Result. Math. 33 (1 998) 274-287 0378-62 ] 8/98/040274-]4 S 1.50+0.20/0 o Birkhauser Verlag, Basel, 1998

I Results in Mathematics

Boundary conditions for quasi-differential operators in LP spaces Hilbert Frentzen Abstr act For a very general class of ordinary quasi-differential expressions M with matrixvalued coefficients and for p, q E [1,00) or p, q E (1, ooJ aU operators T of a subspace of LP into Lq whicb satisfy Tfr e T c Tu and are Fredholm operators with index zero are characterized by suitable boundary conditions, where TM and T~1 are the maximal and minimal operators associated to M . This generalizes a result of Evans and Ibrahim forp = q=2.

Mathematical Subjects Classification 34B05 Key words: differential operator, boundary conditions, adjoint, Fredholm operator

1. Introductio n Given an ordinary differential expression M, the re are two extremal operators associated to M in appropriate Lr spaces, namely the so-caUed minimal operator T~ and t he maximal operator TM, which ate related by T1 c Tu. One problem is to characterize differential operators T with certain nice properties such that Tff e T c TM . For (formally) symmetric expressions M and £2 space much work has been done on the problem of finding selfadjoint T with the aid of boundary conditions, sec for example Naimark [8], Zeitl [12] and Weidmann [11 ] to mention only a few. If M is not symmetric, one can ask for T which are Fredholm operators with index zero. For L2 space and second order scalar differential expressions this question was answered by Evans [2], for scalar expressions of general order on half-open intervals by Evans and Ibrahim [3] and on open intervals by Ibrahim [7]. In this paper the main result of Evans and Ibrahim concerning differential operators is generalized to Lr spaces and arbitrary intervals (see Section 4) . For the convenience of the reader, in Sections 2 and 3 we give the necessary definitions and cite some results of [5]. Section 2 is concerned with closed and weakly closed operators and adjoints of operators in L" spaces. In section 3 we consider a rather general class of quasi-differential expressions

Frentzcn

215

M with matrix-valued coefficients and the associated maximal operator TM and minimal operator 'Itt as maps of a subspace of V into L~ for arbitrary p,q E !1,00] . Section 4 deals with differential operators T with T~t e T c Tu and their adjoints and contains our main result on the characterization of such operators which arc Fredholm operators with index zero as restrictions of the maximal operator by boundary conditions.

2. Preliminaries Throughout this paper let IK denote either lit , the field of real numbers, or C, the field of complex numbers. For positive integers k and m let Mo" m denote the vector space of k x m matrices with K-valued entries and GL m the subset of M". := M".,m consisting of all nonsingular matrices. For A E M.I.,,,. lcl A' denote the transpose and A' the adjoint, i. e. the complex conjugate transpose, of A. We use this nolation also for scalars considering them as elements of MIl. In the following I . I denotes both the euclidean norm in Km and the associated matrix norm. If Z is a subset of M k •m and I is an interval, then B(I, Z) denotes the set of Lebesguemeasurable maps of 1 into Z and AC1oc{1, Z) the scl of locally absolutely continuous maps . Measurable maps arc regarded as equal if they arc equal almost everywhere . Further we define

LP(I, Z) := {y E B(I, Z) IlylP is Lebesgue-integrable}, !i yllp,/: = ( / Iy lp) I / p for all y E LP(I,Z) and p E

P,oo),

I

L""(l,Z):= {y E B(l,Z) y is essentially bounded}, Ily !i ",,'!: = ess suply(x)1 for all y E L""'(I,Z)

""

,nd

Lfoc(l, Z)

:= {y E

I

B(I, Z) ylK E LP(K, Z) for all compact subintervals K of I} for all p E

!1,ooJ.

For a normed linear space we denote the dual of E, i. e . the space of bounded linear maps of E into IK, by E' and the conjugate dual of E, i. e. the space of bounded conjugate linear maps of E into K, by E'. If r E [1,00], then r' E [1,00] is always chosen such that ~ + -:.- = 1. In the following we always assume that p,q E [1 , 00] and + E {t,*}. If V: = LP(1,K') fot some positive integer s, then (£1')+ = V' for p E [1,(0) and £1 is a subspace of (L"")+. Further we define {(y, z), (v, u))o := u+y ~ v+z for all (y, z) E LP x L~ and (u, v) E x Lq' . Then ( ., ' )0 : (V x L~) x (U' xLI") --t K is bilinear for + = t and sesquilinear for + = *. Therefore, we can deftne a weak topology cr(£P x LQ, Lq' x £1") on £I' x U which is given by the family of seminorms {I( " (v, u»o ll (u, v) E V' x L~'}. Then (LP x U, cr(V x L\ L~' x V')) is a locally convex Hausdorff space. For G c £I' x Lq we denote the closure of G with respect to cr(V x L\ Lq' x V') by G'" and the closure of G wilh tespecl lo the norm topology by G<. We say that Gis w-closed if and only if Guo = G and that G is closed if and only if GC = G. Lct A : V ::> D(A) --t Lq be a linear operator. Then R(A) denotes the range and G(A) the graph of A. If B is another linear operator, we write A c B if and only if

