Mathematics

K. M. Fishman

On the Connection of the Method of Nearby Systems in Special Linear Topological Spaces with Certain Questions in the Perturbation Theory of Linear Operators in Banach Spaces

(Presented by Academician V. I. Smirnov on 28 IV 1958)

In the present note we show the possibility of transferring a number of results of the perturbation theory of linear operators in Banach spaces ((^1)) to certain linear topological spaces. In particular, from an abstract point of view the method of proximity of M. A. Evgrafov in analytic space (((^2),) Ch. IV, §§ 3, 4) is elucidated, which makes it possible not only to strengthen known results, but also to broaden the range of their applicability.

1. Let (\mathcal{B}r) ((a<r<b)) be a family of Banach spaces satisfying the conditions: 1) (\mathcal{B}_r) is an everywhere dense linear manifold in (\mathcal{B}), with respect to the norm (|\cdot|{r'}) in (\mathcal{B}), for (r'<r); 2) (|f|{r'} \le |f|_r) for (f\in \mathcal{B}_r) and (r'<r). Let (\widetilde{\mathcal{B}}_r) we denote the complete linear topological space consisting of the elements}) ((\tilde a<\tilde r<\tilde b;\ \tilde r) is a continuous monotonically increasing function of (r)) be another family of Banach spaces with the same properties as (\mathcal{B}_r). By (\mathfrak{A

[
\prod_{r'<r}\mathcal{B}_{r'}
]

with linear operations induced from (\mathcal{B}{r'}), ((r'<r)), and with convergence (f_n\underset{r}{\to} f) equivalent to the convergence (|f_n-f|}\to 0) for all (r'<r). Analogously, starting from the family (\widetilde{\mathcal{B}{\tilde r}), we construct the spaces (\widetilde{\mathfrak{A}}).* In what follows the asterisk sign for spaces will denote passage to conjugate spaces, and for operators passage to conjugate operators. It is obvious that

[
\mathfrak{A}_r^{}=\sum_{r'<r}\mathcal{B}_{r'}^{}.
]

Consider a distributive operator (A), defined on a linear manifold (D_A(\mathcal{B}r)\subset \mathcal{B}_r) ((a<r<b)), with range (R_A(\widetilde{\mathcal{B}}})\subset \widetilde{\mathcal{B}{\tilde r}). It is assumed that (D_A(\mathcal{B}_r)\subset D_A(\mathcal{B}_r) a distributive operator with domain of definition})) for (r'<r) and that the result of the action of the operator (A) does not depend on (r). The operator (A) induces in each space (\mathfrak{A

[
D_A(\mathfrak{A}r)=\prod)}D_A(\mathcal{B}_{r'
]

and range

[
R_A(\widetilde{\mathfrak{A}}{\tilde r})=\prod).}R_A(\widetilde{\mathcal{B}}_{\tilde r'
]

Theorem 1. If (A) is a (\Phi)-operator** from (\mathcal{B}r) into (\widetilde{\mathcal{B}}}) for all (a<r<b), then it is also a (\Phi)-operator from (\mathfrak{Ar) into (\widetilde{\mathfrak{A}}) for all (r\in(a,b)).

* We shall denote the norm in (\widetilde{\mathcal{B}}{\tilde r}) by (|\cdot|).}), and by (\tilde f_n \underset{\tilde r}{\to}\tilde f) convergence in (\widetilde{\mathfrak{A}}_{\tilde r

** The concept of a (\Phi)-operator, introduced in ((^1)) (p. 52) for Banach spaces, includes only topological properties and therefore may also be introduced for operators in linear topological spaces.

