UDC 513.881 + 517.948

MATHEMATICS

M. A. GOLDMAN, S. N. KRACHKOVSKII

ON THE \(d\)-CHARACTERISTIC OF A LINEAR OPERATOR

(Presented by Academician V. I. Smirnov on 9 IV 1965)

Let \(E\) and \(F\) be vector spaces over one and the same field, and let \(\mathscr L(E,F)\) be the set of all possible linear operators mapping \(E\) into \(F\). Denote by \(\mathfrak Z_A\) the set of all zeros of the operator \(A \in \mathscr L(E,F)\). The ordered pair \((\alpha_A,\beta_A)\), where \(\alpha_A=\dim \mathfrak Z_A\), \(\beta_A=\operatorname{codim}_F A(E)\), is called the \(d\)-characteristic of the operator \(A\).

In the present note we shall be interested in the dependence between the \(d\)-characteristics of two operators from \(\mathscr L(E,F)\) connected by certain conditions. We first consider some auxiliary decompositions of the spaces \(E\) and \(F\) and note a number of properties of these decompositions.

\(1^\circ\). Let \(P_1\) and \(P_2\) be the projection operators generated by the decomposition of \(E\) into the direct sum of subspaces \(E_1\) and \(E_2\) \((E=E_1\oplus E_2,\ P_1(E)=E_1,\ P_2(E)=E_2)\), and let \(E_0\) be some subspace in \(E\). Put \(M_1=E_0\cap E_1\), \(M_2=E_0\cap E_2\), and take some subspace \(M_0\) complementary to \(M_1\oplus M_2\) in \(E_0\) \((M_0=\operatorname{comp}_{E_0}(M_1\oplus M_2))\). Let \(M_3=P_1(M_0)\), \(M_4=P_2(M_0)\). Then \(M_i\cap M_j=\{0\}\) for \(i\ne j\) \((i,j=0,1,2,3,4)\), and the spaces \(M_0,M_3,M_4\) are isomorphic to one another.

\(2^\circ\). If \(M_5=\operatorname{comp}_{E_1}(M_1\oplus M_3)\), \(M_6=\operatorname{comp}_{E_2}(M_2\oplus M_4)\), then
\[ E=E_0\oplus M_4\oplus M_5\oplus M_6(=E_0\oplus M_3\oplus M_5\oplus M_6). \]

Indeed, for any \(x\in E\) we shall have
\[ x=\sum_{i=1}^{6} x_i,\quad \text{where } x_i\in M_i. \]
But \(x_3=P_1x_0=x_0-P_2x_0\), where \(x_0\in M_0\); consequently,
\[ x=(x_0+x_1+x_2)+(x_4-P_2x_0)+x_5+x_6, \]
i.e. \(x\in E_0\oplus M_4\oplus M_5\oplus M_6\).

\(3^\circ\). If \(T\in\mathscr L(E,F)\), \(E_0=\mathfrak Z_T\), then: a) \(T(M_3)=T(M_4)\); b)
\[ T(E)=T(M_3)\oplus T(M_5)\oplus T(M_6) \quad(=T(E_1)\oplus T(M_6)); \]
c)
\[ T(E_1)\cap T(E_2)=T(M_3). \]

a) Let \(x_3\in M_3\). Then \(x_3=P_1x_0=x_0-P_2x_0=x_0-x_4\), where \(x_0\in M_0\), \(x_4\in M_4\). Hence \(Tx_3=-Tx_4\in T(M_4)\), \(T(M_3)\subset T(M_4)\). Similarly we obtain the inclusion \(T(M_4)\subset T(M_3)\).

