MATHEMATICS

M. A. GOLDMAN, S. N. KRACHKOVSKII

ON SOME PERTURBATIONS OF A CLOSED LINEAR OPERATOR

(Presented by Academician V. I. Smirnov on 20 IV 1964)

Consider a vector space \(\mathfrak X\) and a linear operator \(T\) acting in it. Let \(\mathfrak D_T\), \(\mathfrak R_T\), and \(\mathfrak Z_T\) denote, respectively, the domain, the range, and the set of all zeros of the operator \(T\). Introduce the linear sets
\[ \mathfrak M_T=\bigcup_{n=1}^{\infty}\mathfrak R_{T^n} \quad\text{and}\quad \mathfrak N_T=\bigcup_{n=1}^{\infty}\mathfrak Z_{T^n}. \]
Obviously, \(\mathfrak M_T\subset \mathfrak D_T\) and
\[ T(\mathfrak D_T\cap \mathfrak M_T)\subset \mathfrak M_T,\qquad T(\mathfrak N_T)\subset \mathfrak N_T. \]
Put
\[ \mathfrak Z_0=\mathfrak Z_T,\qquad \mathfrak Z_n=\mathfrak Z_0\cap \mathfrak R_{T^n} \]
\[ (n=1,2,\ldots),\qquad \mathfrak Z_\omega=\mathfrak Z_0\cap \mathfrak M_T\left(=\bigcap_{n=1}^{\infty}\mathfrak Z_n\right). \]

Theorem 1. If, beginning with some number \(m\), all \(\mathfrak Z_n\) coincide, then
\[ T(\mathfrak D_T\cap \mathfrak M_T)=\mathfrak M_T. \]

Proof. Take an arbitrary element \(y\) of \(\mathfrak M_T\). Then \(y\in\mathfrak R_{T^{n+1}}\) for every \(n\). Denote by \(y_n\) some preimage of the element \(y\) contained in \(\mathfrak R_{T^n}\) \((n=1,2,\ldots)\). This means that \(y_n\in\mathfrak D_T\cap\mathfrak R_{T^n}\) and \(Ty_n=y\). The difference
\[ z_{mk}=y_m-y_{m+k} \]
is contained in \(\mathfrak R_{T^m}\), and \(Tz_{mk}=0\) \((k=1,2,\ldots)\). Consequently, \(z_{mk}\in\mathfrak Z_m\). But \(\mathfrak Z_m=\mathfrak Z_{m+1}=\cdots=\mathfrak Z_\omega\), whence it is clear that \(z_{mk}\in\mathfrak Z_{m+k}\). Thus
\[ y_m=z_{mk}+y_{m+k}\in\mathfrak R_{m+k}. \]
Since this is true for every \(k\) and \(y_m\in\mathfrak D_T\), we have \(y_m\in\mathfrak D_T\cap\mathfrak M_T\). Hence (since \(Ty_m=y\)) we conclude that
\[ T(\mathfrak D_T\cap\mathfrak M_T)=\mathfrak M_T. \]

Let \(\alpha_T\) be the dimension of \(\mathfrak Z_T\), and \(\beta_T\) the dimension of the space complementary to \(\mathfrak R_T\).

Theorem 2. If at least one of the numbers \(\alpha_T\) or \(\beta_T\) is finite, then, beginning with some number, all \(\mathfrak Z_n\) are identical.

Proof. If \(\alpha_T\) is finite, then the assertion is obvious (for the \(\mathfrak Z_n\) do not increase and are contained in \(\mathfrak Z_T\)).

