数理解析研究所講究録 1253 巻 2002 年 152-163

152
INVERSES OF AFAMILY OF
BOUNDED LINEAR OPERATORS
群馬 大 工 斎 藤 三 郎 (Saburou Saitoh)$\text{・}$

Department of Mathematics, Faculty of Engineering
Gunma University, Kiryu 376-8515, Japan
E-mail : ssaitoh@math.sci.gunma-u.ac.jp

Abstract

We considered ageneralization of the Pythagorean theorem with geometric meanings and from the generalization it seems that we were able toobtain ageneral and fundamental concept for the inversion of afamily ofbounded linear operators on aHilbert space into various Hilbert spaces.
After reviewing the applications to linear transforms in the fameworkof Hilbert spaces of the general theory of reproducing kernels, we shallstate the results for the case of operator versions. In the last, we shall add
the prototype example and meaning of the operator versions by figures,
which show clearly ageneralization of the Pythagorean theorem.

1.

REPRODUCING KERNELS

We consider any positive matrix $K(p, q)$ on $E$ ;that is, for an abstract set $E$
and for acomplex-valued function $K(p, q)$ on $E\cross E$ , it satisfies that for any
finite points
of $E$ and for any complex numbers,$\{p_{j}\}$$\sum_{j}\sum_{j’}C_{j}\overline{C_{j’}}K(p_{j’},p_{j})\geqq 0$$\{C_{j}\}$.

Then, by the fundamental theorem by Moore-Aronszajn, we have:

Proposition 1.1([1]) For any positive matrix $K(p, q)$ on $E$ , there exists
uniquely determined functional Hilbert spacecomprising functionson$E$ andadmitting the reproducing kernel $K(p, q)$ (RKHS) satisfying andcharacterized by$a$$H_{K}$$\{f\}$$H_{K}$

K(., $q)\in H_{K}$ for any q $\in E$

(1.1

153
and, for any $q\in E$ and for any $f\in H_{K}$$f(q)=(f(\cdot), K(\cdot, q))_{H_{K}}$

(1.2)

.

For some general properties for reproducing kernel Hilbert spaces and forfrom apositive matrix $K(p, q)$ , see thevarious constructions of the RKHS
recent book [14] and its Chapter 2, Section 5respectively.
$H_{K}$

2.

CONNECTION WITH LINEAR TRANSFORMS

Let us connect linear transforms in the framework of Hilbert spaces with reproducing kernels ([7]).
For an abstract set $E$ and for any Hilbert (possibly finite-dimensional) space$H$ , we shall consider an $H$ -valued function
on $E$
$h$

$h$

:

(2.1)

$Earrow H$

and the linear transform for $H$$f(p)=(f, h(p))_{H}$for$f\in H$

(2.2)

into alinear space comprising functions on $E$ . For this linear transform (2.2),
we form the positive matrix $K(p, q)$ on $E$ defined by$K(p, q)=(h(q), h(p))_{H}$on $E\cross E$ .

(2.3)

Then, we have the following fundamental results:

admitting the reproducing kernel $K(p, q)$ defined by(I) For the RKHS
(2.3), the images $\{f(p)\}$ by (2.2) for $H$ are characterized as the members of$H_{K}$

the RKHS$H_{K}$.

(II) In general, we have the inequality in (2.2)$||f||_{H_{K}}\leqq||f||_{H}$

(2.4)

,

however, for any $f\in H_{K}$ there exists auniquely determined $f^{*}\in H$ satisfying$f(p)=(f^{*}, h(p))_{H}$

on

$E$

(2.5)

and
$||f||_{H_{K}}=||f^{*}||_{H}$

.

(2.6)

In (2.4), the isometry holds if and only if $\{h(p);p\in E\}$ is complete in $H$ .
(III) We can obtain the inversion formula for (2.2) in the form
$farrow f^{*}$

,

(2.7)

154
by using the RKHS

$H_{K}$.

