belongs to \(\Omega_0\).

1. Let \(\mathfrak G\) be a separable Hilbert space with scalar product \((f,g)_{\mathfrak G}\) \((f,g\in\mathfrak G)\), and let \(B\) be a simple Volterra dissipative operator with one-dimensional imaginary component, defined in \(\mathfrak G\). Consider the Hilbert space \(\widetilde{\mathfrak G}\), consisting of sequences
   \[ \tilde f=\|f_1,f_2,\ldots\|\quad \left(f_k\in\mathfrak G,\ \sum_{k=1}^{\infty}(f_k,f_k)_{\mathfrak G}<\infty\right), \]
   in which the scalar product is defined by the equality
   \[ (\tilde f,\tilde g)_{\widetilde{\mathfrak G}} = \sum_{k=1}^{\infty}(f_k,g_k)_{\mathfrak G}, \]
   and define in \(\widetilde{\mathfrak G}\) the operator \(\widetilde B=\widetilde B_l\) by the formula
   \[ \widetilde B\tilde f=\|Bf_1,Bf_2,\ldots\|\qquad (l=\operatorname{sp} B_I). \tag{2} \]

Obviously, \(B\in\Omega_0\) and \(\sigma(\widetilde B)=2l\).

The operator \(\widetilde B\) has the following universal property*: every simple operator \(A\) of the class \(\Omega_0\), for which \(\sigma(A)\le 2l\), is unitarily equivalent to the restriction of the operator \(\widetilde B_l\) to one of its invariant subspaces.

1. Denote by \(\widetilde{\mathfrak G}_{\xi}\), where \(\xi=(\xi_1,\xi_2,\ldots)\) is an arbitrary orthonormal basis of the Hilbert space \(l^{(2)}\), the subspace in \(\widetilde{\mathfrak G}\) consisting of all vectors

* This assertion was originally established by M. S. Brodskii and the author \((^{7})\) (see also \((^{8})\)) for the case when the imaginary component of \(A\) is nuclear, and then generalized by L. E. Isaev.

of the form \(\|\xi_1 f,\xi_2 f,\ldots\|\) \((f\in \mathfrak G)\). The orthoprojector \(\widetilde P_\xi\) onto the subspace \(\widetilde{\mathfrak G}_\xi\) is defined by the formula

\[ \widetilde P_\xi \widetilde f=\|\xi_1 f,\xi_2 f,\ldots\| \quad \left(\widetilde f=\|f_1,f_2,\ldots\|,\ f=\sum_{k=1}^{\infty}\overline{\xi}_k f_k\right). \tag{3} \]

The orthogonal complement \(\widetilde{\mathfrak G}_{\xi}'=\widetilde{\mathfrak G}\ominus \widetilde{\mathfrak G}_{\xi}\), as is easy to see, is the set of vectors of the form

\[ \|\varphi_1,\varphi_2,\ldots\| \quad \left(\varphi_k\in\mathfrak G,\ \sum_{k=1}^{\infty}\overline{\xi}_k\varphi_k=0\right). \]

The following propositions are easily verified.

I. For any \(\xi\), the subspaces \(\widetilde{\mathfrak G}_{\xi}\) and \(\widetilde{\mathfrak G}_{\xi}'\) are invariant with respect to the operator \(\widetilde B\).

II. The operator \(\widetilde B_{\xi}=\widetilde B|\widetilde{\mathfrak G}_{\xi}\) is simple, Volterra, dissipative; its imaginary component is one-dimensional and its trace is equal to \(l\).

III. If \(\xi^{(1)},\xi^{(2)},\ldots\) is a complete orthonormal sequence in \(l^{(2)}\), then the decomposition

\[ \widetilde{\mathfrak G}=\widetilde{\mathfrak G}_{\xi^{(1)}}\oplus \widetilde{\mathfrak G}_{\xi^{(2)}}\oplus\cdots \tag{4} \]

is valid (see (7)).

Theorem 1. Let \(\widetilde{\mathfrak G}_0\) be a subspace invariant with respect to the operator \(\widetilde B\), and let \(\widetilde B_0=\widetilde B|\widetilde{\mathfrak G}_0\). If the operator \(\widetilde B_0\) is nonunivalent, then the subspace \(\widetilde{\mathfrak G}\) has a nonzero intersection with each of the subspaces \(\widetilde{\mathfrak G}_{\xi}'\).

