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IMM-NYU 312

JULY, 1963

NEW YORK UNIVERSITY

COURANT INSTITUTE OF

MATHEMATICAL SCIENCES

On Gentle Perturbations, 11

p. A. REJTO

ST

I '

n

z

H

PREPARED UNDER

CONTRACT NO. DA-ARO-(D)-31-124-Gl59

WITH THE

U.S. ARMY RESEARCH OFFICE

NEW YORK UNIVERSITY ;

COURANT INSTITUTE - LIBRARY ,

^ Washington Place, New Yorfc 3, N. Y^ â€¢;,

i

ir-m-NYU 312

July, 1965

NElÂ«/ YOBK UNIVERSITY

COURANT INSTITUTE OF MATHEMATICAL SCIENCES

ON GENTLE PERTURBATIONS, II*

P. A. Rejto

The research reported in this document has been sponsored "by

the Department of the Army, U.S. Army Research Office (Durham),

Project Number 983-M under contract No. DA-AR0-(D)-31-124-G159.

Reproduction in whole or in part is permitted for an^r purpose

of the U.S. Government.

This vrork is an extended version of the report presented at

the 601st meeting of the American Mathematical Society in

Nev; York, April, I963.

ABSTRACT

This report Is based on the observation that the gentleness

condition defined with the aid of a Holder condition is ''local''

and that for perturbations of finite rank the Friedrichs'

equation admits a ''formal solution''. This formal solution

always can be interpreted as a densely defined, possible

unbounded bilinear form. Then, roughly speaking, a solution

which is gentle over some interval plays the same role in

constructing the spectral transformation of the part of the

aerator over this Interval, as a gentle solution did in

constructing the spectral transformation of the entire

operator.

ii

TABLE OF CONTENTS

Si - Introduction. 1-5

S2 - The space of locally gentle bilinear forms 4-10

Â§3 - Solution of the Friedrichs' equation for locally

gentle perturbations of finite rank 11-21

Â§4 - Locally gentle perturbations of rank one 22-50

s

Â§5 - Locally gentle perturbations of finite rank Jl-'^S

S6 - Other perturbations 49-54

o

Â§7 - Applications 55-67

Appendix 1 69-77

Appendix II 78-86

Appendix III 87-89

Appendix IV 90-95

Bibliography 94-98

1-1

*^1

INTRODUCTION

It is an interesting problem to ask for conditions which

ensiire that two operators acting on an abstract Hilbert space

are unitarily equivalent. One expects that this is the case if

their difference is small in some appropriate norm and thus one

reformulates the question by asking which norms are appropriate?

First this question was considered by von Neumann, [CI], who

investigated the Hilbert-Schmidt norm and showed it to be

inappropriate. Later, Friedrichs, [B2], gave a very simple

example showing that an arbitrary small perturbation of rank

one can produce pointeigenvalues. Hence there is no lanitarily

invariant crossnorm which is appropriate.

In spite of this large variety of inappropriate norms,

he also introduced, [B2], an appropriate norm and called it a

gentleness norm.

In this report we introduce the notion of an operator being

gentle over an interval. For brevity, we shall refer to this

interval as a gentleness interval and to the condition as local

gentleness. This notion of local gentleness, which will be

described in Section 2, is very general and at present we do

not try to work with it. Instead, in Section 2, we also introduce

a rather special space of locally gentle operators whose elements

1-2

are, in particular, operators of finite rank. Then we shall show t

that for such locally gentle perturbations, the Friedrichs method

can be applied to construct a spectral transformation for the part

of the perturbed operator over a gentleness interval of the

perturbation. Here and in the following we call the restriction

of an operator to its eigenspace .aasociated with a given interval,

the part of the operator over the interval. In our construction

we shall combine the gentleness considerations with the resolvent

loop integral formula for the spectral projectors, [E2,c]. The

important role of this formula in connection with gentleness

considerations was emphasized recently by J. Schwartz, [B5], and

according to a verbal communication of L.D. Faddeev, it will be

emphasized in his forthcoming paper.