II

IT

V'

Frcnb:en

276

G(A) C G(E}. The operator A is called w-closable if and on ly if there exists an operator A"': £p :) D(AIU) -+ L~ such that G( A"') = G(A)'" and it is called w-closed if and only if it is w-closable and A'" = A. Analogously, we say that A is closable if and only if there exists an operator AC: LP :> D(AC) _ L9 such that G(A<) = G(A)" and it is called closed if and only if it is closable and A< = A. Then we have LEMMA 2.1. (d. [5, Lemma 3.4])

(a) If A

~

w-closable, /hen A iJ closable and A< C A"' .

(b) If A is w-closed, then A is closed. LEMMA 2.2. (d. [5, Lemma 3.51) Forp,q E [1,00) we have (al if A is closable, then A is w-closabfe and A<= A"',

(b)

if A is closed, th en A is w-closed.

For G

c

V x U we defi ne the orthogonal complement of C as

I {(y,z),(v,u))o = 0 for all (y,.:) E C}. If D(A) is total, i. c. if {z E If I frz+y = 0 for all y E D(A)} = {O}, then (O,v) E G( A ).1. is only possible fo r v = O. In this case wc dcfinc an opcrator A+ by G( A +) = G( A ).1., C.l:= ({v,u) E L9'

X

L"

1,

i . c. A+ : L'f' ~ D(A+) -> V' i . a lincar opcrator such that v+ Ay = !,(A+v)+y fot all y E D(A) and v E D(A+). Wc call At the transpo.!c and A' thc adjoint of A. Fot p E [1,00)

=

the domain D(A) is total if and only if it is dcnse in V. If p q = 2, tlten A' is the usual Hilbert space adjoint. For p, q E [1,(0), the operator At is what Goldberg calls the CO!\jugate of A [6, Definition 11.2 .21, for p,q E (1,001 it is what Goldberg calls the preconjugate of A [6, Definition VI.1.1].

LEMMA 2.3. (d. [5, Lemma 3.G]) If D(A) is total, then we have

(a) A + is w-dosed and dosed, (b) A i.! w-dosable if and only if D(A+) is total,

(c) if A is w-do.!ablc, then A++ = A"'.

3. Maximal and minimal operators corresponding to quasi-differential expressions Let I be an interval with endpoints a, b (-00 -:; a < b :5 (0), let n, $ be positive integers and p,q E [1,00[. DEFINITION 3.1. The set ZPo'f of Shin-ZeW matrice$ is defined to be the set of all lower ',' triangular matrices F of the fo rm

FO•I

FI.I

FI.2

0

F,, _I.I

F,,_1.2 F" ,2

F n _l.n

F= F". I

F".n

Fn,n+t

277

Frcntzcn

where FO•1 E LfoJI, M, ) and ; .....n+l E L~;c(1, M. ),

Fj.k E Lloc(1, M,) for all! 'S j 'S nand 1 'S k 'S min{j

+ 1,n}

(Zl) (Z2)

and

Fj.j+l(t) E CL, for all 0 'S j 'S n _and t E l. For F E Z;:·q we define ,the last column, i. e.

F as

(Z3)

the mat rix obtained from F by removing t he fi rst row and

F:=

(:::,

F I •1

F""'") ,

F" _1,2 F".l

F...l

F",n

DEFINITION 3.2. For F E Z:::~ the associated quasi-derivatives are defined by

"d Dry :=

Fj~l+l [(Df_1Y)' - t

k ", t

Fj.kDLIY] for all 1 'S j 'S nand

I

Y E D{Dfl := {Y E D(Dr_t ) Dr_1Y E AC1oc (1,K'j}

where I denotes differentiation . The associated quasi-differential expression Mp is given by

