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P. A Rejto.

Some absolutely continuous operators, I online

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NEW YORK UNIVERSITY
COURANT LMSTITUTE - LIBRARY



IMM-NYU 329



XEW YORK INIVHRSITY
COURANT INSTITUTF. OF
MATHEMATICAL SCIENCES



Some Absolutely Continuous Operators I

P. A. Rejto



•r
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^



PREPARED UNDER
CONTRACT NO. NONR 285(46)
OFFICE OF NAVAL RESEARCH



IMM-NYU 329
September, 1964



NEW YORK UNIVERSITY
COURANT INSTITUTE OF MATHEMATICAL SCIENCES



SOME ABSOLUTELY CONTINUOUS OPERATORS I
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.
NONR-285(46).

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,



I GnOTAHaiO aUOUKITKOO YJaTUJCjSSA 2M0a



Acknowledgement

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



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



ABSTRACT

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

2
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 ,
o

(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



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^.q ^A On3 _^



- (!



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.noiittC'iJj^-isq ^u?



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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

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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



and



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



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









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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^.



!'rtn->






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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
section.



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sonoil :Ii



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;T . ,P > r.M.



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



. '3 !.. >J J-



- (3}



3.5



12



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,
thus

(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 ,



o



e.n



n' o



n



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



Sq(z) = (z-M)



-1



and observe that according to (5*1)



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



Hence



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



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



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oev :.



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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

2
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) .

2
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

o
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



n



X.-l'



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



-'xJb



2,.



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.!: sx.sj £



;0'yj



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-^Ui



-''ZZj.Q'l iiuXi






4.2 15



CO



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

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

(4.6) Iklig < ~ .

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

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

following:

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

o
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
setting

(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



II I I

lip!!



:;: ■ \ :.o1



j,.4^ :r{5



f :i }1\






i \



i-






/-^«






kU.



« {i:;):I



•3V



)(s)^H



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 ,
€-*+0

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)