However, this inversion formula will depend on, case by case, the realizationsof the RKHS.
$H_{K}$(IV) Conversely, if we have an isometric mappingfrom aRKHS
admitting areproducing kernel $K(p, q)$ on $E$ onto aHilbert space $H$ , then the
mapping is linear and its isometrical inversion
is represented in the form
(2.2). Here, the Hilbert space $H$ -valued function satisfying (2.1) and (2.2)
is given by$\tilde{L}$$H_{K}$$\tilde{L}^{-1}$$h$$h(p)=\tilde{L}K(\cdot,p)$

on

(2.8)

$E$

and, then $\{h(p);p\in E\}$ is complete in H.

When (2.2) is isometrical, sometimes we can use the isometric mapping forarealization of the RKHS
, conversely–that is, if the inverseof thelinear transform (2.2) is known, then we have $||f||_{H_{K}}=||L^{-1}f||_{H}$ .
We shall state some general applications of the results
to several
wide subjects and their basic references:$H_{K}$$L^{-1}$$(\mathrm{I})\sim(\mathrm{I}\mathrm{V})$(1) Linear transforms ([7],[11]).

The fact that the image spaces of linear transforms in the frameworkof Hilbert spaces are characterized as reproducing kernel Hilbert spaces
defined by (2.3) is the most important one in the general theory of repr0ducing kernels. Therefore, the fact will mean that the theory of repr0ducing kernels is fundamental and ageneral concept in mathematics. To
look for the characterization of the image space is astarting point when
we consider the linear equation (2.2). (II) gives ageneralization of the
Pythagorean theorem (see also [6]) and means that in the general linear
mapping (2.2) there exists essentially an isometric identity between the
input and the output. (Ill) gives ageneralized (natural) inverse (solution) of the linear mapping (equation) (2.2). (IV) gives ageneral method
determining and constructing the linear system from an isometric relation between outputs and inputs by using the reproducing kernel in the
output space.
(2) Integral transforms among smooth functions ([18]).

We considered linear mappings in the framework of Hilbert spaces, however, we can consider linear mappings in the framework of Hilbert spaces
comprising smooth functions, similarly. Conversely, reproducing kernel
Hilbert spaces are considered as the images of some Hilbert spaces byconsidering some decomposed representations (2.3) of the reproducing
kernels. Such decomposition is, in general, possible. This idea is important in [18] and also in the following items (6) and (7)155(3) Nonharmonic integral transforms ([8]).

If the linear system vectors $h(p)$ move in asmall way (perturbation of
the linear system) in the Hilbert space $H$ , then we can not calculate
the related positive matrix (2.3), however, we can discuss the inversion
formula and an isometric identity of the linear mapping. The prototype
result is the Paley-Wiener theorem on nonharmonic Fourier series.
(4) Various norm inequalities $([8],[12])$ .

Relations among positive matrices correspond to those of the associatedreproducing kernel Hilbert spaces, by the minimum principle. So, we
can derive various norm inequalities among reproducing kernel Hilbert
spaces. We were able to derive many beautiful norm inequalities.
(5) Nonlinear transforms ([12], [15]).

In avery general nonlinear transform of areproducing kernel Hilbert
space, we can look for anatural reproducing kernel Hilbert space containing the image space and furthermore, we can derive anatural nottn
inequality in the nonlinear transform. How to catch nonlinearity in connection with linearity? It seems that the theory of reproducing kernels
gives afundamental and interesting answer for this question.
(6) Linear integral equations ([19]).

(7) Linear differential equations with variable coefficients ([19]).

In linear integr0-differential equations with general variable coefficients,
we can discuss the existence and construction of the solutions, if thesolutions exist. This method is called abackward transformation method
and by reducing the equations to Fredholm integral equations of the first
type $-(2.2)-$ and we can discuss the classical solutions, in very generallinear equations.
(8) Approximation theory $([3],[2])$ .

Reproducing kernel Hilbert spaces are very nice function spaces, because
the point evaluations are continuous. Then, the reproducing kernels areafundamental tool in the related approximation theory.
(9) Representations of inverse functions ([13]).