MFy:= D..F y for all Y E D(MF): = D(D n•

)·

For y E D( M p) we further define

Clearly the maps Df: D(Df) -+ B(1, K' l,M F : D(MF) -+ B(1 ,K') and QF: D(MF) -+ AC1o <(1, K''') are linear. In the following we always assume that F E zp,q and M := Mp . In analogy to the adjoint and the transpose of :"matrix, there are two different "(formal) adjoinh" of a quasi-differential expression. DEFINITION 3.3. For

+ E {t,*}

we define

GO,I:= F:'n+1,G",n+l: = - Fo~llG: = -( H"FHn)+

278

Frcntzcn

with

H.. :=

0 I,) ( "

E GL""

0

I,

M+:= Me and [y,zIM.+ ;= (QGZ)+HnQFY for all y E D(M) and z E D(M +) . Then M ' is called the adjoint of M, Ml the transpose of M and [ " -IM,+ the Lagrange form of M. REMARK 3.4. (cf. [5, Remark 4.6]) ,p' (al G E zq' n, J ,

(bl M ++ = M, (c) Ml = JM ' J and [y,zIM,1 = [y,Jz]M, . for all Y E D(M) and z E D(Mt) where

(Ju)(x):=

:. (",(Xl')

forallu =

u,(x)

DEFINITION 3.5 . The

qua.si~difTerclltial

("') :

Ell(I,K'jandx E I.

v.. expression 111 is said to be

(a) symmetric if and only if M ' = M,

(b) J -symmetric if and only if

lift

= 111.

In view of Remark 3.4{c) this definition of J-sYlllmetry coincides with the usual one, namely M " = JM}, see for example [4, Definition 2.3].

(a) The maximal operator corresponding to M,p and q is defined as TM,p,qY:= My for all y E D(TM,p,9):= {y E D(M) n LP My E Lq},

DEFINITION 3.6.

I

(b) The preminimal operator corresponding to M,p and q is defined as 1'tr,p,qY ;= My for all Y E D(Tl},p,q} ; = {y E D(TM,p,q) supp(y) is compact and contained in the interior

I

off} . Clearly TM,p,q and Tl},P,9 are unear operators of subspaccs of LP into Lq and Tfr,P,9 C

TM,p,q' THEOREM 3.7, (d,

there exist

[5, Theorem 5Al) For Y E D(TJIf,p,q),z E D(TM t,9',p') and

liml _~[y,z ] M, + (t) = ;

c E {a,b}

[y,Z ]M,+(C ) alld the equation

/, z+My -

/,(!~l+Z )+Y = {y,Z}M,+

holds where {y, Z}M,+ := [y, z]MAb) - [y, z]M,+(a), In [5 , Theorem 5.7 and Corollary 5.8] it is shown that D(T,{l-,p,q) is total and TA~ .p,q is w-c1osable. lIenee we ean define

Frcntzen

279

,

DEFI NITION

< )" . ( TM,p,q

3.8 . The minimal operator eorresponding to M, p and q is defined as T~f

From [5, Corollary 5.8, Theorem 5.12 and Theorem 5.16] we obtain T UEOREM 3.9.

(a) Tff.p.q

-

:=

C TM ,p,q,

(b) Tt.p,q and TM,p,q are w-closed and closed,

(e) T~!.p,q = {TM+.q"p') + ' (d) (T~f .p,q)+ = TM+ ,q'.p"

I {y,Z}M. + = 0 for all Z E D(TM +,q'.P,j}, D(1'/",, ) ~ {, E D(Tu " ,, ) I (QF,)( a) ~ (QF,)(b) ~ OJ if a,b E ],

(e) D(T~I,p,q) = (f)

COROLLARY

{Y E D(TM,p,q)

3 .1 0. (d. [5, Corol!ary 5.13])

(a) If M is symmetric, then

T~f22

is symmetric in th e Hilb ert space swse.

(b) If Mis J-symmdric, th en T~f22 is J-symmctric in th e Hilbert space sense.

4 . Differe ntia l op erators and boundary conditions Let I, n, s be as in Section 3. In this section we assume that, in Goldberg's notation [6], p, q arc admissible, i. c . p, q E [1,00) or p,q E (1 , 00), we consider M: = MF for some F E Z~:~ and define d := dim D(1~r,p,q) / D(Tt.p.q )' LEMMA

4.1. d = dim D(TAf +,q',p') / D(1'~f + ,q',p' ) :5 2ns .