Consider the case where \(\beta_T\) is finite. We shall prove the theorem by contradiction, i.e., suppose that there exists an increasing sequence
\[ k_1,\ k_2,\ldots \]
such that
\[ \mathfrak Z_0=\mathfrak Z_1=\cdots=\mathfrak Z_{k_1}\ne\mathfrak Z_{k_1+1} =\mathfrak Z_{k_1+2}=\cdots=\mathfrak Z_{k_2}\ne\mathfrak Z_{k_2+1}=\cdots. \]
Take
\[ z_i\in\mathfrak Z_{k_i},\qquad z_i\notin\mathfrak Z_{k_i+1}\quad (i=1,2,\ldots). \]
Then \(z_i\in\mathfrak R_{k_i}\), \(z_i\notin\mathfrak R_{k_i+1}\). To each \(z_i\) assign an \(x_i\) such that
\[ T^{k_i}x_i=z_i. \]
The sequence \(x_1,x_2,\ldots\) is linearly independent. Indeed, let
\[ x=\sum_{i=1}^{r}\alpha_i x_i=0. \]
Then
\[ T^{k_r}x=\alpha_rT^{k_r}x_r=\alpha_r z_r=0, \]
and since \(z_r\ne0\), it follows that \(\alpha_r=0\). Similarly we obtain
\[ \alpha_{r-1}=0,\ldots,\alpha_1=0. \]
Consequently, \(x_1,\ldots,x_r\) are linearly independent for every \(r\). Denote by \(\mathfrak L_r\) the linear span of the elements \(x_1,\ldots,x_r\), and show that
\[ \mathfrak L_r\cap\mathfrak R_1=0. \]
Let \(x\in L_r\cap\mathfrak R_1\), i.e.,
\[ \sum_{i=1}^{r}\alpha_i x_i\in\mathfrak R_1. \]
Then
\[ x=Tx',\qquad \alpha_r z_r=T^{k_r}x=T^{k_r+1}x'\in\mathfrak R_{k_r+1}. \]
But \(z_r\notin\mathfrak R_{k_r+1}\), and therefore \(\alpha_r=0\). Similarly,
\[ \alpha_{r-1}=0,\ldots,\alpha_1=0. \]
Thus \(x=0\), i.e.,
\[ \mathfrak L_r\cap\mathfrak R_1=0. \]
This proves (in view of the arbitrariness of \(r\)) that outside \(\mathfrak R_1\) there are infinitely many linearly independent elements, contrary to the assumption that \(\beta_T\) is finite.

Let us note that the finiteness of one of the numbers \(\alpha_T\) or \(\beta_T\) is not a condition necessary for all \(\mathfrak Z_n\), beginning with some index, to coincide. Indeed, let the space \(\mathfrak X\) be decomposed into the direct sum of two infinite-dimensional subspaces. Then each of the projection operators thereby obtained gives an example of an operator for which both numbers \(\alpha\) and \(\beta\) are infinite, while \(\mathfrak Z_1=\mathfrak Z_2=\cdots=\{0\}\).

Further we shall assume that the space \(\mathfrak X\) is a Banach space. Let \(A\) be a closed linear operator acting in \(\mathfrak X\). Fixing \(A\), consider some set \(\{B\}\) of bounded linear operators \((B:\mathfrak X\to\mathfrak X)\), commuting with \(A\) and with one another (commutativity of \(B\) with \(A\) means that \(B\) maps \(\mathfrak D_A\) into itself and \(ABx=BAx\) for \(x\in\mathfrak D_A\)). Denote by \(\mathfrak B\) the closure of the linear hull of the set \(\{B\}\). It is easy to see that all operators belonging to the Banach algebra \(\mathfrak B\) commute with \(A\) and with one another. As \(\{B\}\) one may take, for example, the set consisting of the identity operator \(I\) alone; then \(\mathfrak B\) is a one-dimensional space with elements \(\lambda I\). If the operator \(A\) is defined on all of \(\mathfrak X\), then as \(\{B\}\) one may take the set \(\{I,A,A^2,\ldots\}\); then the elements of \(\mathfrak B\) will be all possible polynomials in \(A\) and their limits.

Suppose now that, together with \(A\), \(\mathfrak B\) is also fixed. In \(\mathfrak B\) distinguish the set \(\Phi\), consisting of operators \(S\) such that: 1) the range \(\mathfrak M_T\) of the operator \(T=A_S=A-S\) is closed in \(\mathfrak X\); 2) at least one of the numbers \(\alpha_T\) or \(\beta_T\) is finite (here \(\beta_T\) denotes the dimension of the space complementary to \(\mathfrak M_T\) in the algebraic sense). Then, by Theorems 2 and 1, for every \(S\) in \(\Phi\) the operator \(T=A-S\) maps \(\mathfrak D_A\cap\mathfrak M_T\) onto \(\mathfrak M_T\). The set \(\Phi\) is open in \(\mathfrak B\) \((^1)\).

Our subsequent task is to study the sets \(\mathfrak M_{A_S}\) and \(\mathfrak N_{A_S}\) as \(S\) varies in \(\Phi\).