b) Since \(E=M_1\oplus\cdots\oplus M_6\), \(T(M_1)=T(M_2)=\{0\}\), \(T(M_3)=T(M_4)\), it follows that
\[ T(E)=T(M_3)+T(M_5)+T(M_6). \]
We prove that the spaces \(T(M_3)\), \(T(M_5)\), \(T(M_6)\) are pairwise disjoint, whence it will follow that \(T(E)\) is their direct sum. Let us carry out the proof, for example, for \(T(M_5)\) and \(T(M_6)\). Let \(y\in T(M_5)\cap T(M_6)\); then \(y=Tx_5=Tx_6\), \(x_5-x_6=x_0+x_1+x_2\), where \(x_i\in M_i\), \(i=0,1,2,5,6\). But
\[ x_0=P_1x_0+P_2x_0=x_3+x_4; \]
therefore,
\[ x_1+x_2+x_3+x_4-x_5+x_6=0. \]
Hence \(x_i=0\), \(i=1,\ldots,6\), and \(y=0\).

c) The equality \(T(E_1)\cap T(E_2)=T(M_3)\) follows from a), b) and the decompositions
\[ T(E_1)=T(M_3)\oplus T(M_5),\quad T(E_2)=T(M_4)\oplus T(M_6). \]

In items 1 and 2 some constructions were performed in the space \(E\). In what follows we shall need the same constructions in the space \(F\). The corresponding notation is obtained by replacing \(E_i\) by \(F_i\) \((i=0,1,2)\), \(M_i\) by \(N_i\) \((i=1,\ldots,6)\), and \(P_i\) by \(Q_i\) \((i=1,2)\).

Let operators \(A\) and \(T\) from \(\mathcal L(E,F)\) be given. With their aid one can carry out the constructions indicated above in the spaces \(E\) and \(F\), putting
\(E_2=\mathfrak Z_A\), \(E_0=\mathfrak Z_T\), \(F_1=A(E)\), \(F_0=T(E)\), and taking as \(E_1\) and \(F_2\) some complements to \(\mathfrak Z_A\) and \(A(E)\), respectively, in \(E\) and \(F\). In what follows we shall always assume that the constructions under consideration in \(E\) and \(F\) have been carried out precisely in this way.

Theorem 1. If \(\mathfrak Z_{Q_1TP_1}=\mathfrak Z_A\) and \(Q_1TP_1(E)=A(E)\), then the following isomorphisms hold:
1) \(\mathfrak Z_A\sim \mathfrak Z_T\oplus M_6\);
2) \(\operatorname{coker}_F A(E)\sim \operatorname{coker}_F T(E)\oplus N_2\);
3) \(M_6\sim N_2\).

Proof. 1) From the condition \(\mathfrak Z_{Q_1TP_1}=\mathfrak Z_A\) it follows that \(\mathfrak Z_{TP_1}=\mathfrak Z_A\), whence \(M_1=\{\theta\}\). Therefore, by item \(1^0\), \(\mathfrak Z_T\sim M_2\oplus M_4\). But
\(\mathfrak Z_A=M_2\oplus M_4\oplus M_6\), and hence
\(\mathfrak Z_A\sim \mathfrak Z_T\oplus M_6\).

2) From the condition \(Q_1TP_1(E)=A(E)\) it follows that \(Q_1T(E)=A(E)\), whence, by item \(1^0\) (as applied to \(F\)), \(N_5=\{\theta\}\). Hence, by item \(2^0\) (for \(F\)),
\(\operatorname{coker}_F T(E)=N_4\oplus N_6\). But
\(\operatorname{coker}_F A(E)=N_2\oplus N_4\oplus N_6\), and therefore
\(\operatorname{coker}_F A(E)=\operatorname{coker}_F T(E)\oplus N_2\).

3) We show that \(T(E)=TP_1(E)\oplus N_2\). Indeed, from the condition
\(\mathfrak Z_{Q_1TP_1}=\mathfrak Z_A\) it follows that
\(TP_1(E)\cap N_2=\{\theta\}\). Moreover, if \(y\in T(E)\), then, putting
\(Q_1y=z\) \((\in A(E))\), we find (taking into account the condition
\(Q_1TP_1(E)=A(E)\)) an element \(y_1\in TP_1(E)\) such that
\(Q_1y_1=z\). Then \(Q_1(y-y_1)=\theta\), i.e. \(y_2=y-y_1\in N_2\). Thus
\(y=y_1+y_2\), where \(y_1\in TP_1(E)\), \(y_2\in N_2\).

