MATHEMATICS

M. A. GOLDMAN, S. N. KRACHKOVSKII

INVARIANCE OF CERTAIN SPACES ASSOCIATED WITH THE OPERATOR \(A-\lambda I\)

(Presented by Academician V. I. Smirnov on 21 IX 1963)

Let \(A\) be a linear closed operator with domain of definition \(\mathfrak D_A\) and range \(\mathfrak R_A\) in a Banach space \(\mathfrak B\). Consider the operator \(A_\lambda=A-\lambda I\), where \(I\) is the identity operator and \(\lambda\) is a complex parameter. Denote by \(\alpha(A_\lambda)\) the dimension of the null space \(\mathfrak Z_{A_\lambda}\) of the operator \(A_\lambda\), and by \(\beta(A_\lambda)\) the dimension of the quotient space \(\mathfrak B/\mathfrak R_{A_\lambda}\). The set \(\Phi\) of all those values \(\lambda\) for which \(\mathfrak R_{A_\lambda}\) is closed and at least one of the numbers \(\alpha(A_\lambda)\), \(\beta(A_\lambda)\) is finite is an open set. For any \(\lambda\in\Phi\) put

\[ \mathfrak N_\lambda=\bigcup_{n=1}^{\infty}\mathfrak Z_{A_\lambda^n},\qquad \mathfrak M_\lambda=\bigcup_{n=1}^{\infty}\mathfrak R_{A_\lambda^n}. \]

The elements of \(\mathfrak N_\lambda\) and \(\mathfrak M_\lambda\) are called, respectively, the nil-elements and the elements of the core of the operator \(A_\lambda\). Obviously, the sets \(\mathfrak N_\lambda\) and \(\mathfrak M_\lambda\) are invariant with respect to \(A_\lambda\), and moreover \(\mathfrak M_\lambda\), being the intersection of closed sets, is closed. It can be proved that the operator \(A_\lambda\) maps \(\mathfrak M_\lambda\) onto itself, i.e. \(A_\lambda(\mathfrak D_A\cap\mathfrak M_\lambda)=\mathfrak M_\lambda\) (for the case when \(\alpha(A_\lambda)<\infty\), this was established in (\(^1\))).

Our task is to study the sets \(\mathfrak N_\lambda\) and \(\mathfrak M_\lambda\) as \(\lambda\) varies. This will be done by representing the elements of \(\mathfrak M_\lambda\) in the form of analytic functions of the parameter \(\lambda\) (see (\(^2\))).

Let \(A_{1\lambda}\) be the restriction of the operator \(A_\lambda\) to \(\mathfrak D_{1\lambda}=\mathfrak D_A\cap\mathfrak M_\lambda\). It induces an operator \(\widetilde A_{1\lambda}\) in the quotient space \(\mathfrak M_\lambda/\mathfrak Z_{A_{1\lambda}}\subset\mathfrak B/\mathfrak Z_{A_{1\lambda}}\) (here \(\mathfrak Z_{A_{1\lambda}}=\mathfrak Z_{A_\lambda}\cap\mathfrak M_\lambda\)). Let \(N_{1\lambda}=\|\widetilde A_{1\lambda}^{-1}\|\). Fix some value \(\lambda_0\in\Phi\) and some element \(y_0\in\mathfrak M_{\lambda_0}\). The equation \(A_{\lambda_0}y=y_0\) has a solution \(y_1\in\mathfrak M_{\lambda_0}\), which may be chosen so that \(\|y_1\|\le (N_{1\lambda_0}+1)\|y_0\|\). Similarly, the equation \(A_{\lambda_0}y=y_1\) has a solution \(y_2\in\mathfrak M_{\lambda_0}\) such that \(\|y_2\|\le (N_{1\lambda_0}+1)\|y_1\|\), and so on. Form the series

\[ y_{1\lambda}=y_1+(\lambda-\lambda_0)y_2+(\lambda-\lambda_0)^2y_3+\cdots \]

and the derived series

\[ y_{1\lambda}^{(n)}=\frac{1}{n!}\frac{d^n}{d\lambda^n}y_{1\lambda},\qquad n=1,2,\ldots, \]

all of which converge in the disk \(|\lambda-\lambda_0|<1/(N_{1\lambda_0}+1)\). Direct verification shows that
\[ A_\lambda^n y_{1\lambda}^{(n-1)}=y_0,\qquad n=1,2,\ldots \]
Consequently, \(y_0\in\mathfrak M_\lambda\). By the arbitrariness of the element \(y_0\) of \(\mathfrak M_{\lambda_0}\), we obtain the inclusion \(\mathfrak M_{\lambda_0}\subset\mathfrak M_\lambda\), where \(\lambda\in\Phi\) and \(|\lambda-\lambda_0|<1/(N_{1\lambda_0}+1)\). Note that termwise application of the operator \(A_\lambda\) to convergent series is possible by virtue of the closedness of this operator.