Let \(A'_S\) be the restriction of the operator \(A_S\) to \(\mathfrak D_{A'_S}=\mathfrak D_A\cap\mathfrak M_{A_S}\). It generates, in the quotient space \(\mathfrak M_{A_S}/\mathfrak Z_{A'_S}\), the invertible operator \(\widetilde A'_S\), defined by the equality \(\widetilde A'_S\widetilde x=A_Sx\), where \(\widetilde x\) is the coset (element of \(\mathfrak M_{A_S}/\mathfrak Z_{A'_S}\)) containing \(x\) \((x\in\mathfrak D_{A'_S})\). The operator \((\widetilde A'_S)^{-1}\) is bounded. Let
\[ N'_S=\|(\widetilde A'_S)^{-1}\|. \]

Theorem 3. For every \(S_0\in\Phi\) there exists a neighborhood in \(\Phi\) such that the inclusions
\[ \mathfrak M_{A_{S_0}}\subset \mathfrak M_{A_S} \quad\text{and}\quad \mathfrak N_{A_S}\cap\mathfrak M_{A_{S_0}}\subset \mathfrak N_{A_{S_0}} \]
hold.

Proof. Fix an element \(y_0\in\mathfrak M_{A_{S_0}}\). The equation \(A_{S_0}y=y_0\) has a solution \(y_1\in\mathfrak M_{A_{S_0}}\) such that \(\|y_1\|<(N'_{S_0}+1)\|y_0\|\). Similarly, the equation \(A_{S_0}y=y_1\) has a solution \(y_2\), for which \(\|y_2\|<(N'_{S_0}+1)\|y_1\|\), etc. Form the series
\[ y_S^0=\sum_{k=0}^{\infty}(S-S_0)^k y_{k+1} \]
and the series
\[ y_S^n=\frac{1}{n!}\frac{d^n}{dS^n}y_S^0\quad(n=1,2,\ldots), \]
which all converge in the ball
\[ \|S-S_0\|<(N'_{S_0}+1)^{-1}. \]
Using the closedness of the operator \(A_{S_0}\), its commutativity with every operator \(S\) from \(\mathfrak B\), and the equality \(A_{S_0}(\mathfrak D_A\cap\mathfrak M_{A_{S_0}})=\mathfrak M_{A_{S_0}}\), one can prove that all terms and sums of the series under consideration belong to \(\mathfrak D_A\cap\mathfrak M_{A_{S_0}}\). A direct verification shows that
\[ A_S^n y_S^{\,n-1}=y_0,\quad n=1,2,\ldots \]
(here one has to apply the operator \(A_S\) to the series term by term, which is permissible because \(A_S\) is closed). Consequently, \(y_0\in\mathfrak M_{A_S}\). In view of the arbitrariness of the element \(y_0\) from \(\mathfrak M_{A_{S_0}}\), we obtain the inclusion \(\mathfrak M_{A_{S_0}}\subset\mathfrak M_{A_S}\), where \(\|S-S_0\|<(N'_{S_0}+1)^{-1}\).

The second inclusion is obtained in a similar way \((^1)\), by considering the series

\[ x_S^0=\sum_{k=0}^{\infty}(S-S_0)^k x_k \quad\text{and}\quad x_S^n=\frac{1}{n!}\frac{d^n}{dS^n}x_S^0\quad (n=1,2,\ldots), \]
where \(x_0,x_1,\ldots\) is a sequence of elements chosen in a definite way from \(\mathfrak M_{A_{S_0}}\cap \mathfrak M_{A_{S_0}}\). The magnitude of the corresponding neighborhood is estimated by the inequality
\[ \|S-S_0\|<\frac12\,(N'_{S_0}+1)^{-1}. \]

Denote by \(\Gamma\) the set of those \(S\) in \(\Phi\) for which the operator \(A_S\) has zeros not contained in \(\mathfrak M_{A_S}\).

Theorem 4. The set \(\Gamma\) is closed in \(\Phi\).

Proof. Suppose the contrary. Then there exists an accumulation point \(S_0\) of the set \(\Gamma\), belonging to \(\Phi\setminus\Gamma\). Let \(S_n\in\Gamma\) \((n=1,2,\ldots)\) and \(\|S_n-S_0\|\to 0\). For each zero \(x_0\) of the operator \(A_{S_0}\) construct, as above, the element
\[ x_S^0=\sum_{k=0}^{\infty}(S-S_0)^k x_k, \]
which is a zero of the operator \(A_S\) in some neighborhood of the point \(S_0\) and belongs to \(\mathfrak M_{A_{S_0}}\subset \mathfrak M_{A_S}\). Such a construction is possible, since every zero of the operator \(A_{S_0}\) belongs to \(\mathfrak M_{A_{S_0}}\). Without loss of generality, one may assume that all \(S_n\) are contained in this neighborhood.