Proof. The operator (A) from (\mathfrak A_r) into (\widetilde{\mathfrak A}r) is, obviously, closed. Denote, following (1) (p. 50), by (\alpha_A(\mathfrak B_r)) ((\alpha_A(\mathfrak A_r))) the dimension of the null subspace (\mathfrak z_A(\mathfrak B_r)) ((\mathfrak z_A(\mathfrak A_r))), and by (\beta_A(\widetilde{\mathfrak B}_r^{})) ((\beta_A(\widetilde{\mathfrak A}_r^{}))) the dimension of the defect subspace (\mathfrak z_A^{}(\widetilde{\mathfrak B}_r^{})) ((\mathfrak z_A^{}(\widetilde{\mathfrak A}_r^{}))) of the operator (A), acting from (\mathfrak B_r) into (\widetilde{\mathfrak B}_r) (from (\mathfrak A_r) into (\widetilde{\mathfrak A}_r)). From the assumption made and from the fact that (\mathfrak z_A(\mathfrak B)) ((r-\delta(r)<r'<r)). From the inclusion (\mathfrak z_A^{}) \supset \mathfrak z_A(\mathfrak B_r)) for (r'<r), it follows that (\alpha_A(\mathfrak B_r)) ((a<r0) (we choose (\delta(r)) maximal possible) that (\alpha_A(\mathfrak B_{r'})=\mathrm{const}) for (r'\in(r-\delta(r),r)). Then (\mathfrak z_A(\mathfrak B_{r'})) is one and the same for all (r'\in(r-\delta(r),r)), and therefore (\mathfrak z_A(\mathfrak A_r)=\mathfrak z_A(\mathfrak B_{r'}(\widetilde{\mathfrak B}_{r'}^{})\subset \mathfrak z_A^{}(\widetilde{\mathfrak B}_{r}^{})) ((r'0) that (\beta_A(\widetilde{\mathfrak B}{r'}^{})=\mathrm{const}) for (r'\in(r-\eta(r),r)) and, consequently, (\mathfrak z_A^{}(\widetilde{\mathfrak B}^{})) is one and the same for all (r'\in(r-\eta(r),r)). Denote by ({\widetilde{\Phi}i}_1^{m_r}) ((m_r=\beta_A(\widetilde{\mathfrak B}^{}))) a basis in the spaces (\mathfrak z_A^{}(\widetilde{\mathfrak B}_{r'}^{})) for (r'\in(r-\eta(r),r)). If (\widetilde h\in R_A(\widetilde{\mathfrak A}r)\subset R_A(\widetilde{\mathfrak B}})), then (\widetilde\Phi_i(\widetilde h)=0) ((i=1,2,\ldots,m_r)). Conversely, let (\widetilde h\in\widetilde{\mathfrak Ar), (\widetilde\Phi_i(\widetilde h)=0) ((i=1,2,\ldots,m_r)); then (\widetilde h\in R_A(\widetilde{\mathfrak B}_r)).})) for all (r'\in(r-\eta(r),r)), and consequently (\widetilde h\in R_A(\widetilde{\mathfrak A

This proves that the operator (A) from (\mathfrak A_r) into (\widetilde{\mathfrak A}_r) is normally solvable and has finite (d)-characteristic ((1), p. 50). Moreover, it has been found that

[
\alpha_A(\mathfrak A_r)=\lim_{r'\to r-0}\alpha_A(\mathfrak B_{r'})
\quad\text{and}\quad
\beta_A(\widetilde{\mathfrak A}_r^{})=\lim_{r'\to r-0}\beta_A(\widetilde{\mathfrak B}_{r'}^{}).
]

The index

[
\varkappa_A(\mathfrak A_r)=\beta_A(\widetilde{\mathfrak A}_r^{*})-\alpha_A(\mathfrak A_r)
]

is an integer-valued nonincreasing function of (r\in(a,b)), and

[
\varkappa_A(\mathfrak A_r)=\lim_{r'\to r-0}\varkappa_A(\mathfrak B_{r'}).
]

Remark. The theorem is also true in the case when (A) is a (\Phi)-operator from (\mathfrak B_r) into (\widetilde{\mathfrak B}_r) for all (r\in(a,b)) except for a discrete set of values.

