Abstract
Full Text
MATHEMATICS
I. V. SUKHAREVSKII
ON (\lambda)-STABILITY OF SOLUTIONS OF OPERATOR EQUATIONS IN BANACH SPACE
(Presented by Academician V. I. Smirnov on 15 VII 1957)
Let, for each (\lambda) from a simply connected domain (\Lambda) of the complex-variable plane, there be defined a linear completely continuous operator (A_{(\lambda)}), mapping a Banach space (E) into (E_{(\lambda)} \subset E), and suppose that (A_{(\lambda)}) depends analytically on (\lambda) in the domain (\Lambda) in the sense of convergence in norm: in a neighborhood of each point (\lambda_0 \in \Lambda)
[
A_{(\lambda)}=\sum_{j=0}^{+\infty}(\lambda-\lambda_0)^j A_j,
\tag{1}
]
where (A_j) are linear (bounded) operators. Suppose that (A_{(\lambda)}) has at least one regular point in (\Lambda). Then (\left(^{1,2}\right)) (R_{(\lambda)}=(I-A_{(\lambda)})^{-1}) is a meromorphic operator in (\Lambda).
Denote by (F_\lambda) the subspace of the space (E) consisting of those elements (f) for which the equation
[
u-A_{(\lambda)}u=f
\tag{2}
]
is solvable for fixed (\lambda \in \Lambda). The solution of equation (2) at a regular point (\lambda) ((F_\lambda=E)) will be denoted by (u_\lambda).
Definition. A value (\lambda_0 \in \Lambda) will be called a stability point of the operator (A_{(\lambda)}) if, for every (f \in F_{\lambda_0}), there exists
[
\lim_{\lambda\to\lambda_0} u_\lambda = u_{(\lambda_0)}
\tag{3}
]
in the sense of strong convergence(^*).
Here, obviously, (u_{(\lambda_0)}) is a solution of equation (2) for (\lambda=\lambda_0), and if the point (\lambda_0) is regular, then (u_{(\lambda_0)}=u_{\lambda_0}). We shall call the solution (u_{(\lambda_0)}) (\lambda)-stable.
The stability points include, obviously, all regular points of the operator (A_{(\lambda)}). But spectral points may also fail to be stable in the sense indicated above. For example, if (A_{(\lambda)}=\lambda A), then no multiple pole of the resolvent is a stability point (this follows directly from Theorem 1 of the present note). The aim of the present note is to clarify criteria for stability of spectral points of the operator (A_{(\lambda)}), depending on the parameter (\lambda) in general nonlinearly, and the properties of (\lambda)-stable solutions.
Let (\lambda_0) be a spectral point of the operator (A_{(\lambda)}), in a neighborhood of which (R_{(\lambda)}) has the expansion
[
R_{(\lambda)}=\sum_{j=-m}^{+\infty}(\lambda-\lambda_0)^j R_j;
]
(^*) It is not difficult to show that, if the limit (3) does not exist in the sense of strong convergence, then it also does not exist in the sense of weak convergence.
({u_j}{j=1}^n) is a basis of the eigensubspace (E_0) of the operator (A);
({V_j}_{j=1}^n) is a basis of the eigensubspace (E_0^) of the adjoint operator (A_{(\lambda_0)}^) ((E_0^ \subset E^;\ E^*) is the space of functionals adjoint to (E)).
From the expansions (1), (4) and the equalities
[
R_{(\lambda)}(I-A_{(\lambda)})=(I-A_{(\lambda)})R_{(\lambda)}=I
\tag{4}
]
it follows that
[
(I-A_0)R_k=
\begin{cases}
0, & \text{for } k=-m;\[4pt]
I+\displaystyle\sum_{j=1}^{m} A_jR_{-j}, & \text{for } k=0;\[8pt]
\displaystyle\sum_{j=1}^{m+k} A_jR_{k-j}, & \text{for } k>-m,\ k\ne 0;
\end{cases}
\tag{5}
]
[
R_k(I-A_0)=
\begin{cases}
0, & \text{for } k=-m;\[4pt]
I+\displaystyle\sum_{j=1}^{m} R_{-j}A_j, & \text{for } k=0;\[8pt]
\displaystyle\sum_{j=1}^{m+k} R_{k-j}A_j, & \text{for } k>-m,\ k\ne 0.