Comparing the decomposition just found,
\(T(E)=T(E_1)\oplus N_2\), with the decomposition
\(T(E)=T(E_1)\oplus T(M_6)\) (item \(3^0\), b), we conclude that
\(N_2\sim T(M_6)\). But \(\mathfrak Z_T\cap M_6=\{\theta\}\), and therefore
\(N_2\sim M_6\).

We note that assertion 1) of the theorem has been proved only with the aid of the first hypothesis in the weakened form \((\mathfrak Z_{TP_1}=\mathfrak Z_A)\), and assertion 2) only with the aid of the second hypothesis, also in the weakened form \((Q_1T(E)=A(E))\). Both hypotheses are used only in the proof of assertion 3).

Theorem 2. Suppose that both hypotheses of Theorem 1 are satisfied and at least one of the numbers \(\alpha_A\) or \(\beta_A\) is finite. Then the spaces \(M_6\) and \(N_2\) are finite-dimensional, and the indices of the operators \(A\) and \(T\) are equal:
\[ \alpha_A-\beta_A=\alpha_T-\beta_T . \]

Proof. Since \(M_6\subset \mathfrak Z_A\) and
\(N_2\subset \operatorname{coker}_F A(E)\), on the basis of assertion 3) of Theorem 1 we conclude that \(M_6\) and \(N_2\) are finite-dimensional. Therefore, by assertions 1) and 2) of Theorem 1, we can write
\[ \dim \mathfrak Z_A=\dim \mathfrak Z_T+\dim M_6,\qquad \operatorname{codim}_F A(E)=\operatorname{codim}_F T(E)+\dim N_2 . \]
But \(\dim M_6=\dim N_2\), whence
\(\dim \mathfrak Z_A-\operatorname{codim}_F A(E)=\dim \mathfrak Z_T-\operatorname{codim}_F T(E)\), i.e.
\(\alpha_A-\beta_A=\alpha_T-\beta_T\).

From assertions 1) and 2) of Theorem 1 it follows that
\(\alpha_T\le \alpha_A\) and \(\beta_T\le \beta_A\). If, however, the conditions of Theorem 2 are satisfied and \(\alpha_A\) is finite, then \(\alpha_T=\alpha_A\); analogously, if \(\beta_A\) is finite, then \(\beta_T=\beta_A\).

Up to this point all the considerations have been carried out for linear operators in vector spaces without topology. Next we shall apply the results obtained to linear operators in Banach and Hilbert spaces.

Let \(X\) and \(F\) be Banach spaces, \(A\) a linear closed operator with domain \(E\subset X\) and range \(A(E)\subset F\), closed in \(F\). Take the operator \(T=A+B\), where \(B\) is a linear bounded operator defined on \(E\).

Theorem 3. If the operators \(P_2\) and \(Q_1\), projecting \(E\) onto \(\mathfrak Z_A\) and \(F\) onto \(A(E)\), are bounded, then there exists an \(\varepsilon>0\) such that
\[ \mathfrak Z_{Q_1TP_1}=\mathfrak Z_A \]
and
\[ Q_1TP_1(E)=A(E), \]
as soon as \(\|B\|<\varepsilon\).

Proof. Let \(A_1\) and \(T_1\) be the restrictions of the operators \(A\) and \(T\) to \(E_1\). We show that, for sufficiently small values of \(\|B\|\), the boundedness of the operator \(P_2\) implies the continuous invertibility of \(T_1\) (simple invertibility of \(T_1\) would mean that \(\mathfrak Z_{TP_1}=\mathfrak Z_A\)). Indeed, from the properties of the operator \(A\) it follows—