In Section 3, we shall introduce a sufficient condition in

order that the Friedrichs' equation admits a locally gentle

solution for locally gentle perturbations of finite rank. We

shall see that, in general, this solution will be a â€¢'-deri;sely defined

bilinear form. In Section k, we shall consider locally gentle

perturbations of rank one and shall show the following: Under

the conditions of Section 3, a spectral transformation of the

part of the disturbed operator over a gentleness interval can be

constructed with the aid of the solution to the Friedrichs'

equation. We shall establish this by explicitly evaluating the

3

1-3

resolvent loop integral. In Section 5Â» we shall show that the

results of Section k concerning locally gentle perturbations

of rank one are typical for perturbations of finite rank. This

will be the statement of the basic Theorem 5.I, of which we shall

give several ramifications. In Section 6, we shall combine the

theorems of Section 5 with ''. Friedrichs' original theorem on

â€¢'small perturbations''. This will give theorems on perturbations

which can be written as the surn of a gentle perturbation and a

locally gentle perturbation of finite rank. This will yield a

generalization of the main theorem of the previous paper, [b6].

In Section 7, we shall illustrate how o\ir abstract theorems apply

to certain differential operators.

Finally let us mention that the gentleness norm can be

replaced by a unitarily invariant cross norm in order to draw

the following slightly weaker conclusion: The absolutely

continuous parts of the perturbed and unperturbed operators are

unitarily equivalent. For, T. Kato, [A2,A5] has shown, using a .

lemma of M. Rosenblum [Al], that this is the case for trace class

perturbations. Then M.G. K^ein and M.S. Birman, [a6], showed

that this also holds for relative trace class perturbations. In

Appendix II, we shall illustrate the connection between the Kato

theorem and the gentleness considerations by establishing a

special case of his theorem using gentleness arguments.

2.1

The space of locclly gentle bilinear forms.

Originally the notion of gentleness was defined for

operators by first defining it for kernels and then saying

that an operator is gentle if it admits a gentle kernel. We

shall proceed analogously in defining the notion of a locally

gentle bilinear form. First, hov/ever, let us formulate some

, â€” â– '

definitions concerning forms on an abstract Hilbert space ')\ .

Let F be a densely defined possibly unbouiided form on

Â«^Â£^- X cZvp, ffnd let B be a bounded operator. Then we

define BF, the product of the operator E and the form F to be

the form determined by EF[f,g] = F[B f,g], where B is the

adjoint of B. This form is defined for those vectors f for

which B f is in Cl")-)* Since the range of a bounded operator

need not be closed, and the intersection of two dense sets

may be empty, it may happen that the form EF is defined for

f = only. Similarly we define the form FE by setting

FB[f,g] = F[f,Bg]. Finally recall that P*, the adjoint of

the form F, is defined by F*[f,g] = F[g,f].

Next we turn to the description of the space of locally

gentle bilinear forms:

1. Let n be a separable Hilbert space and A on ^ in H Â£

strictly selfadjoint operator with continuous spectrvim. Let

[5,, â€¢3^] be a given interval and E^^ the spectral projector of

A over o.

:\XK

2.2

2. Let C^ be s collection of densely defined possibly

unbOTjjnded forms on V] such that: ;? ) The domain of each

G in CS> ^ is of the form cID ^ ^ > where \X) is dense

and possibly dependent on G. b) The finite sum of forms in

(^S^ â€ž is also densely defined.

3. Suppose that for arbitrary E- and G in Ci?,,,^ the form

E^G is in C^ â€¢ Moreover there is a transformation I ^

a ux

which assigns to a form G in (^ ^ snother densely defined

form I 'g, possibly not in Ci

h. Let (a/[ J-j_, 5^], ^ , n )