Proof. (I) In view of T heorem 3.9(f), for p,q E [1,00) the proof of [1, Theorem 9.1 J also works in t he present situation . (II ) For p, q E (1 , 00) we apply (I) to M+, q' ,p' instead of M ,p, q and usc Remark 3.4(b). 0

Following Goldberg [6] we call T a diffe re ntial operatol' corresponding to M, p and q if and only if Tff,p,q e T c TM ,p,q' In particular, the minimal operator 11"p,q and t he maximal operator TM,p,q are differential operators in this sense. LEMMA 4 .2. Let T be a differential operator corresponding to M ,p and

q. Then

(a) Tis w-closed and closed, (b) T + is a differential operator corresponding to M+ , q' and p' . Proof. (a) Lemma 4.1 implies that T is a fini te dimensional extension of T~f ,p,q ' Hence we obtain the assertion from Theorem 3.9(b) and [10, 1.3.3J. (b) follows {rom Theorem 3.9(c) and (d). 0

Thc following result, which is in part due to Coddington and Dijksma [1), shows how differential operators and their transposes and adjoints ean be described as extensions of the minimal operator .

280

Frcnlzcn

11m lS (l nonnegative integer with m $; d, ijyJ, ... ,y", D(TM ~ ,q',p') are such that

THEOREM 4.3 .

z" ... ,Zd_m E

E

D(Tf,f,p,q) and

(a) {Yi> " "Y"'} is linearly independent modulo D(TM" ,p,q ), (b) {Zl",.,Zd_m} is linearly independent modulo D(T~ft,q'.p')' (e) {Yj,z",lu,+ = O/or all I

(d) D(T) = D(Tgf.p,~)

:s:

j:S m and

1::;

k S d-m,

+ span{Ylt ... , Yrn},

th en T := Mlv(T) is a differential operator corresponding to M,p,q and

Conversely, ifT is a differential operator corresponding to M ,p and q, then there exist functions Yl, ... ,Y", E D(TM,p,q) and ZI, .•. ,Z
Proof. (I) For p,q E [1,00) this result follows from [1, Theorem 4.2] and Theorem 3.9(c), (d). (II) For p,q E (1,001 to prove the fltSt part we define

D(S) := D(Tfr+,q',p') + span{zl, . . . , Zd _... } and S := M+ID(S) . Then from (I) for M + instead of M we obtain that S is a differential operator corresponding to M+ ,q',p' and D(S+) = D(T). Since S is w-dosed by Lemrna4.2(a), Lemma 2.3(c) implies S = T + and, thus, (e). To prove the converse, we observe that in view of Lemma 4.2(b) T+ is a differential operator and with m:= d - dimD(T+)j D(Tfr+ ,q ' ,p,) part (I) for M+ instead of M yields the existence of functions Zl, . . . ,Zd_m E D(TMI,q',p') and YI, ... ,y... E D(TM,p,q) such that (a) - (c), (e) hold and D(T++) = D(Tff.p,q)+span{Yl, .. . ,Ym}. Since T is w-dosed by Lemma 4.2(a), Lemma 2.3{e) implies that T++ = T and, thus, m = dim D{T) j D(T~.p,q)' 0 This result can be used to characterize those differential operators which are Fredholm operators with index zero, where for normed Spaces E, F a dosed linear operator A : E ::> D(A) -+ F is called Fredholm if R(A) is dosed and the nullity nul A := dim N(A) and the defect def A := dim Fj R(A) arc finite. For a Fredholm operator the index is defmed as ind A := nul A - def A. To give the said characterization, we need some auxiliary results. Part of the following two lemmata can be found in [6, Theorem VL2.7]. L EMMA 4.4. 1fT is a differential operator corresponding to M,p and q, then

(a) R(Tff.p,q ) is closed if and only if R(T) is closed, (b) R(Tff,p,q) is closed if and only if R(T~+.q"p') is closed.

FrcntzclI

281

Proof. (a) In view of Lemma 4.1 this assertion follows from [6 , Lemma V.1.5 and T lteorem IV .1.l2J. (b)(i) Assume that R (T~' ,p,q) is closed . (I) If p,q E [1,00), tllen tIle closed range theorem [6 , Theorem IV.1.2] and T heorem 3.9(d) imply that R(TM t ,q',p') is closed and, thus, R(T~f+ ,q' .P') is closed by (a) . (II ) If p,q E (1,00], then T heorem 3.9(c) implies that R((TM +,q'.P')+) is closed , hence R(TM+,q'.P' ) is closed by the closed range theorem and again we get that R(T~ + .q"p') is closed by (a). (ii) If R(T~f + ,q " P') is closed, then we use (i) for M + instead of M .

o

L~MMA 4. 5. If R(Tt,p,q) is closed and T is a differential operator corresponding to M,p and q, then T is a Fredholm operator and