Theorem 1. The set \(\Gamma\) of those values \(\lambda\) in \(\Phi\) for which \(\mathfrak Z_{A_\lambda}\ne\mathfrak Z_{A_{1\lambda}}\) is an isolated set.

In fact, every zero of the operator \(A_\lambda\), \(\lambda \ne \lambda_0\), belongs to \(\mathfrak M_{\lambda_0}\) (for, if \(Ax-\lambda x=0\), then \(x=(\lambda-\lambda_0)^{-n}A_{\lambda_0}^n x\) for arbitrary \(n\)), i.e. \(\mathfrak Z_{A_\lambda}\in \mathfrak M_{\lambda_0}\). But, as was shown, \(\mathfrak M_{\lambda_0}\subset \mathfrak M_\lambda\) in the circle \(|\lambda-\lambda_0|<1/(N_{\lambda_0}+1)\); consequently, \(\mathfrak Z_{A_\lambda}\in \mathfrak M_\lambda\). Thus, \(\lambda\in\Gamma\) for all \(\lambda\) from this circle, except, possibly, \(\lambda=\lambda_0\). The theorem is proved.

Consider the set \(\mathfrak N_{\lambda_0}\cap\mathfrak M_{\lambda_0}\) and fix in it some nonzero element \(\bar x\). Let \(n_0\) be the smallest of the numbers \(n\), \(n=1,2,\ldots\), for which \(A_{\lambda_0}^n \bar x=0\). Starting from the element \(\bar x\), denoted henceforth by \(x_{n_0-1}\), construct a sequence \(x_{n_0-1},x_{n_0},x_{n_0+1},\ldots\) of elements from \(\mathfrak M_{\lambda_0}\) such that
\[ A_{\lambda_0}x_{n+1}=x_n,\quad \|x_{n+1}\|\le (N_{\lambda_0}+1)\|x_n\|,\quad n=n_0-1,n_0,n_0+1,\ldots \]
If \(n_0>1\), then put \(x_{n_0-2}=A_{\lambda_0}x_{n_0-1}\), \(x_{n_0-3}=A_{\lambda_0}x_{n_0-2},\ldots, x_0=A_{\lambda_0}x_1\). Obviously, \(x_n\in \mathfrak N_{\lambda_0}\cap\mathfrak M_{\lambda_0}\), \(n=0,1,2,\ldots\), and \(x_0\) is a zero of the operator \(A_{\lambda_0}\). Form the series
\[ x_{0\lambda}=x_0+(\lambda-\lambda_0)x_1+(\lambda-\lambda_0)^2x_2+\cdots \]
and the derived series
\[ x_{0\lambda}^{(n)}=\frac{1}{n!}\frac{d^n}{d\lambda^n}x_{0\lambda} =\frac{1}{n!}\sum_{k=n}^{\infty}\frac{k!}{(k-n)!}(\lambda-\lambda_0)^{k-n}x_k,\quad n=1,2,\ldots \]

All these series converge in the circle \(|\lambda-\lambda_0|<1/(N_{\lambda_0}+1)\). The relation between the sequences \(\{x_n\}\) and \(\{x_{0\lambda}^{(n)}\}\) can also be expressed by the series
\[ x_n=\frac{1}{n!}\sum_{k=0}^{\infty}(-1)^k\frac{(n+k)!}{k!}(\lambda-\lambda_0)^k x_{0\lambda}^{(n+k)},\quad n=0,1,2,\ldots, \]
convergent in the circle \(|\lambda-\lambda_0|<1/2(N_{\lambda_0}+1)\). Hence it follows that all \(x_n\) (and, in particular, \(x_{n_0-1}\)) belong to \(\overline{\mathfrak N}_\lambda\), since the sequence \(\{x_{0\lambda}^{(n)}\}\) forms an infinite chain of zero-elements from \(\mathfrak N_\lambda\):
\[ A_\lambda x_{0\lambda}=0,\quad A_\lambda x_{0\lambda}^{(n)}=x_{0\lambda}^{(n-1)},\quad n=1,2,\ldots \]