Let [j-ijOp] be z given bounded interval. Then \ie first define

the space (^ ^ to be the set of those measure ble 2-forms

c ux

of finite rank which can be written as

G = T~ h; > < g; g; e JToOv, ^ ) â€¢

Next define (5p {a/[oj^,d^], X , H ) to be the subset of

(3z> consisting of those forms to which there is an interval

[6,, 5"^] O [j-,, j^], such thct the form is c-Holder continuous

on [o-,,5p]; that is the functions -^ h^ ^ and ^g^^ sre a-Holder

continuous. Note thet in view of our extension convention,

in the special case S, = [3-i,^o]> this condition requires in

particular that the functions vanish at the end points- To

complete the gentleness structure set

n rfr v1 - G(x,y)

(2.1) I Â£'-lx,y; - ^, _ y ^ ^_

and define

< f, r Gg > = lim < f, P^Gg > , for f,g C o[ g^^n^*

eâ€” >+o

2.6

Finally in cese [5,,2p] is an unbounded interval define

the form G to be gentle over f^^^/Jg] if it is gentle over any

bounded sub interval. In Appendix I, we shall show that the

quadruplet M, p, Cl> .^, Â£nd CS io./[o^,f.^], X , ^7 ) de-

fines a space of gentle forms over the interval [o,,Cp]; that

is we shall establish the validity of the propositions stated

in the beginning of this section.

Note that our space of locally gentle forms overlaps

vjith, but does not contain the corresponding space of gentle

operators. For, vre have required explicitly that our locally

gentle forms are of finite rank, v;hile a gentle operator need

not be of finite ranlc. We made this simplifying assumption

in viev; of the fact that for such forms the condition of

local gentleness and the theorems concerning it, can be

established in a particularly simple manner. Let us also

note that the definition of the form Pg is not restricted

to forms of finite rani:. Nevertheless ii' G is of finite ranlc

the form pG can be describee vrith the aid of the Hilbert

transformation, and we can obtain properties of pG from the

corresponding properties of tne Hilbert transformation. More

specifically we claim that if G = g, > < En then

(2.2) < f^, pG fg ^ = < f^_, SiH_^82i2 "

where

r (sp(y).fp(y))

H.. gpfot^) = / â€” ^ eMir) -1- i7r(fp(x),gp(x)) .

' ^ y-x '^ '^

2.7 ^ ^Â°

Finally v;e shall neeci the following generalization of

the notion of a locall;- Â£;entle function:

The space of locally polite functions ,

^(c:/[5^,62],7.A, f\ ).

Let the function p be measurable with respect to the

measure A and set as before p=0 on the exceptional set.

S\ippose that p is such that there is a point x in [S-^jS^]

and a number C,, satisfying the conditions

a) p is a-Holder continuous in the intervals [5n,x - C]>

b) for every 7 > I7I the function (x-x ) p(x) is a -Holder

continuous in [ 5, , 5p] .

Then v;e define the space ^^^(a/[ 5, , Sg] 57;X, P) ) to be

the set of finite linear combinations of such functions.

11

>"

Solution of the Friedrichs' equation for locally gentle

perturbations of finite rank.

According to the sonsiderations of Friedrichs, [B2,B7]j one

can construct a spectral transformation of the operator

M+K, with the aid of the solution of the equation

(5.1) (l.-rQ)K = Q .

Originally one was seeking a solution Q to this equation

in the class of gentle operators;, a class for which \~Qi

was a bounded operator. As mentioned earlier, this report

is centered around the observation that it is useful to

know whether this equation has a solution in the class of

locally gentle forms and we shall take up this question

in the present section. We start with a limiting case:

a ) Case of arbitrary selfad.loint perturbations of finite rank.

We maintain that for perturbations of finite rank

equation 3-1 always admits s solution Q in Ci::,,,,^- More

specifically v;e maintain that there exists a form Q m _â€ž^

such that the product of "t^^Â®.. form P'Q and the operator K,

I ^ ^^densely defined, moreover equation 3-1 holds on

a dense set.