(a) nul TU+,q',p' = dcfT~"p , q, (b) dcfTM+.q,.P' = nuIT~l,v,q, (e) indTM+.q,.p' = - indT1-,p.q'

Proof. For p,q E [1,00) Theorem 3.9(b), (d) and [6 , Theorem IV.2.3j imply (a) and (b) . In view of Lemma 4.4(b) for p,q E (1,00] Theorem 3.9(b), (c) for M + instead of AI and [6 , T heorem IV.2 .3] imply (a) and (b). liellce defT~f ,p.q = nuITM +. 9 ,.P' :5 nulM+ = 1lS and lIul Tt.p,q :5 nul M = ns , i. e. T~f ,p,q is a .Fredholm operator. T hcrefore, Lemma 4.1 a nd [6, Lemma V.1.5] imply that T is a Fredholm operator and (c) foHow s from (a) and (il) . 0 LEMMA 4.6 . If R(T~f ,p.q) is closed and nul T~f ,p.q = nul T~f + .q'.P' = 0, then d = nul TM,p.q +

nuITM+,q',p" Proof. From [6 , Lemma V.1.5] and Lemma 4.5 we obtain

d = ind l'M,P,9 - ind Tit.p,q = - ind T~f + ,q"p' - ind Tit ,p,q = def11, +,q"p'

+ def Tif ,p,q =

nul TM ,p.q + nul TM+,q',p"

o L E"f MA 4.7 .

D(T+ ) =

(a) If L C D(TM.p.q ), if D{T):= D(T2r,p,q) + span Land T := M ID(T), then {z E D(TM+,q',p') 1 {y,Z}M, + = 0 for all y E L}.

(b) If T is a differential operator corresponding to M,p,q and if L C D(TM+.q"p') is such that D(T+) = D(T~f .p.q) + spanL, then D (T) = {y E D(TM •p.q) {y,Z}M,+ = Oforallz E L }.

I

Proof. (a) From the assumption we obtain that G(T) = G(T~t.p.q) + G( M lu) where U := span L. In view of Theorem 3.9(d) we get G(T+) = G(T) l. = G(T~1,p ,q) 1. n G(M lu) l. = G(TM +,q',p') n {(z,w) E £9' x y' l l w+y - 11 z+ M y = 0 for aU y E L}. Hence Theorem 3.7 implies lhe assertion. (b) Applying (a) to T+ instead ofT yields D(T++ ) {y E D(1 Af ,p,q) \ {y,Z}M, + 0 fo r all Z E L}. Since T is w-closed by Lemma 4.2(a), Lemma 2.3(c) im plies T++ = T. 0

I

=

=

282

Frcntzcn

The following result generalizes [3, T}lcorcm 3.2] and [7, Theorem 3.2] from L2 to L"

.-

spaces.

.

THEOREM 4.8. Assume that nuIT~, = nulT?,+ ~~ ' , = 0 and define m m+:= nul T M +,9',P" If y" . . . , Y",+ E D(TM,p,q) and 2:11" " Zm E D(TMt ,q',p') are such thal

:=

fl,

nul TM -, and

(i) R(TXf,I',q) is closed, (ii) {YI,'" ,Ymt} is linearly independent modulo D(Tf,,p,q)' (iii) {Zl, ... ,Zm} is linearly independent modulo D(1'MO + ,q, .P,I,

(iv) {Yj,Z .. }M,+ = 0 for aliI::; j::: m+ and 1 :S: k:::;

TIl,

(v) D(T) = {Y E D(Tr.r,p,q) I {y, ZIt}M,+ = 0 for all 1 ::; k ::;

TIl},

then T := M lv(T) is a differential operator correspollding to M,p,q which is a Fredholm operator with index zero and

(vi) D(T+) = {z E D(Tu+,q' ,p')

I {Yj,Z}M,+ = 0 for aliI :S j

::: m +}.

Conversely, If T 1S a differential operator cOrl'espoftdill9 to AI, p, q which is a Fredholm operator with index zero, then there exist functions Yl, ... ,Y",+ E D(TM,p,q) and Zl,.,. ,Z", E D(Tf>[+,q',p') sueh that (i) - (vi) hold.

Proof. To prove the first part, we define

D(S)

:=

D(T~f,p , q)

+ span{Yl, '"

,y,.,+}.