For any mapping, we discussed the problem of representing its inversein term of the direct mapping and we derived aunified method for thisproblem. As asimple example, we can represent the Taylor coefficientsof the inverse of the Riemann mapping function on the unit disc on thecomplex plane in terms of the Riemann mapping functions. This fact
was important in the representation of analytic function in terms of local
data in ([22],[23])156(10) Various operators among Hilbert spaces ([16]).

Among various abstract Hilbert spaces, we can introduce various operators of sum, product, integral and derivative by using the linear mapping(2.2) or very general nonlinear transforms. The prototype operator isconvolution and we discussed it from awide and general viewpoint withconcrete examples.
(11) Sampling theorems ([14], Chapter 4, Section 2; [5]).

The Whittaker-KoteFnikov-Shannon sampling theorem may be interpretated by (I) and (II), very well and we can discuss the truncation errors inthe sampling theory. J. R. Higgins [5] established afully general theory
for [14].

(12) Interpolation problems of Pick-Nevanlinna type $([8],[9])$ .

General and abstract theory of interpolation problems of Pick-Nevanlinna
type may be discussed by using the general theory of reproducing kernels.
(13) Analytic extension formulas and their applications ([20],[10]).

We were able to obtain various analytic extension formulas and theirapplications from various isometrical identities (II). For their applications
to nonlinear partial differential equations, see the survey article by N.
Hayashi [4].
In this survey article, we shall present also new results on(14) Inversions of afamily of bounded linear operators on aHilbert space into
various Hilbert spaces,

which are generalizations of [21] and [6].

3.

OPERATOR VERSIONSWe shall give operator versions of the fundamental theory (I)whichmay be expected to have many concrete applications. In particular, for fullgeneralizations of the Pythagorean theorem with geometric meanings, see [6].
Some special versions were given in [21].
For an abstract set , we shall consider an operator-valued functionon$\sim(\mathrm{I}\mathrm{V})$$\Lambda$$L_{\lambda}$$\Lambda$

,

A

(3.1)

$arrow L_{\lambda}$

whereare bounded linear operators from aHilbert space H into various,
Hilbert spaces$L_{\lambda}$$\mathrm{H}_{\lambda}$$L_{\lambda}$:H$arrow \mathrm{H}_{\lambda}$

.

(3.2)

In particular, we are interested in the inversion formula$L_{\lambda}xarrow x$,$x\in H$

.

(3.1)

157
;A
Here, we consider {} as informations obtained from and we wish
belongto determine from the informations. However, the informations, and so, in order to unify the informations in a
to various Hilbert spaces
and consider the linear mapping
sense, we shall take fixed elements$\in\Lambda$$L_{\lambda}x$$x$$L_{\lambda}x$$x$$\mathrm{H}_{\lambda}$$\mathrm{b}_{\lambda,\iota v}\in \mathrm{H}_{\lambda}$

from $H$$X_{\mathrm{b}}(\lambda, \omega)$$=$$(L_{\lambda}x, \mathrm{b}_{\lambda,\mathrm{t}v})_{\mathrm{H}_{\lambda}}$$=$$(x, L_{\lambda}^{*}\mathrm{b}_{\lambda,\omega})_{H}$

,

(3.4)

$x\in H$, weinto alinear space comprising functions on A4). For the informations
as observations (measurements, in fact) for depending
shall consider
on Aand . For this linear transform (3.4), we form the positive matrix
defined byon$L_{\lambda}x$$x$$X_{\mathrm{b}}(\lambda, \omega)$$\omega$$\mathrm{A}\cross\Omega$$K_{\mathrm{b}}(\lambda, \omega;\lambda’, \omega’)$$K_{\mathrm{b}}(\lambda,\omega;\lambda’, \omega’)$Then, as in (I)$=$$(L_{\lambda}^{*},\mathrm{b}_{\lambda’,\omega’}, L_{\lambda}^{*}\mathrm{b}_{\lambda,\omega})_{H}$$=$$(L_{\lambda}L_{\lambda}^{*},\mathrm{b}_{\lambda’,\omega’}, \mathrm{b}_{\lambda,\mathrm{t}v})_{\mathrm{H}_{\lambda}}$o$\mathrm{A}\cross\Omega$$\mathrm{n}$

.