First let us assume that this is the case and let us

.t -t

12

3-2

determine Q. It is v;ell toovm that an arbitrary selfadjoint

operator of finite rank, K, can be written as;

n

(3.2) K = ^ A ic k ,

i=l ^ ^ 1

where the fi^-T are the normalized eigenvectors and the

/a^ V are the corresponding eigenvalues. Insertion of this

fact in (3-1) yields that,

(3.3) Q = ZI h^ >< \A< k.)k. can be identified with a measurable

J J -'-

function, moreover this measurable function can be expressed

in terms of the augmented Hilbert transformation;

(3.5) Rh. >,.. H (k, {â€¢ ),k,(- ))(x)

(k.(y),k.(y)). . ;

(3-7) = 5.. , -1- 7.. { â€”^ â€”J dA(y)-i7r(k. (x),k,.(x)

v/here 5. , denotes the Kronecker sj^inbol. Hence in (3.3) and

(3-6) we have found an equation for the solution Q.

Note that the vectors k(x): ^k.(x) | , h(x): )h (x)|

are elements of the n-orthogonal sum of );(^ with itself, vjhich

v;e denote by j| , Equation (3.5) is an equation on ){ ,

A(x) maps yi into )'( and the associated matrix is a

numerical valued matrix ivhich we denote by A(x) again. Also

note tnat this matrix is defined almost everywhere only as a

Ciuchy principle value.

Next V7e claim that the form Q defined by equations (3-3)

and (3.6) is in fh , which amounts to the statement that the

'^ "-^aux

functions < h, / entering it are measurable. In order to

establish this fact v;e first note that the matrix A(x) is

measurable. For, according to the Â£, -version of the Plemelj-

14

3.4

Prlvalov theorem, [ EJ> ], this matrix is the a.e. boundary

value of the matrix

ij ij J J "^ d.A(y) .

y-z

Hence almost everywhere it is the pointwise- limit of a family of

â– measurable matrices end so it is measiArable. Next we note

that this matrix admits a.e. an inverse which is measurable

in view of the fact that A(x) is measurable. At this point we

make essential use of the fact that A(z) can be identified

with the matrix of the perturbation, (M, M+K), and according to

S.T. Kuroda, the boundary value of the perturbation matrix is

almost everywhere invertible, [ a8 1 â€¢ Therefore, the functions

? h. ( defined by equation (3Â«5) are measurable and hence

the form Q defined by (30) is in (J7 ^â€ž^-

The argujnents leading to equations (3' 3) snd (3-S) show

that in order that a form Q in Cb^ â€ž should satisfy equation

cjUX

(3.1) it is necessary that (3-5) end (3-6) should hold. The

same argument also shows that for o form Q in (^ ^ this

condition is also sufficient. Therefore, from the statement

of the previous paragraph we conclude that for an arbitrary

self adjoint perturbation of finite rank the Freidrichs' equation.

15

3-5

(3-11), admits a solution in U^ ^^,.* Most likely this is sll

that one can say about the solution in general; therefore, we

Introduce additional assumptions in the following.

b) Case of locally gentle perturbations of finite rank.

First let us assiome that the previous operator K is gentle

over some bounded interval [o, ,5p]. Then we ask the question.

Does this ensure that the solution h(x) to (3-^) is gentle over

[5,,52]? From the local gentleness of the eigenf unctions we infer

in view of (J.?) the local Holder continuity of the matrix A(x).

Now this matrix has tv;o types of exceptional points.

Those at Which the integrals deflng the the matrix ..

elements fail to exist in the sense of Cauchy's principle

value and those at which the matrix A(x) is not Invertible.

For brevity let us call them exceptional points of the first

and second type. Novr the gentleness of K over [5,,52] ensures

that the matrix A(x) has no exceptional points of the first

type in [5,,5p]. Nevertheless we maintain that exceptional

points of the second type may exist in the Interval i^-^^'o^]'

For, In the previous paper [ b6 ] we gave an example of a

gentle perturbation of rank one for which the ira trix A(x) was

not invertible, moreover the Frledrlchs' equation had no gentle

Arc)A

16

3.6

solution. V/e rise introduced a condition there, which excluded

such examples. Wow we observe that this condition is local,

for, in a slightly generalised form, it reads as follows:

Condition [ 5, , o^ ]

