P. A Rejto.

Some absolutely continuous operators, I online

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Some Absolutely Continuous Operators I

P. A. Rejto

\ -




September, 1964


P. A. Rejto

This work represents results obtained at the Courant
Institute of Mathematical Sciences, New York University, under
the sponsorship of the Office of Naval Research, Contract No.

Reproduction In whole or In part is permitted for any
purpose of the United States Government.

■301:R12Q j:iOITA^iHHTAr.l W JTJTITBV!! T,



The author Is indebted to Professor K, 0. Friedrichs
for his valuable criticism.

.:iiiilolil^io a.Ld.:^t .i.iiv o.trf 'lo'-.


In this report abstract conditions are formulated v/hich
assure that the part of an operator over a given interval is

absolutely continuous » Tliese conditions are verified for the

operators - — 5- + q(x) acting on the Intervals (- «, + ] •

For convenience set

A, = A + P ,
1 o

A and A, being strictly self adjoint on the same domain. For brevity
o 1

we call A and /-> perturbed and unperturbed operators respectively;

P is the perturbation. Distinguishing the corresponding resolvents by

subscripts, for non real z we have the elementary relation,

R, ( 2) - R^( z) = R, ( z) PR ( z) ,
1 o 1 o

which is called the second resolvent equation. In case ( 1 - PR ( z) )

admits an inverse this equation allows one to express R { z) in terms of

R ( z) and P ,

(2.2) R^(z) =R^{z)(l - PR^(z))"^ .

Next we assume that the unperturbed resolvent has property [ *J , and
ask for conditions on the family of operators ( 1 - PR ( C + i« ) ) which
ensure that the perturbed resolvent also has property [ *] . First one is
tempted to say tliat all that one needs is the strong convergence of this
sequence on S , arguing that the product of a strongly and a weakly
convergent sequence of

\ .t


yM^'-t' ''^-'^ .cdziuob sm;


. o I

:J pfiii

:t08 vX'isirio,/:'.

a ...V ;;

^.q ^A On3 _^

- (!


.noiittC'iJj^-isq ^u?

'.:i07 aor: ".'Oi ^a:

' .■ *

, 1 bll:


.K- SH'.

2.5 5

operators converges weakly. Then according to (2o2) the per-
turbed resolvent would converge weakly. ?Iowever this arguunent
does not apply to our operators since v/e do not have convergence
on the entire space. Another pathology concerning vjeak conver-
gence of operators on a dense set Is that the space of bounded
operators Is not complete with reference to this convergence. In
order to exclude such pathologies vje need some additional condi-
tions and definitions.

Consider an abstract Banach space B and a form F on
B "x B which is conjugate linear in its first and linear in its
second argument, in short, Hermitian. Let < f", Fg > denote
the value of F for the pair (f^g). The complex conjugate
sign above f does not imply that the conjugate of an element
of an abstract Banach space is defined; it merely emphasizes that
F is conjugate linear in its first argument. In analogy to the
notion of the norm of an operator we define the norm of the form
F by setting

(2.5) |||| = sup ^ . I '^^^H .

f,S llfllB-llsllB

In analogy to weak convergence of a family of operators, v;e say
that the family of forms F(e) converges v;eakly to the form
F(0) if for every f and g in B

(2.4) lim < f, F(e)g > = < f,F(0)g> .

sono'ii'i'-'Viico 9Y,£irf d-orr ob sv; ,^icqo nuo oct "c'lq'-.r.G :)on

•^••....■■vr..;o "'-.c^s;; ;;: '■ ;.', -grBqe; C':'I:vno erl3 no

•x/^i'OL' iBaoJt;! .'.5jb.G 0;:to3 Ibo^er, sv 8e'.:"o.Co.l"'j:j';: :io^? ebislnxo ■'■d' ••

2A 6

Note that if P(e) Is a weak Cauchy sequence and the
correspondlnG norms |||| , remain bounded independently
of e then F(e) converges to some form, say F(0). Let A
be a bounded operator on B and define, FA , the product of the
form F v;ith the operator A , to be the form determined by

(2.5) < f , FAg > = < f, FlAg) > .

