MATHEMATICS

A. S. MARKUS

ON HOLOMORPHIC OPERATOR-FUNCTIONS

(Presented by Academician V. I. Smirnov on 6 XII 1957)

Let \(G\) be an open connected domain of the complex plane and let \(A_\lambda\) be an operator-function holomorphic in the domain \(G\),* whose values are linear closed operators acting from one complex Banach space \(\mathfrak B_1\) into another \(\mathfrak B_2\). Let, further, \(\lambda_0\) be an arbitrary point of the domain \(G\), and suppose that for \(|\lambda-\lambda_0|<\rho\) \((\rho>0)\) the operator-function \(A_\lambda\) admits an expansion in an operator-norm convergent series:

\[ A_\lambda=A_{\lambda_0}+\sum_{i=1}^{\infty}(\lambda-\lambda_0)^i C_i . \tag{1} \]

We note that the convergence of this series for \(|\lambda-\lambda_0|<\rho\) is equivalent to the fact that for every positive number \(\rho_1\) less than \(\rho\) there is a number \(M\) such that

\[ |C_i|<M\rho_1^{-i}\quad (i=1,2,\ldots). \tag{2} \]

Consider an arbitrary vector \(x_0\in\mathfrak Z(A_{\lambda_0})\). By \(\mu(x_0,A_{\lambda_0})\) we denote the largest of all nonnegative integers \(\mu\), for each of which there exist vectors \(x_{\mu0}=x_0, x_{\mu1}, x_{\mu2},\ldots,x_{\mu\mu}\) such that

\[ \sum_{i=0}^{k} C_i x_{\mu,k-i}=0\quad (k=0,1,\ldots,\mu), \]

where \(C_0=A_{\lambda_0}\). If among such numbers there is no largest one, then we put \(\mu(x_0,A_{\lambda_0})=\infty\). The linear set consisting of all vectors \(x\in\mathfrak Z(A_{\lambda_0})\) for which \(\mu(x,A_{\lambda_0})=\infty\) will be denoted by \(\mathfrak N(A_{\lambda_0})\), and the dimension of this linear set by \(n(A_{\lambda_0})\).

Put, further, \(k(\lambda)=\sup_y \inf_x |x|\), where the infimum is taken over all \(x\) solving the equation \(A_\lambda x=y\), and the supremum over all \(y\in\mathfrak R(A_\lambda)\), \(|y|=1\).

In 1954 I. Ts. Gokhberg suggested to the author that Theorem 1 of \((^2)\) (see also Theorem 3.6 of \((^1)\)) be generalized to the case of a holomorphic operator-function whose values are \(\Phi_+\)-operators. In the present note the indicated theorems are generalized and some other results are obtained, related to the same circle of questions \((^1)\), Theorem 8.2; \((^3)\), Theorem 12; \((^4,^5)\).

§ 1. Theorem 1. Suppose that for every point \(\lambda\in G\) the operator \(A_\lambda\) is a \(\Phi_+\)- (or \(\Phi_-\)) operator. Then there exists a set \(\Gamma\subset G\) such that \(G-\Gamma\) is isolated in \(G\) and such that for all \(\lambda\in\Gamma\) the function \(\alpha(A_\lambda)\) has the constant value: \(\alpha(A_\lambda)=\alpha_0\). If \(\lambda\in G-\Gamma\), then \(\alpha(A_\lambda)>\alpha_0\). Moreover, for all \(\lambda\in G\) the function \(n(A_\lambda)\) has the same constant value: \(n(A_\lambda)=\alpha_0\).

* The terminology and notation are borrowed by us from \((^1)\). For convenience, instead of \(\mathfrak Z_A\) we shall write \(\mathfrak Z(A)\), etc.

Proof. First of all, let us note that it is enough to prove the following assertion: if \(\lambda_0\) is an arbitrary point of \(G\), then there exists a positive number \(r\) such that, for \(0<|\lambda-\lambda_0|<r\), the equality
\[ \alpha(A_\lambda)=n(A_\lambda)=n(A_{\lambda_0}) \]
holds. Indeed, from this assertion, by means of the usual arguments (see, for example, the proof of Theorem 3.3 in \((^1)\)), Theorem 1 is easily obtained.