The set of pointeigenvelues of M+K on^'.;'-J(a, g-l, >>,

JULY, 1963

NEW YORK UNIVERSITY

COURANT INSTITUTE OF

MATHEMATICAL SCIENCES

On Gentle Perturbations, 11

p. A. REJTO

ST

I '

n

z

H

PREPARED UNDER

CONTRACT NO. DA-ARO-(D)-31-124-Gl59

WITH THE

U.S. ARMY RESEARCH OFFICE

NEW YORK UNIVERSITY ;

COURANT INSTITUTE - LIBRARY ,

^ Washington Place, New Yorfc 3, N. Y^ â€¢;,

i

ir-m-NYU 312

July, 1965

NElÂ«/ YOBK UNIVERSITY

COURANT INSTITUTE OF MATHEMATICAL SCIENCES

ON GENTLE PERTURBATIONS, II*

P. A. Rejto

The research reported in this document has been sponsored "by

the Department of the Army, U.S. Army Research Office (Durham),

Project Number 983-M under contract No. DA-AR0-(D)-31-124-G159.

Reproduction in whole or in part is permitted for an^r purpose

of the U.S. Government.

This vrork is an extended version of the report presented at

the 601st meeting of the American Mathematical Society in

Nev; York, April, I963.

ABSTRACT

This report Is based on the observation that the gentleness

condition defined with the aid of a Holder condition is ''local''

and that for perturbations of finite rank the Friedrichs'

equation admits a ''formal solution''. This formal solution

always can be interpreted as a densely defined, possible

unbounded bilinear form. Then, roughly speaking, a solution

which is gentle over some interval plays the same role in

constructing the spectral transformation of the part of the

aerator over this Interval, as a gentle solution did in

constructing the spectral transformation of the entire

operator.

ii

TABLE OF CONTENTS

Si - Introduction. 1-5

S2 - The space of locally gentle bilinear forms 4-10

Â§3 - Solution of the Friedrichs' equation for locally

gentle perturbations of finite rank 11-21

Â§4 - Locally gentle perturbations of rank one 22-50

s

Â§5 - Locally gentle perturbations of finite rank Jl-'^S

S6 - Other perturbations 49-54

o

Â§7 - Applications 55-67

Appendix 1 69-77

Appendix II 78-86

Appendix III 87-89

Appendix IV 90-95

Bibliography 94-98

1-1

*^1

INTRODUCTION

It is an interesting problem to ask for conditions which

ensiire that two operators acting on an abstract Hilbert space

are unitarily equivalent. One expects that this is the case if

their difference is small in some appropriate norm and thus one

reformulates the question by asking which norms are appropriate?

First this question was considered by von Neumann, [CI], who

investigated the Hilbert-Schmidt norm and showed it to be

inappropriate. Later, Friedrichs, [B2], gave a very simple

example showing that an arbitrary small perturbation of rank

one can produce pointeigenvalues. Hence there is no lanitarily

invariant crossnorm which is appropriate.

In spite of this large variety of inappropriate norms,

he also introduced, [B2], an appropriate norm and called it a

gentleness norm.

In this report we introduce the notion of an operator being

gentle over an interval. For brevity, we shall refer to this

interval as a gentleness interval and to the condition as local

gentleness. This notion of local gentleness, which will be

described in Section 2, is very general and at present we do

not try to work with it. Instead, in Section 2, we also introduce

a rather special space of locally gentle operators whose elements

1-2

are, in particular, operators of finite rank. Then we shall show t

that for such locally gentle perturbations, the Friedrichs method

can be applied to construct a spectral transformation for the part

of the perturbed operator over a gentleness interval of the

perturbation. Here and in the following we call the restriction

of an operator to its eigenspace .aasociated with a given interval,

the part of the operator over the interval. In our construction

we shall combine the gentleness considerations with the resolvent

loop integral formula for the spectral projectors, [E2,c]. The

important role of this formula in connection with gentleness

considerations was emphasized recently by J. Schwartz, [B5], and

according to a verbal communication of L.D. Faddeev, it will be

emphasized in his forthcoming paper.