Concerning such products we have a simple but important proposi-
tion: if the fam.ily of forms FCe) is bounded in norm and
convernes weakly to F(0) and if the family of operators A(e)
convers;es stronp;ly to A(0) , then the family of forms F(e)A(e)
conver?;es weakly to the form F(0)A(0). Tnis an immediate con-
sequence of definitions (2.4) and (2.5) if we remember that for
the family of operators ^A(e) i the principle of imiform bound-
edness holds.

After this digression on abstract Hermitian forms v;e
return to the question of finding conditions v/hich ensure that
property [*] holds for the perturbed resolvent. Recall that
we already assumed that [*] holds for the unperturbed resol-
vent and we consider the set S entering it. Then roughly
speaking v;e shall require the existence of a norm such that: if
B is the completion of S with reference to this norm then one
can set


F(e) = R^(C+le)

A(£) = 'i-PR^((^+ie))"^,

'.T.-r ; .^ ...

iwuriCwO/. n-^ol edcf srf

< -J

( .: > ■•! n-;I J

. -^ (riA)': ,M

■■*■: cfr- on:; 101}- iyis-Ci/j [-^rrVS 'io e-r5.:?:olo e;lci llii^^ g:; r .jri'j'' «iK'xon-S
K no 'io:f.y^Lrp hebrvjcd b sd i rial •^I'XJiIxfrrxB ,A \u hanS-jy . sJ i:,b

/li^ol Gilo :ji'r'o svioqqjjs bas

K it

H n -) .-; (H - a) no (y;'\:i;


2.6 8

converp;G strongly on B as £ converc:es to from above, and
this Is uniform In C.

V/e see from the previous proposition and from the second resol-
vent equation (2.2) that If the perturbation problem A^,? ,
satisfies Condition l-I^C^^^Cg] then property [*] holds for
the perturbed resolvent, R,(i;±le). Thus these conditions assure
the absolute continuity of the part of A^ over the Interval

In the Appendix v/e shall verify this condition for the
Sturm-Llouvllle operator -D + M(q) , provided that the
potential q dies out at infinity. In doing this, v/e shall need
detailed information on the perturbed resolvent, in spite of the
fact that it does not enter the condition explicitly. For this
reason v;e introduce two more conditions, vjhich are more restric-
tive then this one. Nevertheless for a large class of perturba-
tion problems we can verify them without reference to the per-
turbed resolvent. They read as follows:

Condition 2 .[^j^,^g]

For every ^ in this Interval the two families of operators
determined by PR (^±ie) on B , conver,Q;e in norm to compact
operators, as e converp;es to from above, and this is
uniform in ^ .

Condition ^ .fCj^^C^I

For every t, in this interval the nullspacec of 1-PR ({;±)
on B are trivial.

bnr:- ■ iTOrfi C

■~I.-.'.:3'-i b:i'joe-6 sr'rJ irro'il -Mis:, n - ^'

^ -^o

/\ j'j aoi jjjdnuvi'ing

'■xol a/Mo/! [■*] \j'i3C/ri'] noilf [^^ < |-'*] -^ - . - ' ■ c^llyJ-Cs^.




tirlo diii':d- jc.jj.i:v/.wcr ^ ^i.;;.! -r "V.;.. - ;;

2.7 9

If this condition is satisfied at a given point C only-
then v/e refer to it as Condition J.C- Similarly v/e shall speak
of Condition l-C and 2.^.

'v/e claim that Conditions 2-5 imply Condition I. For,
according to Condition 2, PR (^4-) is compact and so the
Fredholm alternative applies to the operator 1-PR (C+)« Tlius
from Condition 5 v;e can conclude that 1-PR^(C+) admits a boimd-
ed Inverse defined on all of B. From this in turn, using the
other part of Condition 2 v;e conclude that for small enough e
the operators 1-PR (C+le) admit Inverses; moreover
d-PR (^+le)) converges in the B-operator norm to
d-PR (C+))" . Thus Conditions 2 and p imply Condition I. In
conclusion let us remark that if v;e replace norm convergence by-
strong convergence in Condition 2 this implication may possibly
be false.

