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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^ •;,

i



ir-m-NYU 312

July, 1965



NEl«/ 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

§3 - Solution of the Friedrichs' equation for locally

gentle perturbations of finite rank 11-21

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

s

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

S6 - Other perturbations 49-54

o

§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 •'-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 5» 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
•'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

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

Let F be a densely defined possibly unbouiided form on
«^£^- 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 £
strictly selfadjoint operator with continuous spectrvim. Let
[5,, •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^ „ is also densely defined.

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

E^G is in C^ • 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, ^ ) •



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 £'-lx,y; - ^, _ y ^ ^_

and define

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

e— >+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> .^, £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^) = / — ^ eMir) -1- i7r(fp(x),gp(x)) .

' ^ y-x '^ '^



2.7 ^ ^°

Finally v;e shall neeci the following generalization of
the notion of a locall;- £;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 _„^

such that the product of "t^^®.. 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, {• ),k,(- ))(x)



(k.(y),k.(y)). . ;
(3-7) = 5.. , -1- 7.. { —^ —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 £, -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
■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 • Therefore, the functions

? h. ( defined by equation (3«5) are measurable and hence
the form Q defined by (30) is in (J7 ^„^-

The argujnents leading to equations (3' 3) snd (3-S) show
that in order that a form Q in Cb^ „ 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, >>,


1 3 4 5

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