In Section 3, we shall introduce a sufficient condition in

order that the Friedrichs' equation admits a locally gentle

solution for locally gentle perturbations of finite rank. We

shall see that, in general, this solution will be a â€¢'-deri;sely defined

bilinear form. In Section k, we shall consider locally gentle

perturbations of rank one and shall show the following: Under

the conditions of Section 3, a spectral transformation of the

part of the disturbed operator over a gentleness interval can be

constructed with the aid of the solution to the Friedrichs'

equation. We shall establish this by explicitly evaluating the

3

1-3

resolvent loop integral. In Section 5Â» we shall show that the

results of Section k concerning locally gentle perturbations

of rank one are typical for perturbations of finite rank. This

will be the statement of the basic Theorem 5.I, of which we shall

give several ramifications. In Section 6, we shall combine the

theorems of Section 5 with ''. Friedrichs' original theorem on

â€¢'small perturbations''. This will give theorems on perturbations

which can be written as the surn of a gentle perturbation and a

locally gentle perturbation of finite rank. This will yield a

generalization of the main theorem of the previous paper, [b6].

In Section 7, we shall illustrate how o\ir abstract theorems apply

to certain differential operators.

Finally let us mention that the gentleness norm can be

replaced by a unitarily invariant cross norm in order to draw

the following slightly weaker conclusion: The absolutely

continuous parts of the perturbed and unperturbed operators are

unitarily equivalent. For, T. Kato, [A2,A5] has shown, using a .

lemma of M. Rosenblum [Al], that this is the case for trace class

perturbations. Then M.G. K^ein and M.S. Birman, [a6], showed

that this also holds for relative trace class perturbations. In

Appendix II, we shall illustrate the connection between the Kato

theorem and the gentleness considerations by establishing a

special case of his theorem using gentleness arguments.

2.1

The space of locclly gentle bilinear forms.

Originally the notion of gentleness was defined for

operators by first defining it for kernels and then saying

that an operator is gentle if it admits a gentle kernel. We

shall proceed analogously in defining the notion of a locally

gentle bilinear form. First, hov/ever, let us formulate some

, â€” â– '

definitions concerning forms on an abstract Hilbert space ')\ .

Let F be a densely defined possibly unbouiided form on

Â«^Â£^- X cZvp, ffnd let B be a bounded operator. Then we

define BF, the product of the operator E and the form F to be

the form determined by EF[f,g] = F[B f,g], where B is the

adjoint of B. This form is defined for those vectors f for

which B f is in Cl")-)* Since the range of a bounded operator

need not be closed, and the intersection of two dense sets

may be empty, it may happen that the form EF is defined for

f = only. Similarly we define the form FE by setting

FB[f,g] = F[f,Bg]. Finally recall that P*, the adjoint of

the form F, is defined by F*[f,g] = F[g,f].

Next we turn to the description of the space of locally

gentle bilinear forms:

1. Let n be a separable Hilbert space and A on ^ in H Â£

strictly selfadjoint operator with continuous spectrvim. Let

[5,, â€¢3^] be a given interval and E^^ the spectral projector of

A over o.

:\XK

2.2

2. Let C^ be s collection of densely defined possibly

unbOTjjnded forms on V] such that: ;? ) The domain of each

G in CS> ^ is of the form cID ^ ^ > where \X) is dense

and possibly dependent on G. b) The finite sum of forms in

(^S^ â€ž is also densely defined.

3. Suppose that for arbitrary E- and G in Ci?,,,^ the form

E^G is in C^ â€¢ Moreover there is a transformation I ^

a ux

which assigns to a form G in (^ ^ snother densely defined

form I 'g, possibly not in Ci

h. Let (a/[ J-j_, 5^], ^ , n )