(3.5)

, we have the following fundamental results:$\sim(\mathrm{I}\mathrm{V})$deadmitting the reproducing kernel(F) For the RKHSby (3.4) for $H$ are characterized as the
fined by (3.5), the images$K\mathrm{b}(\lambda,\omega;\lambda’,\omega’)$$H_{K}\mathrm{b}$$\{X_{\mathrm{b}}(\lambda, \omega)\}$members of the RKHS$(\mathrm{I}\mathrm{I}’)$$H_{K}\mathrm{b}$.

In general, we have the inequality in (3.4)$||X_{\mathrm{b}}||_{H\kappa_{\mathrm{b}}}\leqq||x||_{H}$however, for any
fying

(3.6)

,

$X_{\mathrm{b}}\in H_{K}\mathrm{b}$there exists auniquely determined $x’\in H$ satis-$X_{\mathrm{b}}(\lambda, \omega)=(x^{l}, L_{\lambda}^{*}\mathrm{b}_{\lambda,\omega})_{H}$

on

(3.7)

$\mathrm{A}\cross\Omega$

and$||X_{\mathrm{b}}||_{H\kappa_{\mathrm{b}}}=||x||_{H}’$

.

(3.8)

In (3.6), the isometry holds if and only if
in $H$ .

$\{L_{\lambda}^{*}\mathrm{b}_{\lambda,\omega}; (\lambda, \omega)\in \mathrm{A}\cross\Omega\}$

is complete(III’) We can obtain the inversion formula for (3.4) and so, for the mapping(3.3) as in (III), in the form
$L_{\lambda}xarrow(L_{\lambda}x, \mathrm{b}_{\lambda,\omega})_{\mathrm{H}_{\lambda}}=X\mathrm{b}(\lambda,\omega)arrow x’$by using the RKHS

$H_{K}\mathrm{b}$

(3.9)

,

.

Conversely, if we have an isometric mappingon
mitting areproducing kernel$(\mathrm{I}\mathrm{V}’)$$K_{\mathrm{b}}(\lambda, \omega;\lambda’, \omega’)$$\tilde{L}$adfrom aRKHS
in the form (3.5) usin
$H_{K}\mathrm{b}$$\mathrm{A}\cross\Omega$$\mathrm{g}$158bounded linear operatorsand fixed vectors
onto aHilbert space H,then the mapping is linear and the isometric inversion
is represented in
the form (3.4) by using$L_{\lambda}$$\mathrm{b}_{\lambda,\omega}$$\tilde{L}$$L_{\lambda}^{*}\mathrm{b}_{\lambda,\omega}=\tilde{L}K_{\mathrm{b}}(\cdot,$

Further, then$\tilde{L}^{-1}$.;$\lambda,\omega)$on$\Lambda\cross\Omega$$\{L_{\lambda}^{*}\mathrm{b}_{\lambda,\omega}; (\lambda,\omega)\in \mathrm{A}\cross\Omega\}$.

(3.10)

is complete in H.

Acknowledgments
The author wishes to express my thanks Professor J. Kawabe for his kind
invitation to the research meeting.

References[1] N. Aronszajn. Theory of reproducing kernels. I}uns. Amer. Math. Soc,
68(1950), 337-404.
[2] D.-W. Byun and S. Saitoh. Approximation by the solutions of the heat
equation. J. Approximation Theory, 78 (1994), 226-238.
[3] D.-W. Byun and S. Saitoh. Best approximation in reproducing kernel
Hilbert spaces. Proc. of the 2th International Colloquium on Numerical
Analysis, $VSP$-Holland, (1994), 55-61.
[4] N. Hayashi. Analytic function spaces and their applications to nonlinear
evolution equations. Analytic Extension Formulas and their Applications,(2001), Kluwer Academic Publishers, 59-86.