Then Lemma 4.6 and Theorem 4.3 imply that S := M ID(s) is a differential operator corrcsponding to M, 1), q and

D(S+) = D(T~+ ,q',p') + span{ Zl, .. . ,z>n}' Thereforc, applying Lemma 4.7(aJ and (b) we obtain D(S) = D(T) and D(S+) = D(T+). Hence, from Lemma 4.5 and [6, Lemma V.1.5] we get that T is a Fredholm operator and

+ dim D(TJ/ D(T~f.p,q) def Tft,p,q + dim D(S)/ D(T~f'P,q) =

ind T = ind T~(,p,q

= -

- null'M+ ,q'.P'

+ m+

= O.

For the proof of the converse we first notice that since T is a Fredholm operator, Lemma 4.4 implies (i) . With I : = dim D(T)/ D(T~f.p,q) Theorem 4.3 yields the existence of Yh' , ., YI E D(TM,p,q) and Zh .. . ,ZJ_I E D(TM+ ,q',p') such that (ii) - (iv) Iwld with l instead of m+ , d ~ I instead of m and

D(T) = D(T~f,p,q)

+ span{Yl, '"

D(T+) = D(T~f+ ,q',p')

,YI},

+ span{zh" , ,zJ-d.

283

Frcntzcn

From Lemma 4.5 and [6, Lemma V.l.51 we obtain nul TM ,p,~ = ind TM.~.q = ind T

+ dim D(TM,p,q) j D(T)

= dim D(TM.p,q) j D('1~f.",q) - dim D('1') j D(T~!.".~) = d - I

,nd nul TM+,q',p' = dcfT~,.p,q = - ind T~ ,p.q

= - ind T + dim D(T) j D(TXr,p.q) = I. o

Hence Lemma 4.7 implies (vi) and (v) .

Theorem 4.8 describes a certain class of differential operators as restrictions of the max· imal operator given by boundary conditions. It yields well·known results for symmetric and J .symmetric differential expressions. COROLLARY 4.9 . Assume that M is symmetric, that). E K is such that and R(T~r .2.2 - .x) is closed. Define m:= nul(TM,2,2 - -X) .

lIul{'1'f,.2,1-).) = 0

IfY I"",Ym E D(TM,2,1) are such that

(a) nul(TM ,1.2 - ). ' ) =

Tn,

(b) {Y I," " Y"'} is linearly independent modulo D(Tff12)'

(e) {yj,Ykht. , = 0 for alii :5 j, k :5 tn, (d) D(T) , ~

{y E D(T".,.,) I (y,y . }".<

~ 0 j" nil 1 S k S m),

then T: = M lv(T} is a selfadjoint differential operator corresponding to M,2 and 2. Collt'erse/y, if '1' is a selfadjoint differential operator corresponding to M , 2 alld 2, then there exist fune/ ions Yi>' . . , Ym E D(TM,l.1) such that (a) . (d) hold. Proof. First wc notice that nul{Tg, 2 2- ,\ ' ) = O. This is dear if ). is real and follows from the symmet ry of Tgr ,1,2 if .\ is nonreal. 'Choosing Zj := Yj for all 1 :5 j :5 m, Theorem 4.8 applied to M - ). instead of M clearly yields the first part of this result. For the proof of the conVerse we first notice that T - ). is a Fredholm operator by Lemma 4.5. III view of [6 , Theorem IV.2 .3] and the selfadjointness ofT we obtain ind(T - .\) nul(T - .\) - nul(l' - ).') O. The last equality is trivial for real). and for non real ). we have nul(T - ).) = 0 = nu!('1' - .\' ) since T is symmetric. Thus Theorem 4.8 and Lemma 4.7 yield the existence of y" ... , y"" Z" •• • , z", E D(TM.2•2 ) such that (b) holds, {yj, ZdM" = 0 for all 1 :5 j, k :5 m and D(T) = {y E D(TM.2.d {y, Zk}M" = 0 for all 1 :5 k :5 Tn} = {z E D(TM •2,2) {Yj, Z }u" = ofor aU 1 :5 j :5 m} . Hence (c) and (d) are also satisfied. Condition (a) is dear for real .\ and for nonreal .\ the selfadjointness of T implies def(Tff ~ 2 - ,\) = dcf(Tff12 - ). ' ) and, thus, nul(Tu ,2.2 - ).0) = llul(TM.2.2 _).) = m . .. .. 0

=

I

=

I

Analogously we obtain COROLLARY 4.10. Assume that M is J-symmetric, that). E Ii{ is such thatnul(TX, 2 2- ).) =

o and R(Tfl,2,2 -).)

is closed. Define m: = nul(TM •2 •2 If Yi>' .. , y", E D(TM,~,d are such that

'\) .