Summarizing our results we arrive at the follov/lng:

Lemma 2»1

Suppose that the perturbation problem A .P , satisfies condi -
tions l-l,[^-^,^rj or. Conditions 1-2-3. [^3_. ^2^ • "Jl^en the part
of the perturbed operator A + P over the Interval [^ ,4^] is
absolutely continuous *

In the applications that we consider we can verify the first
two of these conditions directly. In establishing Condition
5«[C-i>Coi * hov/ever, it is convenient to proceed indirectly.
I.e., to assume that it is violated and to derive a false con-
clusion from this. Such a conclusion is stated in the next

• : \ ■ - ■

••)ifi-c;3 Ii^dE ov. ■.';I':tj;I.i.;:lC .^.o ',o ':.■.:. b..r>0 5\: cfx ....

. -■ . S '.jaa

.^:iC'i »I -:o.: - ;.j:iinc-0 v;Iqi;..L c-=:- ;Dr!C..7X!:iKX ;ri:
£..i'j ./:, ;;nr; JCJ^'-irivOu j.i; (^^i) HI ^G no.:;'!;:. ■ . '- . ^.-

o::j cjfi-f^^^-' t: '^-v^ fi - . strict ;:;•. \L'd «o 'lo il^- no 'jenlloij St.. .

' o

^- f .'


V;Id'l:roq -^iEr; r;oJ;>: -1 '

iooB'j:-^";o ^shj xC JoenlrrnOoOij ul CI .■■' ii no (.-^J;-^) ..,.n ;.i':^o'i

sonoil :Ii

k "'.^

;T . ,P > r.M.

< (3)B(.>X-^)^^fi ,

. '3 !.. >J J-

- (3}



lim [im < f , R^(C-le)Cl-PR (^+)'^S. >] = < f , ^J^-)^ =^ •
e-^ +o ■- ° o E 4

On the other hand according to (3 ..6) the left member equals zero,

(5.8) Im < f , RqIC - )^ > = .

How from this relation v;e can easily derive conclusion
(5,4). For, according to Condition 1,^, for an arbitrary func-
tion f in B,

(5.9) < f , R (t;-)f > = lim ff^,R^(C-l£)0 ,



n' o


the limit on the ri:;ht being a double limit. Next set

Sq(z) = (z-M)


and observe that according to (5*1)

R (z) = V S (z)V .
o o


Im(f^ , R^(C-lB)f^) = (Vf^,

S^it^-le) - S(C+ie)]vf^) .

From this fact and from the assumption that V maps vectors in
B n H into continuous functions v;e can conclude

lim (Vf ,
£-^ +o

S^(C-i£) - SQ(C+ie)

Vf^) = 7r|5^Vf^r ,

if we use a theorem of Fatou. Since V is a bounded map of
B H into C , this yields


• Ciij'': alfij.;p9 nsditiSv/ Jlsl G^i-y (^^a-^) o;/ y.nJ:&icoos 'xTBrJ lerf'o

:^ < ^(.^^ii^hl t T




f.I 1 nolo

('^,-i(..f:-'^)^.n,^;:;') .nij; • < ■'-J^^h . i > i'i.O

as;;: ^rs'^t .ctlrffJ:! •-/.[c^u^^b r :-r:.i:i'0 d'ri,:j:';i er.i no JJ:n;;.I 8ri

•- I .V ;


(.r.;.'j o:;' ■\n-':b'-.oooa J^-il:i- 6VT©scfo Ijks


U:) !T


O J ;^ Jk O a

oev :.

1 ' v; J.


5.^ 15

(3.9) llm [lm(fj^,R^(C-lE)f^)] = Trl&^Vfl^

Finally combining relations (3.8), (3.9) and (3.10 v;e obtain
(3.^), v;hich establishes the lemma.