1. Consider the Hilbert space \(\vec L=L^{(2)}(0,l)\) of vector-functions

\[ \mathbf f=\|f_1(x),f_2(x),\ldots\| \quad \left(f_k(x)\in L^{(2)}(0,l)\right) \]

with scalar product

\[ (\mathbf f,\mathbf g)=\sum_{k=1}^{\infty}\int_{0}^{l} f_k(x)\overline{g_k(x)}\,dx \tag{5} \]

and the integration operator acting in it

\[ I\mathbf f=\|If_1(x),If_2(x),\ldots\| \quad \left(If(x)=2i\int_{0}^{x} f(t)\,dt\right). \]

According to a theorem of M. S. Livshits \((^9)\), the operators \(\widetilde B_{\xi}^{(k)}=\widetilde B|\widetilde{\mathfrak G}_{\xi^{(k)}}\) \((k=1,2,\ldots)\) and \(I\) are unitarily equivalent. Therefore there exists an isometric mapping \(U_k\) of the subspace \(\widetilde{\mathfrak G}_{\xi^{(k)}}\) onto the subspace \(\vec L_k=\{\|0,\ldots,f_k(x),0,\ldots\|\}\) such that

\[ \widetilde B_{\xi^{(k)}}=U_k^{*}I_kU_k \quad \left(I_k=I|\vec L_k,\ k=1,2,\ldots\right), \]

\[ \widetilde B=U^{*}IU \quad \left(U=\sum_{k=1}^{\infty}U_k\widetilde P_{\xi^{(k)}}\right). \]

1. Let \(\mathfrak H_0\) be an invariant subspace of the operator \(A\). We shall agree to call the subspace \(\mathfrak H_0\) cyclic with respect to the operator \(A\) if there exists a vector \(g\in\mathfrak H_0\) such that the closure of the linear span of the sequence \(A^n g\) \((n=0,1,\ldots)\) coincides with \(\mathfrak H_0\). In this case we shall call the vector \(g\) a generating vector of the operator \(A\) in the subspace \(\mathfrak H_0\).

In the paper \((^2)\) it was proved that the function \(g(x)\) is generating for the operator \(I\) in the space \(L^{(2)}(0,l)\) if and only if the measure of the set of all \(x\in[0,\delta]\) for which \(g(x)\ne0\) is positive for every \(\delta>0\).

Lemma. Let \(\mathfrak H_1,\mathfrak H_2,\ldots\) be a sequence of invariant subspaces of a unicellular operator \(A\), and let \(g_k\) be a generating vector of the operator \(A\) in the subspace \(\mathfrak H_k\) \((k=1,2,\ldots)\), with

\[ \sum_{k=1}^{\infty}(g_k,g_k)<\infty . \]

Then there exists a sequence of numbers \(\xi_1,\xi_2,\ldots\)

\[ \left(\sum_{k=1}^{\infty}|\xi_k|^2=1\right), \]

such that the vector

\[ g=\sum_{k=1}^{\infty}\xi_k g_k \]

will be generating for the operator \(A\) in the subspace

\[ \mathfrak H=\bigcup_{k=1}^{\infty}\mathfrak H_k . \]

Theorem 2. Let \(A\) be a simple operator of class \(\Omega_0\), and let \(\mathfrak H_0\) be its invariant subspace. In order that the subspace \(\mathfrak H_0\) be cyclic with respect to the operator \(A\), it is necessary and sufficient that the restriction \(A_0=A|\mathfrak H_0\) be unicellular.

Proof. Sufficiency is obvious. Let us proceed to establish necessity.

Put \(l=\frac12\sigma(A_0)\). Using the proposition given in Sec. 3, we find an invariant subspace \(\widetilde{\mathfrak G}_0\) with respect to the operator \(\widetilde B=\widetilde B_l\) such that the operators \(A_0\) and \(\widetilde B_0=\widetilde B|\widetilde{\mathfrak G}_0\) are unitarily equivalent. By the necessity condition, some vector \(\widetilde g=\|g_1,g_2,\ldots\|\) is generating for the operator \(\widetilde B\) in the subspace \(\widetilde{\mathfrak G}_0\). Each of the subspaces \(\mathfrak G_k\), spanned respectively by the vectors \(g_k,Bg_k,Bg_k^2,\ldots\) \((k=1,2,\ldots)\), is invariant with respect to the operator \(B\). Since the operator \(B\) is unicellular \((^2)\), applying the lemma we choose a sequence of numbers

\[ \xi_1,\xi_2,\ldots\left(\sum_{k=1}^{\infty}|\xi_k|^2=1\right) \]

so that the vector

\[ g=\overline{\xi}_1g_1+\overline{\xi}_2g_2+\cdots \]

is generating for the operator \(B\) in the subspace

\[ \mathfrak G=\bigcup_{k=1}^{\infty}\mathfrak G_k . \]

We shall show that \(\mathfrak G_0=\mathfrak G\). Indeed, assuming the contrary, we obtain that \(\sigma(\widetilde B_0)<2l\), and this contradicts the definition of \(l\), in view of the unitary equivalence of the operators \(\widetilde B_0\) and \(A_0\).