Thus, \(x_{n_0-1}\in\overline{\mathfrak N}_\lambda\). But \(x_{n_0-1}\) is an arbitrary element of \(\mathfrak N_{\lambda_0}\cap\mathfrak M_{\lambda_0}\), and, consequently, \(\overline{\mathfrak N}_{\lambda_0}\cap\mathfrak M_{\lambda_0}\subset \overline{\mathfrak N}_\lambda\). Taking also into account that \(\mathfrak M_{\lambda_0}\subset \mathfrak M_\lambda\), we obtain the inclusion \(\overline{\mathfrak N}_{\lambda_0}\cap\mathfrak M_{\lambda_0}\subset \overline{\mathfrak N}_\lambda\cap\mathfrak M_\lambda\), valid for any \(\lambda\) from the circle \(|\lambda-\lambda_0|<1/2(N_{\lambda_0}+1)\). If \(\lambda_0\in\Gamma\), then in the indicated circle \(\overline{\mathfrak N}_{\lambda_0}\subset\overline{\mathfrak N}_\lambda\). Thus, the following holds.

Lemma 1. Let \(\lambda_1,\lambda_2\) be two points of \(\Phi\setminus\Gamma\) such that
\[ |\lambda_1-\lambda_2|<\frac12 (N_{\lambda_i}+1),\quad i=1,2. \]
Then \(\overline{\mathfrak N}_{\lambda_1}=\overline{\mathfrak N}_{\lambda_2}\) and \(\mathfrak M_{\lambda_1}=\mathfrak M_{\lambda_2}\).

This result will be used below in the proof of Theorem 2.

Lemma 2. Let \(\lambda\in\Phi\) and \(N_\lambda=\|\widetilde A_\lambda^{-1}\|\). If \(F\) is a bounded closed set contained in \(\Phi\setminus\Gamma\), then
\[ \sup_{\lambda\in F} N_\lambda<\infty . \]

Suppose the contrary. Then in \(F\) there is a convergent sequence \(\{\lambda_n\}\) \((\lim \lambda_n=\lambda_0\in F)\) such that \(\lim N_{\lambda_n}=\infty\). In the quotient space \(\mathfrak B/\mathfrak Z_{A_{\lambda_n}}\), \(n=1,2,\ldots\), there exists an element \(\widetilde x_n\), \(\|\widetilde x_n\|=1\), for which \(\|\widetilde A_{\lambda_n}\widetilde x_n\|<1/(N_{\lambda_n}-1)\to0\). Let \(x_n\in\widetilde x_n\) and \(\|x_n\|<2^{1/n}\). Since \(\rho(x_n,\mathfrak Z_{A_{\lambda_0}})\le \|x_n\|<2^{1/n}\), there exists a sequence \(\{\delta_n\}\), \(\delta_n>0\), \(\delta_n\to0\), and a sequence \(\{x_0^n\}\), \(x_0^n\in \mathfrak Z_{A_{\lambda_0}}\;(=\mathfrak Z_{A_{\lambda_0}}\subset\mathfrak M_{\lambda_0})\), satisfying the condition
\[ \|x_n-x_0^n\|<\rho(x_n,\mathfrak Z_{A_{\lambda_0}})+\delta_n<2^{1/n} \]
(with this \(\|x_0^n\|<4\)).

For each \(x_0^n\) construct an element \(x_{\lambda_n}^n \in \mathfrak Z_{A_{\lambda_n}}\), setting

\[ x_{\lambda_n}^n=x_0^n+(\lambda_n-\lambda_0)x_1^n+(\lambda_n-\lambda_0)^2x_2^n+\cdots, \]

where \(A_{\lambda_0}x_{k+1}^n=x_k^n\), \(\|x_{k+1}^n\|\le (N_{\lambda_0}+1)\|x_k^n\|\), \(k=0,1,2,\ldots\). From the boundedness of the set \(\{\|x_0^n\|\}\) it follows that \(\|x_{\lambda_n}^n-x_0^n\|\to0\). Since

\[ \rho(x_n,\mathfrak Z_{A_{\lambda_0}})+\delta_n>\|x_n-x_0^n\|\ge \|x_n-x_{\lambda_n}^n\|-\|x_{\lambda_n}^n-x_0^n\|\ge \]

\[ \ge \rho(x_n,\mathfrak Z_{A_{\lambda_n}})-\|x_{\lambda_n}^n-x_0^n\| =1-\|x_{\lambda_n}^n-x_0^n\|, \]