On the basis of this theorem and Theorem 2.2 of (1), we obtain:

Theorem 2. Let (A) be an operator satisfying the conditions of Theorem 1. Then there is a positive function (\rho(r)) ((a<r<b)) such that, whatever the linear bounded operator (B) mapping (\mathfrak B_r) into (\widetilde{\mathfrak B}_r), independently of (r\in(a,b)), for which* (\operatorname{ps}|B|_r<\rho(r)), the operator (A+B), acting from (\mathfrak A_r) into (\widetilde{\mathfrak A}_r) ((a<r<b)), will also be a (\Phi)-operator, and

[
\varkappa_{A+B}(\mathfrak A_r)=\varkappa_A(\mathfrak A_r)\quad (a<r<b).
]

With the help of analogous arguments and Theorem 2.4 of (1) we obtain:

Theorem 3. Let the operator (A) satisfy the conditions of Theorem 1. Then there is a positive function (\rho(r)) ((a<r<b)) such that for all linear bounded operators (B), acting from (\mathfrak B_r) into (\widetilde{\mathfrak B}_r) independently of (r), for which (|B|_r<\rho(r)) ((a<r<b)), the operator (A+B), acting from (\mathfrak A_r) into (\widetilde{\mathfrak A}_r) ((a<r<b)), will also be a (\Phi)-operator,

[
\varkappa_{A+B}(\mathfrak A_r)=\varkappa_A(\mathfrak A_r)\quad (a<r<b),
]

and, moreover,

[
\alpha_{A+B}(\mathfrak A_r)\leqslant \alpha_A(\mathfrak A_r)\quad (a<r<b).
]

Theorem 4. If (A) is a (\Phi_{\pm})-operator ((1), p. 89) from (\mathfrak B_r) into (\widetilde{\mathfrak B}r) for (a<r<b), then it is respectively also a (\Phi_r) for (a<r<b).})-operator from (\mathfrak A_r) into (\widetilde{\mathfrak A

Using Theorems 7.1 and 7.2 of (1), we obtain:

* By (\operatorname{ps}|B|_r) we denote the pseudonorm of the operator (B) from (\mathfrak B_r) into (\widetilde{\mathfrak B}_r) (see (1), p. 55).

Theorem 5. Let (A) be a (\Phi_+)-operator ((\Phi_-)-operator) acting from (\mathfrak{B}r) into (\widetilde{\mathfrak{B}}_r), independently of (r \in (a,b)). Then there exists a function (\rho(r)>0) such that, whatever the bounded linear operator (B) may be ((B\mathfrak{B}_r \subset \widetilde{\mathfrak{B}}_r)), whose action does not depend on (r \in (a,b)), if (|B|_r<\rho(r)) ((a<r<b)), the operator (A+B), acting from (\mathfrak{A}_r) into (\widetilde{\mathfrak{A}}_r), will also be a (\Phi+)-operator ((\Phi_-)-operator), and moreover
[
\alpha_{A+B}(\mathfrak{A}r)\leq \alpha_A(\mathfrak{A}_r)\quad \text{for } r\in(a,b)
]
[
\bigl(\beta r\in(a,b)\bigr).}(\widetilde{\mathfrak{A}}_r)\leq \beta_A(\widetilde{\mathfrak{A}}_r)\quad \text{for
]

Remark. The theorems stated will remain valid if (A) is assumed to be a bounded linear operator mapping (\mathfrak{B}r) into (\widetilde{\mathfrak{B}}_r) ((ar}\mathfrak{B}_{r''}\) with the following definition of convergence: \(f_n\to f\) if and only if there exists an \(r''>r) such that ({f_n}_1^\infty) and (f) are contained in (\mathfrak{B}) and (|f_n-f|{r''}\to 0). In this case the conjugate space (\overline{\mathfrak{A}}_r^{\,}) is equal to
[
\prod_{r''>r}\mathfrak{B}_{r''}^{},\qquad
\alpha_A(\overline{\mathfrak{A}}_r)=\lim)}\alpha_A(\mathfrak{B}_{r''
]
and
[
\beta_A(\widetilde{\mathfrak{A}}_r^{\,})=\lim_{r''\to r+0}\beta_A(\widetilde{\mathfrak{B}}_{r''}^{}).
]