'NMV.oi-vf - I (^^-^Isx-'^ . r i':.0

4.1 14

4. The operator p + q(x) on (-~, + oo)

Let D be the set of tv;ice continuously dlfferentlable
functions on [-oo, + oo] with bounded support. We define the
u. perturbed operator to be the D-closure of the formal operator

- 2__ and denote it by -D , For the perturbation we take the
operator defined by

(4.1) M(q)f(x) = q(x)f(x) .

As is well known the operator -D is strictly self adjoint.

Hence if the function q is bounded so is the operator Il(q)

and we see that -D + M(q) is also strictly self adjoint and

\ie take it to be the perturbed operator.

VJe formulate additional conditio- s on the function q which

assure that the perturbation problem -D , M(q) , satisfies the

abstract conditions of Section 2. Specifically v;e assume that

q is a continuous function for v.'hlch

(4.2) 11m q(x) = .

Ixl-i CO

Moreover, q is integrable v;ith respect to a weight function,

(4.3) / |q(x) lw(x)dx < 00 ,

>-' -co

such that w(x) ^ 1 and

(4.4) lim w(x) = w ,

|xM w

Define the Banach space B to be the closure of D with refer-
ence to the norm



( 00 -i , ^ii- ) cfo { .: ] .



ftr.'c* sr-lBct ew nold'sd'Awo'isc: si'.i lOu

V J-

• d.ixo'cbB llscj •'rlc^ci'r:; - ;.;.r. "'G- '-loci-.s-xsao ario m.'ca^' IXs'.: ':■.■: r.A

.'iocf.n'ioo;o iiS'a'X0;J*"r3q ?ri.:f sd od" j.!: £


'(-)pi ^^

X -• \:uu



-''ZZj.Q'l iiuXi

4.2 15


(4.5) llfMB= suplf(x)| + / |f(x)|v;(x)doc .

Note that assumptions (4t3) and (4.4) imply that

(4.6) Iklig < ~ .

We maintain that the perturbation problem -D , W(q) satisfies

the conditions of Section 2. More specifically v;e maintain the


Sub theorem 4.1

Suppose that the function q satisfies conditions ( 4 . 3 ) * (4.4)

and that the bounded interval [^, ,Cp] does not contain the

poin t or a point eigenvalue of the operator -D + M(q) in

Lp. Then the perturbation problem -D , M(q) satisfies condi -
tions 1-2-3' [C-] jCpl with reference to the Banach space B,

The proof of this subtheorem makes essential use of the
representation of the unperturbed resolventt, Specifically

(4.7) k(u) = e^'^' ,
v;e have

(4.8) R (z)(x,y) = ik(/i-(x-y)) ,

\/z being defined in the plane cut along the positive real axis
and having positive imaginary part,

a) Condition 1 . [ C •■ > C ^ 3 » Let [Ci^^o^ ^® ^ closed and bounded
interval not containing the point 0. First observe that for

:d:>(::h'! (:•)•:! \ -i- !(>•

f ;i li

^farvj '\

rS.qril (f^.!-) tit^ (■.%■!') ?^noi:oqfnf;C2S d-j3f{J ^:>oVI



:;: ■ \ :.o1

j,.4^ :r{5

f :i }1\

i \




« {i:;):I




':' -J •_

4.3 16

t, In [?, , ?, ] the kemol of the unperturbed resolvent converges pointwise.

Specifically if on the positive real axis

six = llm \/t, + ie ,

then according to (4. 7) ,

(4.9) lim R (^ + iO(x,y) = — k(\T(x - y) ) .

Since this convergence is uniform in any bounded (x, y) region, we see
that setting

(4.10) RCi+)(x,v) = —k{^{x-Y)),

for any pair of functions ( f, g) In D ,

(4.11) lim = .

Note that R ( C+) is a possibly unbounded Hermitian fonTi, defined by the

1 3

Online LibraryP. A RejtoSome absolutely continuous operators, I → online text (page 1 of 3)