Let [j-ijOp] be z given bounded interval. Then \ie first define

the space (^ ^ to be the set of those measure ble 2-forms

c ux

of finite rank which can be written as

G = T~ h; > < g; g; e JToOv, ^ ) â€¢

Next define (5p {a/[oj^,d^], X , H ) to be the subset of

(3z> consisting of those forms to which there is an interval

[6,, 5"^] O [j-,, j^], such thct the form is c-Holder continuous

on [o-,,5p]; that is the functions -^ h^ ^ and ^g^^ sre a-Holder

continuous. Note thet in view of our extension convention,

in the special case S, = [3-i,^o]> this condition requires in

particular that the functions vanish at the end points- To

complete the gentleness structure set

n rfr v1 - G(x,y)

(2.1) I Â£'-lx,y; - ^, _ y ^ ^_

and define

< f, r Gg > = lim < f, P^Gg > , for f,g C o[ g^^n^*

eâ€” >+o

2.6

Finally in cese [5,,2p] is an unbounded interval define

the form G to be gentle over f^^^/Jg] if it is gentle over any

bounded sub interval. In Appendix I, we shall show that the

quadruplet M, p, Cl> .^, Â£nd CS io./[o^,f.^], X , ^7 ) de-

fines a space of gentle forms over the interval [o,,Cp]; that

is we shall establish the validity of the propositions stated

in the beginning of this section.

Note that our space of locally gentle forms overlaps

vjith, but does not contain the corresponding space of gentle

operators. For, vre have required explicitly that our locally

gentle forms are of finite rank, v;hile a gentle operator need

not be of finite ranlc. We made this simplifying assumption

in viev; of the fact that for such forms the condition of

local gentleness and the theorems concerning it, can be

established in a particularly simple manner. Let us also

note that the definition of the form Pg is not restricted

to forms of finite rani:. Nevertheless ii' G is of finite ranlc

the form pG can be describee vrith the aid of the Hilbert

transformation, and we can obtain properties of pG from the

corresponding properties of tne Hilbert transformation. More

specifically we claim that if G = g, > < En then

(2.2) < f^, pG fg ^ = < f^_, SiH_^82i2 "

where

r (sp(y).fp(y))

H.. gpfot^) = / â€” ^ eMir) -1- i7r(fp(x),gp(x)) .

' ^ y-x '^ '^

2.7 ^ ^Â°

Finally v;e shall neeci the following generalization of

the notion of a locall;- Â£;entle function:

The space of locally polite functions ,

^(c:/[5^,62],7.A, f\ ).

Let the function p be measurable with respect to the

measure A and set as before p=0 on the exceptional set.

S\ippose that p is such that there is a point x in [S-^jS^]

and a number C,, satisfying the conditions

a) p is a-Holder continuous in the intervals [5n,x - C]>

b) for every 7 > I7I the function (x-x ) p(x) is a -Holder

continuous in [ 5, , 5p] .

Then v;e define the space ^^^(a/[ 5, , Sg] 57;X, P) ) to be

the set of finite linear combinations of such functions.

11

>"

Solution of the Friedrichs' equation for locally gentle

perturbations of finite rank.

According to the sonsiderations of Friedrichs, [B2,B7]j one

can construct a spectral transformation of the operator

M+K, with the aid of the solution of the equation

(5.1) (l.-rQ)K = Q .

Originally one was seeking a solution Q to this equation

in the class of gentle operators;, a class for which \~Qi

was a bounded operator. As mentioned earlier, this report

is centered around the observation that it is useful to

know whether this equation has a solution in the class of

locally gentle forms and we shall take up this question

in the present section. We start with a limiting case:

a ) Case of arbitrary selfad.loint perturbations of finite rank.

We maintain that for perturbations of finite rank

equation 3-1 always admits s solution Q in Ci::,,,,^- More

specifically v;e maintain that there exists a form Q m _â€ž^

such that the product of "t^^Â®.. form P'Q and the operator K,

I ^ ^^densely defined, moreover equation 3-1 holds on

a dense set.