2. We shall show one possible application of the preceding theorems. Consider the Banach space (\mathfrak{B}r) ((0<r<R)), consisting of all
[
f(z)=\sum a_n z^n,}^{\infty
]
for which
[
|f|r=\sum_n |a_n|r^n<\infty.
]
(\mathfrak{B}_r) is a normed ring. The space (\mathfrak{B}_r^{*}) may be realized as the set of functions
[
\Phi=\Phi(\zeta)=\sum\quad (|\zeta|>r)}^{\infty} c_n\zeta^{-n-1
]
with
[
|\Phi|r=\sup_n \frac{|c_n|}{r^n}<\infty .
]
The action of the functional is given by the formula
[
\Phi(f)=\sum_n a_n c_n
=\frac{1}{2\pi i}\int\Phi(\zeta)f(\zeta)\,d\zeta
]
(in the case where the integral does not exist, its value is taken to be equal to (\Phi(f))). (\mathfrak{A}_r) and (\overline{\mathfrak{A}}_r) are spaces of analytic functions in (|z|<r), respectively (|z|\leq r), with their usual topologies.

The space (\mathfrak{B}r) possesses in the domain (|z|\leq r) a reproducing family of functionals
[
{\Phi_z},}=\left{\frac{1}{\zeta-z}\right}_{|z|<r
]
so that (\Phi_z(f)=f(z)) for (f\in\mathfrak{B}_r) and (|z|\leq r). Moreover, for any (|z|<r) and natural (n\geq 1), there exists a linear functional
[
\Phi_z^{(n)}=\frac{n!}{(\zeta-z)^{n+1}}
]
such that
[
\Phi_z^{(n)}(f)=f^{(n)}(z).
]

To every bounded linear operator (B), (B\mathfrak{B}r\subset\mathfrak{B}_r), there corresponds a kernel
[
B^{}\Phi_z=B^{}\frac{1}{\zeta-z}=\mathfrak{B}(z,\zeta)
=\sum_n\sum_m \varepsilon}z^m\zeta^{-n-1
]
such that:

1) (\mathfrak{B}(z,\zeta)), for (|z|\leq r), belongs to (\mathfrak{B}_r^{*});

2) (\mathfrak{B}(z,\zeta)\in\mathfrak{B}_r\subset\mathfrak{B}_r^{**}) for (|\zeta|>r);

3)
[
\sup_n\sum_m |\varepsilon_{mn}|r^{m-n}=|B|_r;
]

4) for all (f(z)\in\mathfrak{B}r) and (\Phi(\zeta)\in\mathfrak{B}_r^{}) we have
[
Bf(z)=\frac{1}{2\pi i}\int_{|\zeta|=r}\mathfrak{B}(z,\zeta)f(\zeta)\,d\zeta,\qquad
B^{}\Phi(\zeta)=\frac{1}{2\pi i}\int}\mathfrak{B}(z,\zeta)\Phi(z)\,dz, \tag{1
]
where the integrals are understood as the actions of functionals represented by the corresponding kernel.

Consider in (\mathfrak{B}_r), (r\in(0,R)), the operator (A) of multiplication by a function (p(z)), analytic in (|z|<R). Exclude the discrete set of values (r) of the interval ((0,R)) for which (p(re^{i\theta})) vanishes. For the remaining values

of the interval ((0,R))

[
p(z)=\prod_{i=1}^{k_r}(z-z_i)^{\nu_i}q_r(z)
\quad
\left(|z_i|