guarantees the existence of such an \(m>0\) that for every \(y\in A(E)\) there is its preimage \(x\in E\) \((Ax=y)\), satisfying the inequality \(\|x\|\leq m\|y\|\). Putting \(x_1=P_1x\), we shall have \(\|x_1\|\leq \|P_1\|\|x\|\leq \|P_1\|m\|y\|=m_1\|y\|\) \((m_1>0)\). Since \(y\) was taken arbitrarily in \(A(E)=A_1(E_1)\) and \(A_1x_1=y\), this means that \(A_1\) is continuously invertible. Now take any element \(z\in T_1(E_1)\). Then \(z=Ax_1+Bx_1\) \((x_1\in E_1)\),

\[ \|z\|\geq \|Ax_1\|-\|Bx_1\|\geq (1/m_1-\|B\|)\|x_1\|, \]

whence it is clear that if \(\|B\|<1/m_1=\varepsilon_1\), then \(T_1\) is continuously invertible.

Let us further note that, since \(\mathfrak Z_{TP_1}=\mathfrak Z_A\) and \(\mathfrak Z_{Q_1}=\operatorname{codim}_F A(E)\), in order to prove the equality \(\mathfrak Z_{Q_1TP_1}=\mathfrak Z_A\) it suffices to establish that \(T_1(E_1)\cap \operatorname{codim}_F A(E)=\{0\}\). First of all, from the boundedness of \(Q_2\) we conclude that

\[ \|y_2-y_1\|\geq \frac{1}{\|Q_2\|}\|y_2\| \]

for any \(y_1\in A(E)\) and \(y_2\in \operatorname{codim}_F A(E)\). Hence, for any \(z\in T_1(E_1)\), we shall have

\[ \begin{aligned} \|y_2-z\|&=\|y_2-Ax_1-Bx_1\|\geq \\ &\geq \|y_2-Ax_1\|-\|Bx_1\|\geq \frac{1}{\|Q_2\|}\|y_2\|-\|B\|\|x_1\| \\ &\geq \frac{1}{\|Q_2\|}\|y_2\|-\frac{\|B\|}{\varepsilon_1-\|B\|}\|z\|, \end{aligned} \]

where \(x_1=T_1^{-1}z\). This shows that, for a sufficiently small value of \(\|B\|\) \((\|B\|<\varepsilon_2)\), the elements \(y_2\) and \(z\) cannot coincide if they are different from zero, i.e. \(T_1(E_1)\cap \operatorname{codim}_F A(E)=\{0\}\).

To prove the equality \(Q_1TP_1(E)=A(E)\), write the operator \(Q_1TP_1\) in the form

\[ Q_1TP_1=Q_1(A+B)P_1=A+Q_1BP_1. \]

Since the operator \(Q_1BP_1\) is small when \(B\) is small \((\|B\|<\varepsilon_3)\) and \(Q_1BP_1(E)\subset A(E)\), the required equality is a consequence of the stability of the epimorphism property of a closed operator.

The \(\varepsilon\) indicated in the theorem is equal to \(\min(\varepsilon_1,\varepsilon_2,\varepsilon_3)\).

If \(X\) and \(F\) are Hilbert spaces, then, by virtue of the closedness of \(\mathfrak Z_A\) and \(A(E)\), the existence of bounded operators \(P_2\) and \(Q_1\) is ensured. Therefore, for Hilbert spaces the theorem on stability of the index (i.e. the fact that, if at least one of the numbers \(\alpha_A\) or \(\beta_A\) is finite, then the addition of a small operator \(B\) to the operator \(A\) does not change the index) follows completely from our preceding theorems. For Banach spaces, bounded operators \(P_2\) and \(Q_1\), in any case, exist if both numbers \(\alpha_A\) and \(\beta_A\) are finite, and in this case the stability of the index still follows from the same theorems. However, as is known, for Banach spaces the stability property of the index also holds when only one of the numbers \(\alpha_A\) or \(\beta_A\) is finite (see, for example, I. Ts. Gohberg and M. G. Krein (1), where results concerning the \(d\)-characteristic of operators in a Banach space are set out in detail), but this fact does not follow from the theorems of the present note.

Received
27 III 1965

CITED LITERATURE

1. I. Ts. Gohberg, M. G. Krein, UMN, 12, no. 2 (74), 43 (1957).