we have \(\rho(x_n,\mathfrak Z_{A_{\lambda_0}})\to1\). Let \(\tilde x_n\) denote the coset in \(\mathfrak B/\mathfrak Z_{A_{\lambda_0}}\) containing \(x_n\). Then \(\|\tilde x_n\|=\rho(x_n,\mathfrak Z_{A_{\lambda_0}})\to1\), and \(\|A_{\lambda_0}x_n\|=\|\tilde A_{\lambda_0}\tilde x_n\|\ge \|\tilde x_n\|/N_{\lambda_0}\to 1/N_{\lambda_0}\), which contradicts the equalities

\[ \lim\|A_{\lambda_0}x_n\|=\lim\|A_{\lambda_n}x_n\|=\lim\|\tilde A_{\lambda_n}\tilde x_n\|=0. \]

Remark. If \(\mathfrak Z_{A_{1\lambda}}=\mathfrak Z_{A_\lambda}\) (which occurs at points \(\lambda\in\Phi\setminus\Gamma\)), then the operator \(\tilde A_{1\lambda}\) is a restriction of the operator \(\tilde A_\lambda\), whence \(N_{1\lambda}\le N_\lambda\). Therefore

\[ \sup_{\lambda\in F} N_{1\lambda}\le \sup_{\lambda\in F} N_\lambda<\infty. \]

Theorem 2. If \(\lambda\) ranges over the set \(G\setminus\Gamma\), where \(G\) is any connected component of the set \(\Phi\), then the spaces \(\mathfrak R_\lambda\) and \(\mathfrak M_\lambda\) are constant.

Indeed, let \(\lambda_1,\lambda\) be any two points of \(G\setminus\Gamma\), and let \(F\) be a polygonal line in \(G\setminus\Gamma\) joining \(\lambda_1\) to \(\lambda\). Take on \(F\) a finite number of points \(\lambda_i\), \(i=1,2,\ldots,m\); \(\lambda_m=\lambda\), for which

\[ |\lambda_i-\lambda_{i+1}|<1/2(1+\sup_{\lambda\in F}N_{1\lambda}). \]

Then (see Lemma 1)

\[ \overline{\mathfrak R}_{\lambda_i}=\overline{\mathfrak R}_{\lambda_{i+1}},\qquad \mathfrak M_{\lambda_i}=\mathfrak M_{\lambda_{i+1}},\qquad i=1,2,\ldots,m-1, \]

whence

\[ \overline{\mathfrak R}_{\lambda_1}=\overline{\mathfrak R}_{\lambda},\qquad \mathfrak M_{\lambda_1}=\mathfrak M_{\lambda}. \]

As follows from the preceding, all elements belonging to \(\mathfrak M_\lambda\) (\(\lambda\in G\)) can be obtained by analytic continuation of power series generated by infinite chains of elements from \(\mathfrak M_{\lambda_0}\) (\(\lambda_0\) is any fixed point in \(G\)). All elements obtained by continuation belong to the space of elements of the original chain and, in the case that \(G\) is simply connected, are single-valued analytic functions of the parameter \(\lambda\).

Take the series considered above,

\[ x_{0\lambda}=x_0+(\lambda-\lambda_0)x_1+(\lambda-\lambda_0)^2x_2+\cdots. \]

It can be continued to the whole component \(G\) and defines in it an analytic vector-valued function of \(\lambda\), which is a zero of the operator \(A_\lambda\). If one describes any closed curve in \(G\) (returning to the initial point \(\lambda_0\)), then as a result of analytic continuation along this curve we again obtain a zero of the operator \(A_{\lambda_0}\). Since this zero must belong to the space determined by the elements \(x_0,x_1,x_2,\ldots\), it may differ from \(x_0\) only by a numerical factor. Thus, whatever the component \(G\) of the set \(\Phi\), the zeros of the operator \(A_\lambda\) belonging to \(\mathfrak M_\lambda\) are, up to a numerical factor, single-valued analytic functions in \(G\). This is also true for the zero-elements of the operator \(A_\lambda\), i.e. for all elements of \(\mathfrak R_\lambda\).

Received
11 VIII 1963

References

1. I. Ts. Gokhberg, M. G. Krein, UMN, 12, no. 2 (74) (1957).
2. M. A. Gol’dman, S. N. Krachkovskii, DAN, 86, No. 1 (1952).