We proceed to the proof of the assertion formulated above. First consider the case when
\[ \mathfrak N(A_{\lambda_0})=\mathfrak Z(A_{\lambda_0}). \]
Let
\[ x_{01},x_{02},\ldots,x_{0\alpha}\quad \bigl(\alpha=\alpha(A_{\lambda_0})\bigr) \]
be a normalized basis in \(\mathfrak Z(A_{\lambda_0})\). For each vector \(x_{0k}\) \((k=1,2,\ldots,\alpha)\) there is a sequence \(x_{1k},x_{2k},\ldots\) such that
\[ \sum_{i=0}^{j} C_i x_{j-i,k}=0\qquad (j=0,1,\ldots). \tag{3} \]
The vectors \(x_{ik}\) \((k=1,2,\ldots,\alpha;\ i=1,2,\ldots)\) may be chosen so that the series
\[ x_k(\lambda)=\sum_{i=0}^{\infty}(\lambda-\lambda_0)^i x_{ik}\qquad (k=1,2,\ldots,\alpha) \tag{4} \]
converge for \(|\lambda-\lambda_0|<r_1\) \((r_1>0)\). For this it is sufficient, for example, to ensure that the inequalities
\[ |x_{ik}|\le k(\lambda_0)|C_0x_{ik}|\qquad (k=1,2,\ldots,\alpha;\ i=1,2,\ldots) \tag{5} \]
are satisfied. In fact, from relations (2), (3), and (5) it is not difficult to obtain that
\[ |x_{ik}|\le \bigl(Mk(\lambda_0)+1\bigr)^i\rho_1^{-i} \qquad (k=1,2,\ldots,\alpha;\ i=0,1,\ldots), \]
whence follows the convergence of the series (4) for
\[ |\lambda-\lambda_0|<\rho_1\bigl(Mk(\lambda_0)+1\bigr)^{-1}. \]
Further, from the equalities (3) it follows that
\[ A_\lambda x_k(\lambda)=0\qquad (k=1,2,\ldots,\alpha). \]

It is not difficult to establish the existence of such a positive number \(r_2\) \((\le r_1)\) that, for \(|\lambda-\lambda_0|<r_2\), the vectors \(x_k(\lambda)\) \((k=1,2,\ldots,\alpha)\) are linearly independent and, consequently,
\[ \alpha(A_\lambda)\ge \alpha=\alpha(A_{\lambda_0}). \]
But, on the other hand, there is a number \(r_3\) \((>0)\) such that, for \(|\lambda-\lambda_0|<r_3\), the reverse inequality
\[ \alpha(A_\lambda)\le \alpha(A_{\lambda_0}) \]
holds \((^1)\), Theorem 7.1. Consequently,
\[ \alpha(A_\lambda)=\alpha(A_{\lambda_0})=n(A_{\lambda_0}) \]
for
\[ |\lambda-\lambda_0|<r=\min(r_2,r_3). \]

Now consider the case when
\[ n(A_{\lambda_0})<\alpha(A_{\lambda_0}). \]
Denote by \(\mathfrak N_k\) the subspace of \(\mathfrak Z(A_{\lambda_0})\) consisting of those vectors \(x\) for which
\[ \mu(x,A_{\lambda_0})\ge k. \]
By virtue of the finite-dimensionality of \(\mathfrak Z(A_{\lambda_0})\), there is a natural number \(m\) such that
\[ \mathfrak N_m=\mathfrak N(A_{\lambda_0}). \]
Denote by \(\mathfrak M_k\) the direct complement to the subspace \(\mathfrak N_{k+1}\) in the subspace \(\mathfrak N_k\) \((k=0,1,\ldots,m-1)\). Let
\[ x_{01},x_{02},\ldots,x_{0n},\ y_{01},y_{02},\ldots,y_{0,\alpha-n} \quad \bigl(n=n(A_{\lambda_0}),\ \alpha=\alpha(A_{\lambda_0})\bigr) \]
be a normalized basis of \(\mathfrak Z(A_{\lambda_0})\), composed of a basis
\[ x_{01},x_{02},\ldots,x_{0n} \]
of the subspace \(\mathfrak N(A_{\lambda_0})\) and bases of the subspaces
\[ \mathfrak M_k\qquad (k=0,1,\ldots,m-1). \]
Denote \(\mu(y_{0t},A_{\lambda_0})\) by \(\mu_t\) and choose elements
\[ y_{1t},y_{2t},\ldots,y_{\mu_t t} \]
such that
\[ \sum_{i=0}^{l} C_i y_{l-i,t}=0 \qquad (l=0,1,\ldots,\mu_t;\ t=1,2,\ldots,\alpha-n). \tag{6} \]