Denote by \(\Omega_n\) the set of all zeros of the operator \(A_{S_n}\) of the form \(x_{S_n}^0\), corresponding to all possible \(x_0\in \mathfrak Z_{A_{S_0}}\). The sets \(\Omega_n\) are linear and \(\Omega_n\subset \mathfrak M_{A_{S_n}}\). Since \(\overline{\Omega}_n\subset \mathfrak Z_{A_{S_n}}\), \(\overline{\Omega}_n\subset \mathfrak M_{A_{S_n}}\), and \(S_n\in\Gamma\), there is an element \(z_n\) in \(\mathfrak Z_{A_{S_n}}\) such that \(\|z_n\|=1\), \(\rho(z_n,\overline{\Omega}_n)>1/2\). Hence we conclude that \(\rho(z_n,\mathfrak Z_{A_{S_0}})>1/3\), starting from some index. This follows from the fact that \(\rho(z_n,\mathfrak Z_{A_{S_0}})=\rho(z_n,K)\), where \(K\) is the ball in \(\mathfrak Z_{A_{S_0}}\) with center at the zero point and radius 2, and from the fact that every point \(x_0\in K\) is arbitrarily little distant from the corresponding point \(x_{S_n}^0\in \mathfrak Z_{A_{S_n}}\), beginning with some index, uniformly in \(K\). But
\[ A_{S_0}z_n=A_{S_0}z_n-A_{S_n}z_n=(S_n-S_0)z_n\to 0 \]
as \(n\to\infty\), i.e. \(y_n=A_{S_0}z_n\to 0\). Consequently,
\[ y_n=\widetilde A_{S_0}\widetilde z_n\to 0, \]
where \(\widetilde A_{S_0}\) is a continuously invertible operator acting in \(\mathfrak X/\mathfrak Z_{A_{S_0}}\). Hence
\[ \widetilde z_n=(\widetilde A_{S_0})^{-1}y_n\to 0, \]
in contradiction to the fact that
\[ \|\widetilde z_n\|=\rho(z_n,\mathfrak Z_{A_{S_0}})>1/3. \]

Theorem 5. If \(S\) runs through the set \(G\setminus\Gamma\), where \(G\) is some connected component of the set \(\Phi\), then the spaces \(\overline{\mathfrak R}_{A_S}\) and \(\mathfrak M_{A_S}\) are constant.

Proof. Let \(S_1\) and \(S\) be any two points of \(G\setminus\Gamma\), and let \(F\) be a polygonal line in \(G\setminus\Gamma\) joining \(S_1\) with \(S\). Take on \(F\) a finite number of points \(S_i\) \((i=1,2,\ldots,m)\), \(S_m=S\), for which
\[ \|S_i-S_{i+1}\|<\frac12\left(1+\sup_{S\in F}N'_S\right)^{-1}. \]
The finiteness of \(\sup_{S\in F}N'_S\) can be proved by the same method as the analogous assertion in Lemma 2 \((^2)\).

On the basis of Theorem 3 (from the estimates for \(\|S-S_0\|\)) there follow the equalities
\[ \mathfrak M_{A_{S_i}}=\mathfrak M_{A_{S_{i+1}}} \quad\text{and}\quad \overline{\mathfrak R}_{A_{S_i}}=\overline{\mathfrak R}_{A_{S_{i+1}}}, \quad i=1,2,\ldots,m-1, \]
whence
\[ \mathfrak M_{A_{S_1}}=\mathfrak M_{A_S} \]
and
\[ \overline{\mathfrak R}_{A_{S_1}}=\overline{\mathfrak R}_{A_S}. \]

Let us note that assertions analogous to Theorems 3 and 5 were obtained in paper \((^2)\) for the case when \(\mathfrak B\) is generated by a single operator \(I\). As for Theorem 4, instead of it we had there a more special result—the isolation of \(\Gamma\) in \(\Phi\).

Received
29 III 1964

CITED LITERATURE

\(^1\) M. A. Gol’dman, DAN, 100, No. 2 (1955). \(^2\) M. A. Gol’dman, S. N. Krachkovskii, DAN, 154, No. 3 (1964).