• '

284

Frentzcn

(a) {Y!,' " ,'Ym} is linearly independent modulQ D(T~I.1.1)' (b) {Yi,YIt}M,I=O!orafllSi,k S m,

(c) D{T);= {Y E D(TM ,2 ,2) I {y,YIt}M,1 = 0 for all I S: k S: m}, then T: = Mlv(T) is a J-sel/adjoint differential operator corresponding to M,2 and 2. Conversely, if T is a J -selfadjoint differential operator corresponding to M, 2 and 2, then th ere exist functions YI, ... , 'Ym E D(TM,2,2) such that (a) - (c) hold.

Race [9, Theorem 4.7 and Corollary 4.51 showed that Corollary 4.10 holds without the assumptions that nul(T~f 22 - A) = 0 and R(T~ 2 2 - >.) is dosed for some). E IK if m is dLOsen as } clim{y E N(MtM + 1) n £2 My E ill. For applications of Theorem 4.8 it seems desirable to provide a recipe for a possible choice of the functions YI,'" ,Ym+ and ZI, . .. , Zm. To this end we assume that

I

R(T~"'9) is closed alld lIuITgr"

,q

= nul Tgf +,9',,' = 0,

(4.1)

and define

( 4.2) Lemma 4.4 yields that R(T""p,q) and R(T;\f+,q',p') are closed and, thus, [0, Theorem IV.1.2[

and Theorem 3.9(c), (d) imply that

R(TM."v) = {z E LV

11

z+u = 0 for all u E N(11f +,v"p')} = LV

and

Hence we call make the following choices:

{Ul! ... ,u"'} is a basis of N(TM.p.v), WI,···, W... E LP' are such t hat

(4.3)

1

wtUi = oi.k for all 1

S i, k S m,

(4.4)

where 0i.k is the Kronecker delta, Vi>.'"

v'"

E D(TM + ,v'.p') are such that TM + ,q'.p,vi =

Wi

foralll SiSm,

{vm+!, .. . , v... +m+ } is a basis of N(TM +,9' ,1',), w".+1,' ••

'w m+m+ E LV are such that

1

vtWi = 0i,k

(4.5) ( 4.6)

(4 .7)

forallm+l S i,k $ m+m+, Um+I, ..• , U m+m"'

E D(TM,p,v) are such that TM,p,qUi = Wi

forallm +l$i$m+m+.

(4.8)

Frcntzen

285

If p = q = 2, then we can choose {uJ, ' . . , um} and {vmtJ," " V",tm t } as orthonormal bases and, thus, Uj

wi:=

{ Vj

for 1 ::£ j::£ m, 'f form + l ::£j::£m + m t , 1 += *,

._ {JUi

Wj. _

}vi

for 1 ::£ j ::;: m, .[ ] for m + 1 ::£ j ::£ m + mt,

For + = * this is just the choice made in [3]. An application of Theorem 3.7 shows that (4.3)

~

+ __ t.

(4.8) imply

l~. )

( 4.9)

with C2 ,] E M.nt,m and, thus, with Theorem 3.9(e) we obtain

{UI, "" Um+mt } is linearly independent modulo D(Tk,,,,q), {VI>' .• ,

Vmtm+} is linearly independent modulo D(Tft +,q',,,')'

(4.10) (4.11)

From Lemma. 4.G we obtain m + m+ = dim D('lf,f.p,q) / D(T~ 'M) and, hence,

D(Tu,p,q) = D(T~f 'M) + span{uJ,"" Umtmt },

(4.12)

D(TM + ,q',,,') = D(T~t ,q'.9' ) + span{v], . .. , V",+mt }.

(4 .13)

Further, T heorem 3.9(e) and Lemma 4.7 imply

{y E D(TM,,,,q) I {y, vkhr,t = 0 for all 1 ::£ k ::£ m + mt}, D(Tfft ,q',,,') = {z E D(TM+,q'.9') I {Uj,Z}M,+ = 0 for all 1 ::£ j ::£ m + m t }.

D(TXf,p,q) =

(4 .14)

(4.15)

THEOREM 4. 11. Assume that (1 .1) holds and define m,m t as in (4 .2). Furth er choose UJ, ... ,U.... t m+'vJ, . .. 'v m+m+ and C as in (4 .3) - (1.9). If A E M",+,m+mt and B E l\.1l"..mt mf are such that

(a) rank A = m + and rank B = m,

(b) ACE' = 0,

«)

D(T)

,= (y E D(TM , •• ) I B

(y,v,H•. , ) : = ( {y, v!+rn+}u,t

O},

then T: = M I1J(T) is a differential operator corresponding to M,p and q which is a Fredholm operator with index zero and

286

Frcntzcn

Conversely, itT is a differential operator corresponding to M,p alld q which is a Fredholm operator with index zero, then there exist A E M...+ ,m+m+ and n E M....m+ ... + such that (a)(d) hold.