Next, let \(\mathfrak K\) be some \((\alpha-n)\)-dimensional normed space, and let
\[ z_1,z_2,\ldots,z_{\alpha-n} \]
be a normalized basis in \(\mathfrak K\). Denote by \(\mathfrak B_1\) the direct sum of the spaces \(\mathfrak B_1\) and \(\mathfrak K\), in which the norm is defined by the equality
\[ |y+z|=|y|+|z|\qquad (y\in\mathfrak B_1,\ z\in\mathfrak K). \]

Denote by $\widetilde C_i$ $(i=0,1,\ldots)$ the operators acting from $\widetilde{\mathfrak B}_1$ into $\mathfrak B_2$, coinciding in $\mathfrak B_1$ with $C_i$ and defined on $\mathfrak R$ by the equalities

\[ \widetilde C_i z_t=-\sum_{j=1}^{\mu_t+1} C_{i+j}y_{\mu_t+1-j,t} \qquad (t=1,2,\ldots,\alpha-n;\ i=0,1,\ldots). \tag{7} \]

It is not hard to show that $\widetilde C_0$ vanishes on $\mathfrak R$ only at zero; hence it follows that $\mathfrak z(\widetilde C_0)=\mathfrak z(C_0)$. Next put

\[ \widetilde A_\lambda=\widetilde C_0+\sum_{i=1}^{\infty}(\lambda-\lambda_0)^i\widetilde C_i . \]

It is not hard to establish that this series, together with the series (1), converges for $|\lambda-\lambda_0|<\rho$ and that, for the indicated values of $\lambda$, the operator $\widetilde A_\lambda$, being an extension by $\alpha-n$ dimensions of the operator $A_\lambda$, is likewise a $\Phi_+$- (or $\Phi_-$) operator.

We now show that $\mathfrak R(\widetilde A_{\lambda_0})=\mathfrak Z(\widetilde A_{\lambda_0})$. For this it is enough to establish that for every vector $y_{0t}$ $(t=1,2,\ldots,\alpha-n)$ there exists a sequence $\widetilde y_{1t},\widetilde y_{2t},\ldots$ such that

\[ \sum_{i=0}^{j-1}\widetilde C_i\widetilde y_{j-i,t}+\widetilde C_j y_{0t}=0 \qquad (j=0,1,\ldots). \tag{8} \]

But it suffices to put $\widetilde y_{it}=y_{it}$ $(i=1,2,\ldots,\mu_t)$; $\widetilde y_{\mu_t+1,t}=z_t$; $\widetilde y_{it}=0$ $(i=\mu_t+2,\mu_t+3,\ldots)$, and, by virtue of equalities (6) and (7), relations (8) will be fulfilled. Applying to the operator $\widetilde A_\lambda$ the result of the first part of the proof, we obtain that for $|\lambda-\lambda_0|<r$ the equality
$\alpha(\widetilde A_\lambda)=\alpha(\widetilde A_{\lambda_0})=\alpha(A_{\lambda_0})$ holds, and the basis of the subspace $\mathfrak Z(A_\lambda)$ consists of the vectors

\[ x_k(\lambda)=\sum_{i=0}^{\infty}(\lambda-\lambda_0)^i x_{ik} \quad (k=1,2,\ldots,n); \qquad y_t(\lambda)=\sum_{i=0}^{\mu_t}(\lambda-\lambda_0)^i y_{it} +(\lambda-\lambda_0)^{\mu_t+1}z_t \]

\[ (t=1,2,\ldots,\alpha-n). \]

Since $\mathfrak Z(A_\lambda)=\mathfrak Z(\widetilde A_\lambda)\cap \mathfrak B_1$, we have
$\dim \mathfrak Z(\widetilde A_\lambda)/\mathfrak Z(A_\lambda)\leq
\dim \widetilde{\mathfrak B}_1/\mathfrak B_1=\alpha-n$.
On the other hand, it is easy to see that for $\lambda\ne\lambda_0$ the subspace spanned by the elements $y_t(\lambda)$ $(t=1,2,\ldots,\alpha-n)$ intersects $\mathfrak B_1$ only at zero, and, consequently,
$\dim \mathfrak Z(\widetilde A_\lambda)/\mathfrak Z(A_\lambda)\geq \alpha-n$.