Consider the decomposition (4) of the space \(\widetilde{\mathfrak G}\), where \(\xi^{(1)}=(\xi_1,\xi_2,\ldots)\), and \(\xi^{(2)},\xi^{(3)},\ldots\) are arbitrary unit vectors supplementing \(\xi^{(1)}\) to an orthonormal basis in \(l^{(2)}\), and let \(U\) be an isometric mapping of the space \(\widetilde{\mathfrak G}\) onto the space \(\widetilde L\), defined in Sec. 5. The vector \(U\widetilde g=\dot g=\|g_1(x),g_2(x),\ldots\|\) will evidently be generating for the operator \(I\) in the subspace \(\widetilde L_0=U\widetilde{\mathfrak G}_0\). Since, according to formulas (3) and (6),

\[ \widetilde g^{(1)}=\widetilde P_{\xi^{(1)}}\widetilde g=\|\xi_1g,\xi_2g,\ldots\| \]

and the vector \(g\) is generating for the operator \(B\) in \(\mathfrak G\), it follows that \(\widetilde g^{(1)}\) is a generating vector of the operator \(\widetilde B\) in \(\widetilde{\mathfrak G}_{\xi^{(1)}}\). Therefore the function \(g_1(x)\) will be generating for the operator \(I\) in the space \(L^{(2)}(0,l)\).

Denote by \(\mathfrak M\) the linear span of the sequence

\[ I^n g=\|I^n g_1(x),\, I^n g_2(x),\ldots\|\quad (n=0,1,\ldots). \]

Since

\[ I^n g(x)=\frac{(2i)^n}{(n-1)!}\int_0^x (x-t)^{n-1}g(t)\,dt =\frac{(2i)^n}{(n-1)!}\int_0^x t^{\,n-1}g(x-t)\,dt \quad (n=1,2,\ldots), \]

then \(\mathfrak M\) is the linear span of the vectors \(\mathbf g\) and \(\|p*g_1, p*g_2,\ldots\|\), where \(p*g\) denotes the convolution of the functions \(p(x)\) and \(g(x)\), while \(p(x)\) ranges over the set of all polynomials \((0\le x\le l)\). We shall show that the intersection of the subspaces \(\mathfrak M=\vec L_0\) and \(\vec L_1^{\,\prime}=\vec L\ominus \vec L_1=\{\|0,f_2(x),f_3(x),\ldots\|\}\) consists of the zero vector. To this end suppose that

\[ (\lambda_n g_1+p_n*g_1)\to 0,\qquad (\lambda_n g_k+p_n*g_k)\to h_k\quad (k=2,3,\ldots) \tag{7} \]

in the sense of convergence in \(L^{(2)}(0,l)\). Using the known properties of convolution \({}^{(10)}\) and the equalities (7), we obtain \(g_1*h_k=0\) \((k=2,3,\ldots)\). Taking into account that the function \(g_1(x)\) is generating for the integration operator in \(L^{(2)}(0,l)\), and applying Titchmarsh’s theorem \({}^{(11)}\), we arrive at the conclusion that \(h_k=0\) \((k=2,3,\ldots)\). Thus, by Theorem 1, the operator \(I_0\) is unicellular and, consequently, the operator \(A_0\) is also unicellular.

Let us note that without the requirement of dissipativity the necessity in Theorem 2 is, generally speaking, false. Indeed, the operator \(\mathbf J=\|I f_1(x),-I f_2(x)\|\), acting in the space \(\vec L=\{\|f_1(x),f_2(x)\|\}\) \((f_k(x)\in L^{(2)}(0,l))\), is not unicellular. However, if the function \(g(x)\) is generating for the operator \(I\) in \(L^{(2)}(0,l)\), then the vector-function \(\mathbf g=\|g(x),g(x)\|\) will be generating for the operator \(\mathbf J\) in \(\vec L\).

Zhitomir Pedagogical Institute
named after I. Franko

Received
8 VI 1966

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