First let us assume that this is the case and let us

.t -t

12

3-2

determine Q. It is v;ell toovm that an arbitrary selfadjoint

operator of finite rank, K, can be written as;

n

(3.2) K = ^ A ic k ,

i=l ^ ^ 1

where the fi^-T are the normalized eigenvectors and the

/a^ V are the corresponding eigenvalues. Insertion of this

fact in (3-1) yields that,

(3.3) Q = ZI h^ >< \A< k.)k. can be identified with a measurable

J J -'-

function, moreover this measurable function can be expressed

in terms of the augmented Hilbert transformation;

(3.5) Rh. >,.. H (k, {â€¢ ),k,(- ))(x)

(k.(y),k.(y)). . ;

(3-7) = 5.. , -1- 7.. { â€”^ â€”J dA(y)-i7r(k. (x),k,.(x)

v/here 5. , denotes the Kronecker sj^inbol. Hence in (3.3) and

(3-6) we have found an equation for the solution Q.

Note that the vectors k(x): ^k.(x) | , h(x): )h (x)|

are elements of the n-orthogonal sum of );(^ with itself, vjhich

v;e denote by j| , Equation (3.5) is an equation on ){ ,

A(x) maps yi into )'( and the associated matrix is a

numerical valued matrix ivhich we denote by A(x) again. Also

note tnat this matrix is defined almost everywhere only as a

Ciuchy principle value.

Next V7e claim that the form Q defined by equations (3-3)

and (3.6) is in fh , which amounts to the statement that the

'^ "-^aux

functions < h, / entering it are measurable. In order to

establish this fact v;e first note that the matrix A(x) is

measurable. For, according to the Â£, -version of the Plemelj-

14

3.4

Prlvalov theorem, [ EJ> ], this matrix is the a.e. boundary

value of the matrix

ij ij J J "^ d.A(y) .

y-z

Hence almost everywhere it is the pointwise- limit of a family of

â– measurable matrices end so it is measiArable. Next we note

that this matrix admits a.e. an inverse which is measurable

in view of the fact that A(x) is measurable. At this point we

make essential use of the fact that A(z) can be identified

with the matrix of the perturbation, (M, M+K), and according to

S.T. Kuroda, the boundary value of the perturbation matrix is

almost everywhere invertible, [ a8 1 â€¢ Therefore, the functions

? h. ( defined by equation (3Â«5) are measurable and hence

the form Q defined by (30) is in (J7 ^â€ž^-

The argujnents leading to equations (3' 3) snd (3-S) show

that in order that a form Q in Cb^ â€ž should satisfy equation

cjUX

(3.1) it is necessary that (3-5) end (3-6) should hold. The

same argument also shows that for o form Q in (^ ^ this

condition is also sufficient. Therefore, from the statement

of the previous paragraph we conclude that for an arbitrary

self adjoint perturbation of finite rank the Freidrichs' equation.

15

3-5

(3-11), admits a solution in U^ ^^,.* Most likely this is sll

that one can say about the solution in general; therefore, we

Introduce additional assumptions in the following.

b) Case of locally gentle perturbations of finite rank.

First let us assiome that the previous operator K is gentle

over some bounded interval [o, ,5p]. Then we ask the question.

Does this ensure that the solution h(x) to (3-^) is gentle over

[5,,52]? From the local gentleness of the eigenf unctions we infer

in view of (J.?) the local Holder continuity of the matrix A(x).

Now this matrix has tv;o types of exceptional points.

Those at Which the integrals deflng the the matrix ..

elements fail to exist in the sense of Cauchy's principle

value and those at which the matrix A(x) is not Invertible.

For brevity let us call them exceptional points of the first

and second type. Novr the gentleness of K over [5,,52] ensures

that the matrix A(x) has no exceptional points of the first

type in [5,,5p]. Nevertheless we maintain that exceptional

points of the second type may exist in the Interval i^-^^'o^]'

For, In the previous paper [ b6 ] we gave an example of a

gentle perturbation of rank one for which the ira trix A(x) was

not invertible, moreover the Frledrlchs' equation had no gentle

Arc)A

16

3.6

solution. V/e rise introduced a condition there, which excluded

such examples. Wow we observe that this condition is local,

for, in a slightly generalised form, it reads as follows:

Condition [ 5, , o^ ]

The set of pointeigenvelues of M+K on^'.;'-J(a, g-l, >>,