[5] J. R. Higgins. Asampling principle associated with Saitoh’s fundamental
theory of linear transformations. Analytic Extension Formulas and their
Applications, (2001), Kluwer Academic Publishers, 73-86.
[6] Th. M. Rassias and S. Saitoh. The Pythagorean theorem and linear mappings.
American Math. J. (to appear).
[7] S. Saitoh. Hilbert spaces induced by Hilbert space valued functions. Proc.
Amer. Math. Soc, 89 (1983), 74-78.

[8] S. Saitoh. Theory of Reproducing Kernels and its Applications. Pitman
Research Notes in Mathematics Series, 189(1988), Longman Scientific&
Technical UK.

[9] S. Saitoh. Interpolation problems of Pick-Nevanlinna type. Pitman Research Notes in Mathematics Series, 212(1989), 253-262

159[10] S. Saitoh. Representations of the norms in Bergman-Selberg spaces on
strips and half planes. Complex Variables, 19 (1992), 231-241.

[11] S. Saitoh. One approach to some general integral transforms and its applications. Integral Transforrms and Special Functions, 3(1995), 49-84.
[12] S. Saitoh. Natural norm inequalities in nonlinear transforms. General
Inequalities 7, (1997), 39-52. Birkhauser Verlag, Basel, Boston.

[13] S. Saitoh. Representations of inverse functions. Proc. Amer. Math. Soc,
125 (1997), 3633-3639.
[14] S. Saitoh. Integral Transforms, Reproducing Kernels and their Applications. Pitman Research Notes in Mathematics Serries, 369 (1997). Addison
Wesley Longman, UK.

[15] S. Saitoh. Nonlinear transforms and analyticity of functions. Nonlinear
Mathematical Analysis and Applications, (1998), 223-234. Hadronic Press,
Palm Harbor.
[16] S. Saitoh. Various operators in Hilbert space induced by transforms. Intemational J. of Applied Math., 1(1999), 111-126.

[17] S. Saitoh. Applications of the general theory of reproducing kernels. Reproducing Ke nels and their Applications, (1999), Kluwer Academic Publishers, 165-188.
[18] S. Saitoh and M. Yamamoto. Integral transforms involving smooth functions. Reproducing Ke nels and their Applications, (1999), Kluwer Academic Publishers, 149-164.
[19] S. Saitoh. Linear integr0-differential equations and the theory of repr0ducing kernels. Volterrra Equations and Applications. C. Corduneanu and
I.W. Sandberg (eds), Gordon and Breach Science Publishers (2000), Amsterdam.
[20] S. Saitoh. Analytic extension formulas, integral transforms and reproducing kernels. Analytic Extension For mulas and their Applications, (2001),
Kluwer Academic Publishers, 207-232.
[21] S. Saitoh. Applications of the reproducing kernel theory to inverse problems. Comm. Korean Math. Soc, 16 (2001), 371-383.

[22] S. Saitoh and M. Mori. Representations of analytic functions in termsof local values by means of the Riemann mapping function. Complex
Variables, (to appear).

[23] S. Saitoh. Principle of telethoscope. Proceedings of the Graz Workshop on”Functional-analytic and complex methods, their interaction and applications to Partial Differential Equations” (to appear).

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em

160

.

IEl
—-

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$\mathrm{f}_{1}(\chi_{\mathfrak{l}y}x_{2})$—-$f_{2}\zeta\#\mathrm{l}$$\mathfrak{z}\not\subset)$—-$j_{3}(\mathit{2}_{1J}\mathit{2}_{\mathit{2}})$$\mathrm{V}\lambda(_{i}\mathrm{d}$$\alpha\tau \mathrm{e}$$\mathrm{f}_{3}$$\mathrm{Y}$es$f_{0\Gamma}$$\xi_{\lambda}$

vt$\mathrm{C}\mathrm{t}$$\iota^{1}\mathrm{o}\mathrm{m}S$$f_{1}$$’+_{2y}$$\mathrm{P}^{\mathrm{e}\mathrm{c}^{-}\mathrm{t}_{i}}\nu\Leftrightarrow$$\backslash -$$(\delta$$X_{j’}$$2_{j}$$\mathrm{s}\mathrm{o}\mathrm{m}\mathrm{e}$$\overline{-}$$\underline{-}$$[[$$\mathrm{K}$($\mathrm{K}_{J}$$|\lfloor_{J}$)$\mathrm{X}$$J$$-(\kappa_{J}1\mathrm{b}_{\mathrm{j}})$$–$$1\mathrm{b}_{J}\backslash ||$.