\end{cases}
\tag{6}
]
From the equalities (5), (6) it is clear that the operators (R_{-1}, R_{-2}, \ldots, R_{-m}) are of finite rank, and moreover
[
R_{-m}f=\sum_{i,j=1}^{n}\gamma_{ij}u_iV_j(f)
\tag{7}
]
((\gamma_{ij}) are scalars). With the aid of the same equalities (5), (6), it is easy to establish the following proposition.
Theorem 1. Every simple pole (\lambda_0) of the resolvent is a point of stability; moreover, the (\lambda)-stable solution (u_{(\lambda_0)}) is uniquely singled out from the (n)-parameter family of solutions of the equation
[
u-A_{(\lambda_0)}u=f \qquad (f\in F_{\lambda_0})
\tag{8}
]
by the conditions
[
V_j\bigl(A_1u_{(\lambda_0)}\bigr)=0 \qquad (j=1,\ldots,n).
\tag{9}
]
If, however, (\lambda_0) is a multiple pole of the resolvent and, at the same time, the functionals (\omega_j(f)=V_j(A_1f)) ((j=1,2,\ldots,n)) are linearly independent in (E), then (\lambda_0) does not belong to the points of stability.
In connection with this theorem, it is of interest to obtain a necessary and sufficient condition under which the eigenvalue (\lambda_0) is a simple pole of the resolvent. Such a condition is contained in the following theorem.
Theorem 2. In order that the pole (\lambda_0) of the resolvent (R_{(\lambda)}) be simple ((m=1)), it is necessary and sufficient that
[
\det{V_j(A_1u_i)}\ne 0,
\tag{10}
]
or, equivalently, that the eigensubspaces (E_0) and (E_0^) have bases ({u_i^0}), ({V_i^0}), biorthogonal with respect to the operator (A_1):*
[
V_i^0(A_1u_j^0)=\delta_{ij}.
\tag{11}
]
Theorem 2 is a generalization of a known result of Goursat (³) on resolvent kernels corresponding to Fredholm equations.
Theorem 3. Let (\lambda_0) be an eigenvalue of rank (n=1) (with total multiplicity of the pole (m\geqslant 1)); let (u_1) be an eigenvector of the operator (A_{(\lambda_0)}=A_0); and let (V_1) be an eigenfunctional of the adjoint operator (A^*_{(\lambda_0)}). Then a necessary and sufficient condition for (\lambda_0) to belong to the stability points is that
[
V_1(A_su_1)\ne 0,
\tag{12}
]
where the index (s) is defined by the condition: (V_1(A_sf)) is the first functional, not identically equal to zero, in the sequence*
[
V_1(A_1f),\quad V_1(A_2f),\ldots,\quad V_1(A_kf),\ldots
\tag{13}
]
In this case, the characteristic property distinguishing the (\lambda)-stable solution (u_{(\lambda_0)}) from the family of solutions of equation (2) for (\lambda=\lambda_0) and (f\in F_{(\lambda_0)}) is the equality
[
V_1(A_su_{(\lambda_0)})=0.
\tag{14}
]
Proof. First of all, note that (\lambda_0) is a stability point if and only if (R_{-k}f=0) ((k=1,2,\ldots,m)) for every (f\in F_{\lambda_0}). But for this it is necessary and sufficient that, for all (f\in E),
[
R_{-k}f=z_{m-k}V_1(f)\quad (k=1,2,\ldots,m),
\tag{15}
]
where (z_{m-k}) are certain fixed elements of the space (E). The sufficiency of such representations of the operators (R_{-k}) is obvious. The necessity can easily be established by means of the equalities (5).
Suppose first that (s=1), i.e.
[
V_1(A_1f)\ne 0.
]
If, in addition, (V_1(A_1u_1)\ne 0), then, by Theorems 1 and 2, (\lambda_0) is a stability point and the (\lambda)-stable solution satisfies condition (14) for (s=1). Now let (V_1(A_1u_1)=0) and, consequently (by Theorem 2), (m>1). Then (see (7)) (R_{-m}f=\gamma u_1V_1(f)) ((\gamma\ne 0)) and
[
R_{-m+1}(I-A_0)f=\gamma u_1V_1(A_1f),
]
and, since (V_1(A_1f)\ne 0), (R_{-m+1}) is not representable in the form (15), and thus (\lambda_0) is not a stability point.
Consider the case (s>1). Then (m>1) and
[
R_{-m+j}f=z_jV_1(f)\quad [1\leqslant j<\mu=\min{s,m}].