Thus,
$\dim \mathfrak Z(\widetilde A_\lambda)/\mathfrak Z(A_\lambda)=\alpha-n$
for $0<|\lambda-\lambda_0|<r$, and therefore
$\alpha(A_\lambda)=\dim \mathfrak Z(A_\lambda)=\alpha-(\alpha-n)=n(A_{\lambda_0})$.

It remains for us to prove that $n(A_\lambda)=n(A_{\lambda_0})$ for $|\lambda-\lambda_0|<r$. Suppose that for some point $\lambda'$ $(|\lambda'-\lambda_0|<r)$
$n(A_{\lambda'})<n(A_{\lambda_0})$. But then there exists a number $r'(>0)$ such that
$\alpha(A_\lambda)=n(A_{\lambda'})<n(A_{\lambda_0})$
for $0<|\lambda-\lambda'|<r'$. Since there are points simultaneously satisfying the inequalities
$0<|\lambda-\lambda_0|<r$ and $0<|\lambda-\lambda'|<r'$, we have arrived at a contradiction. The theorem is completely proved.

Theorem 2. Suppose that for every point $\lambda\in G$ the operator $A_\lambda$ is a $\Phi_-$-operator. Then there exists a set $\Gamma\subset G$ such that $G-\Gamma$ is isolated in $G$ and such that for all $\lambda\in\Gamma$ the function $\beta(A_\lambda)$ has the constant value: $\beta(A_\lambda)=\beta_0$. If, however, $\lambda\in G-\Gamma$, then $\beta(A_\lambda)>\beta_0$. Moreover, $\mathfrak R(A_\lambda)=\mathfrak Z(A_\lambda)$ for all $\lambda\in\Gamma$.

Proof. Let $\mathfrak D=\mathfrak D(A_\lambda)$ for $\lambda\in G$. Denote by $\hat A_\lambda$ the operator acting in the same way as $A_\lambda$, from the space $\widehat{\mathfrak B}=\overline{\mathfrak D}$ into the space $\mathfrak B_2$, and let $\hat A_\lambda^{+}$ be the operator adjoint to it. Applying

Theorem 1 to the holomorphic operator-function \(\hat A_\lambda^+\), whose values are \(\Phi_+\)-operators, and taking into account that \(\alpha(\hat A_\lambda^+)=\beta(A_\lambda)\) and that from the equality \(\mathfrak R(\hat A_\lambda^+)=\mathfrak Z(\hat A_\lambda^+)\) there follows the equality \(\mathfrak R(A_\lambda)=\mathfrak Z(A_\lambda)\), we immediately obtain Theorem 2.

§ 2. Let \(\Gamma\) be the set of complex numbers referred to in the formulation of Theorem 1 or Theorem 2. It can be proved that the function \(k(\lambda)\) is continuous on \(\Gamma\). With the aid of this assertion and the method of proof from \(\left({}^{5}\right)\), the results of § 2 of the note \(\left({}^{4}\right)\) carry over completely to the case under consideration.

Let us note in conclusion that, by means of the general device indicated by B. Sz.-Nagy \(\left({}^{6,7}\right)\) (see also \(\left({}^{1}\right)\)), all the results of the present note can be carried over to the case where the holomorphy condition on \(A_\lambda\) is replaced by the following more general condition: the values of the operator-function \(A_\lambda\) are linear closed operators having, for all \(\lambda\in G\), one and the same domain of definition \(\mathfrak D=\mathfrak D(A_\lambda)\), and for each point \(\lambda_0\in G\) there exist a positive number \(\rho\) and linear operators \(C_1,C_2,\ldots\) such that \(\mathfrak D(C_i)\supseteq\mathfrak D\) \((i=1,2,\ldots)\), and such that, for \(|\lambda-\lambda_0|<\rho\), for every \(x\in\mathfrak D\),

\[ A_\lambda x=A_{\lambda_0}x+\sum_{i=1}^{\infty}(\lambda-\lambda_0)^i C_i x . \]

The author takes this opportunity to express his gratitude to I. Ts. Gohberg and I. A. Fel'dman for a number of valuable remarks.

Kishinev State
University

Received
22 XI 1957

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