$t$ $||\mathrm{b}_{l}+\mathrm{t}_{\mathrm{a}}|\mathrm{b}_{qr}$

161

,

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|

k

,

$\alpha\prime r\backslash$$1\Leftrightarrow+_{\mathrm{W}\mathrm{C}\mathrm{C}f\mathrm{L}}$

,

$\{*$

$\mathrm{C}+_{\mathrm{o}\mathrm{v}^{\backslash }S}$$\infty$ka$\mathfrak{n}$.

$\alpha r$

.

–

(Y

$\nearrow[\mathrm{b}_{J}\cdot$)$\mathrm{A}_{\mathrm{J}}\backslash \}.\dagger \mathrm{L}$${ }$\lrcorner $=[$

e to r$0?^{\mathrm{e}_{\mathrm{Y}^{\wedge}\alpha}^{- \mathrm{t}_{\mathrm{o}r}}}\mathrm{v}\mathrm{e}r$$\mathrm{s}\acute{\iota}\Lambda \mathrm{e}r$$\{\not\equiv_{\sqrt}\}_{\grave{\mathrm{J}}|}\mathfrak{n}=a$$\mathit{5}1\mathrm{o}m$$s$3$\iota’\grave{\mathrm{n}}\prec \mathrm{h}\mathrm{e}$

1

$|S’$

$\vee\cdot\lambda_{\mathrm{C}\sim \mathrm{L}}+_{\iota^{\backslash }}\mathrm{t}_{\mathrm{J}}$$\mathrm{t}\grave{\iota}\mathrm{S}$0e$\mathrm{c}a_{\mathrm{U}}$$\mathrm{V}^{-}\mathrm{I}\backslash \triangleright$

X
tk a

e

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form .
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163

K —-

$\uparrow 1$

K$+\cdot$$K_{1}$$\mathrm{K}_{2}$r7e、$\lambda$

$–$

$+$

$\mathrm{r}_{2}\pi_{[}$$\overline{-}$$\mathrm{K}_{[}$

2 ノゝ$\kappa_{Q}$村。

$\uparrow|$$\overline{-}$$+\cdot$$\mathrm{K}_{3}$

$d$

$1$
$\mathrm{e}_{\nearrow}\sim\backslash \backslash |\cdot\sim\backslash 1$

乞$\mathfrak{n}$

$e$
。
、

$\mathrm{m}\epsilon \mathrm{e}$$j$

伽 $S_{j}$$\grave{\iota}\mathit{5}$$4_{\mathrm{e}}+_{\epsilon\Gamma’\mathrm{W}\mathrm{t}1\mathfrak{n}\mathrm{e}\mathrm{X}\mathrm{t}}^{\phi?^{\mathrm{J}_{\alpha Y}}}n\mathrm{v}\backslash \wedge,\mathrm{r}\backslash \mathrm{Y}^{-}\mathrm{o}\mathrm{c}\mathrm{e}$$\mathrm{t}P\iota\grave{\psi}\in’ \mathrm{w}_{\grave{\iota}\mathit{5}}‘ \mathrm{L}\not\subset_{\mathfrak{O}}\mathrm{f}_{\grave{\iota}\prime\nu\backslash }\prime \mathrm{t}_{\mathrm{e}}\mathrm{t}_{ff}$$\mathrm{A}S$$\grave{\iota}^{\gamma 1}\partial-$$\oint-$$\mathrm{s}\mathrm{t}\mathrm{e}$$\mathrm{s}\mathrm{t}\mathrm{a}$$\circ+_{\mathrm{c}^{\backslash }o}$$+_{\omega 0}\uparrow^{Y^{\backslash }\mathrm{O}}\mathrm{J}^{\backslash }\mathrm{e}$り$s_{j}$『$\mathrm{b}_{\delta^{\backslash }}$Vゝ .