\tag{16}
]
Indeed,
[
(I-A_0)R_{-m+1}f=\gamma A_1u_1\cdot V_1(f),
]
whence
[
R_{-m+1}f=u_1V(f)+x_1V_1(f)\quad (V\in E^*,\ x_1\in E).
]
But
[
R_{-m+1}(I-A_0)f=\gamma u_1V_1(A_1f)=0,
]
therefore (V=\alpha V_1) ((\alpha) is a scalar), and representation (16) for (j=1) holds. Similarly, if (\mu>2), then
[
(I-A_0)R_{-m+2}f=\gamma A_2u_1\cdot V_1(f)+A_1z_1\cdot V_1(f)=x_2\cdot V_1(f),
]
[
R_{-m+2}(I-A_0)f=\gamma u_1\cdot V_1(A_2f)+z_1\cdot V_1(A_1f)=0,
]
whence it follows that (16) is also valid for (j=2), and so on.
* It is easy to see that such an index (s) always exists: if, for all (k=1,2,\ldots) and every (f\in E), (V_1(A_kf)=0), then in some disk (|\lambda-\lambda_0|<\rho) one would have (V_1(A_{(\lambda)}f)=V_1(A_{(\lambda_0)}f)=V_1(f)), which is impossible, since the spectrum of the operator (A_{(\lambda)}) is discrete.
We shall now show that (m \ge s) and, consequently, (\mu=s).
If it were the case that (m<s), then from (6) and (16) it would follow that
[
R_0^\circ (I-A_0)u_1
=
u_1+\sum_{j=1}^{m} R_{-j}A_j u_1
=
u_1+\sum_{j=1}^{m} z_{m-j}V_1(A_j u_1)
=
u_1,
]
which is impossible, since ((I-A_0)u_1=0). Thus, (m \ge s).
If (m=s), then
[
0=R_0(I-A_0)u_1
=
u_1+\sum_{j=1}^{m-1} z_{m-j}V_1(A_j u_1)+R_{-m}A_m u_1
]
[
u_1{1+\gamma V_1(A_m u_1)},
]
whence
[
\gamma V_1(A_m u_1)=\gamma V_1(A_s u_1)=-1,
]
i.e. (V_1(A_s u_1)\ne 0).
If, however, (m>s), then
[
0=R_{-m+s}(I-A_0)u_1
=
\sum_{j=0}^{s-1} R_{-m+j}A_{s-j}u_1
]
[
\gamma u_1 V_1(A_s u_1)+\sum_{j=1}^{s-1} z_j V_1(A_{s-j}u_1)
\gamma u_1 V_1(A_s u_1),
]
and thus (V_1(A_s u_1)=0).
Thus, if (V_1(A_s u_1)\ne 0), then (m=s); if (V_1(A_s u_1)=0), then (m>s). In the first case, as is seen from (16), (\lambda_0) is a point of stability. In the second case ((m>s)),
[
R_{-m+s}(I-A_0)\hat f=\gamma u_1 V_1(A_s\hat f)\ne 0.
]
Consequently, (R_{-m+s}) cannot be represented in the form (15), and (\lambda_0) is not a point of stability.
Further, if (m=s), then, obviously,
[
u_{(\lambda_0)}=R_0\hat f \qquad (\hat f\in F_{\lambda_0}).
]
Moreover, from (5) it follows that
[
(I-A_0)R_s f
=
\sum_{j=1}^{2s} A_j R_{s-j} f
=
\sum_{j=1}^{s-1} A_j R_{s-j} f + A_s R_0^\circ f
\qquad (f\in F_{\lambda_0}),
]
whence
[
V_1(A_sR_0 f)
=
V_1((I-A_0)R_s f)
-
\sum_{j=1}^{s-1} V_1(A_jR_{s-j}f)
=
0
\qquad (f\in F_{\lambda_0}).
]
Thus, (V_1(A_su_{(\lambda_0)})=0), and, since (V_1(A_su_1)\ne 0), this condition uniquely singles out the (\lambda)-stable solution (u_{(\lambda_0)}) from the one-parameter family of solutions, which completes the proof of the theorem.
Kharkov Polytechnic Institute
named after V. I. Lenin
Received
12 VII 1957
REFERENCES CITED
- I. Ts. Gokhberg, DAN, 78, No. 4, 629 (1951).
- D. R. Kharazov, Tr. Tbilissk. matem. inst. AN GruzSSR, 19, 163 (1953).
- E. Goursat, Course of Mathematical Analysis, 3, part II, 1934, pp. 83–84.