Proof· Let A = (a,i,k) E M".t ,...+m+ and B = (bj,k) E M.n,m+.nt be such

that (a) - (c) hold.

Define m+",+

Yj: =

L

aj.I'u>, (1

:s: j :s: m+) and

m+mt

Zk: =

1-'= 1

L

h,,,v,, (1 S k:S m) .

,,= 1

Then dearly Yj E D(TM,p,q) and Ii E D(TMt,q',p')' From (al, (4 .10) and (4.11) we obtain (ii) and (iii) of Theorem 4.8 . Furthermore, = ACB+ = O.

Hence Theorem 4.8 implies the first part of the assertion. For the converse, from T heorem 4.8 we obtain the existence of fundions Yl,' .. , Ym+ E D(TM.p,q) and Ii>"" Z", E D(1!.ft ,q',p') such that (ii) - (vi) of that theorem hold . In view of (4.12) and (4 .13) there exist A = (aj,k) E M". ~.m+m t and n = (b j ,... ) E M".,m+m+ such that Yi =

for some

Yj,O

Yj,O

+

m + m~

L

aj,l-
E D(Tfr,p,q) and

In order to show that rank for all 1 S; k S; m 3.9{e) we obtain

(1 S; j S;

Zj,O

m+)

and

Zk

=

Zk.O

+

m+ mf

L

h ...v" (1 S; k S; m)

E D(Tfr~,q',p')' Then Theorem 3.9(e) implies (e), (d) and

A= m+, we consider al, . .. ,

+m + and define Yo: = {YO,Vk}M.+

=

ami

E K such that L7=~1

ajaj, k

= 0

L::~ Ct;y;. T hen Yo E D(l M.p,q) and with Theorem

m~

.... +mf

;=1

; =1

L L

Ct;a;,j{Uj,V k }M.+ ;::;

0

for alll S k S m + m+ . Hence (4.14) implies that yo E D(Tf'.p,q) and, thus, Ct; ;::; 0 for all 1 S; i S m + in view of (ii) of Thcorcm 4.8. Analogously rank D = m follows from (4 .Hi), 0 Of course there are other possible choices for the functions U; and Vi> which lead to different forms of Theorem 4.11, see for example [31 and [7] for the case p = q;::; 2. REMARK 4. 1 2. The results of this section also hold fo r weighted L" spaces and can even be reduced to those for unweighted ones (d. [5, Section 6]).

Frcntzcn

287

References 1. E. A. Coddington and A. Dijksma. Adjoint subspaces in Banach spaces with applications to ordinary differential subspaces . Ann. Mat. Pura Appl. 118 (1978), 1-118. 2. W. D. Evans. Regularly solvable extensions of non·self-adjoint ordinary differential opC!fators. Proc. Roy. Soc. Edinburgh Sect. A 97 (1984),79-95. 3. W. D. Evans and S. E. Ibrahim. Boundary conditions for general ordinary differential operators and their adjoints. Proc. Roy. Soc. Edinburgh Sect. A 114 (1990),99-117. 4. W. N. Everitt and O. Race. Some remarks on liJlear ordinary quasi-differential expressions. Proc. London Math. Soc. (3) 54 (1987),300·320. 5. H. Frentzen. Quasi-differential operators in £1' spaces. J. London Math. Soc ., accepted for publication. 6. S. Goldberg. Unbounded linear operators (New York: McGraw-Hill, 1966). 7. S. E. Ibrahim. SinguJar non-selfadjoint differential operators. Proc. Roy. Soc. Edinburgh Sec!. A 124 (1991), 825-841. 8. M. A. Naimark . Linear differential opemtors, T,art II (New York: Ungar, 1968). 9. O. Race. The theo ry of J -selfadjoint utensions of J -symmetric operators. J. Differential Equations 57 (1985), 258-271. 10. H. H. Schaefer. Topological vector spaces (New York: Springer, 1971). 11. J . Weidmann. Spectral theory of ordinary differential operators. Lecture Notes in Mathematics 1258 (Berlin: Springer, 1987) . 12. A. Zettl. Formally self-adjoint quasi· differential operators. Rocky Mountain J. Math . 5 (1975), 453-474.

Fachbereich 6 - Mathematik und Informatik, UniversitiLt GliS Essen, D·45117 Essen, e-mail address:[email protected]

Eingegangcn am 13. August 1997