RELATIVELY BOUNDED AND COMPACT PERTURBATIONS OF NTH ORDER DIFFERENTIAL OPERATORS

A perturbation theory for nth order differential operators is developed. For certain classes of operators L, necessary and sufficient conditions are obtained for a perturbing operator B to be relatively bounded or relatively compact with respect to L. These perturbation conditions involve explicit integral averages of the coefficients of B. The proofs involve interpolation inequalities.

DEFINITION B. Let be a differential expression of the form Wilp ai(t) (D ), where W is a positive Lebesgue measurable function defined on l and each a, is a complex-valued function on I. Then the maximal operator L corresponding to has domain D(L) { y I.(I)" y-" AC(1), /[y] e /,(I)} ,__oa,(t) (y D(L)).If a, .Cl(1) for 0 < < n and a, 0onl, then the minimal operator L corresponding to is defined to be the minimal closed extension of L restricted to those y D(L) which have compact support in the interior of L In the Hilbert space setting of L=(1), most of the smoothness requirements on the coefficients a, (0 < < n) are not needed, and the theory is developed in Naimark [7, sect.17].
We consider perturbations a P," D' l=O in the setting of L(a, ,o), where < p < and W is a positive Lebesgue measurable function defined on (a, *,,).Definitions and conditions for P and P, are given in the hypotheses of Theorems 1.1 and 1.2, respectively.We give conditions on certain averages of the perturbation coefficients bj (0 < j < n-1) which are sufficient and, in some cases necessary, for B to be T-bounded or T-    PERTURBATIONS OF nTH ORDER DIFFERENTIAL OPERATORS compact.These results rely heavily on Theorems A and B, which are special cases of Theorem 2.1 in Brown and Hinton [3].These two theorems give sufficient conditions for weighted interpolation inequalities of the form: there exist dj > 0, r/> 0, K > 0, and e > 0 such that for all e (0, e o) and y in a class D of functions, where0 <j < n-landl < p < ,,*.
Theorem 1.1 gives integral average conditions on bj (0 < j < n-1) which are necessary and sufficient for B to be T-bounded or T-compact in the case when < p < and P and W satisfy the conditions in Theorem 5 in Kwong and Zettl [6].When W =-1, these conditions imply that the coefficients of T are bounded above by the corresponding coefficients of an Euler operator.Furthermore, the perturbation conditions for T-compactness of B are sufficient for the essential spectrum and Fredholm index to be invariant under perturbations of T by B.   By definition (Goldberg [4,), the essential spectrum of T, written tr,(T), is the set of all complex numbers 2 such that the range R(21 T) of 21 T is not closed.The essential resolvent of T, written p, (T), is the complement of this set.By definition (Goldberg [4, p.   102]), the Fredholm index to(T) is given by to(T) ct (T) fl (T), where o(T) is the dimension of the null space of T and fl(T) is the dimension of/.(I)R (T). or (T) is called the kernel index of T, and fl(T) is called the deficiency index of T.
In Theorem 1.2, the results in Theorem 1.1 for the single-term operator T are extended to the multi-term operator L.An nth order perturbation of L is considered in Corollary 1.1.Sufficient conditions are given for invariance of the essential spectrum and Fredholm index of L under such perturbations.
Theorems 1.1 and 1.2 and Corollary 1.1 provide generalizations of results of Balslev and   Gamelin [2] as presented in Goldberg [4, pp.166-175].Their work deals with bounded coefficient and Euler operators in the unweighted setting of LP(a, .,,)for < p < oo.In Theorem 2.1, the sufficiency conditions in Theorem 1.1 are generalized for operators T with arbitrarily large coefficients.Again, these conditions involve integral averages of the perturbation coefficients bj (0 <_ j _< n-l).Theorem 2.2 gives pointwise conditions on bj (0 _< j _< n-l) under which the conclusions of Theorem 2.1 hold.The case in which p is covered by Theorem 2.2.Also, perturbation conditions which are sufficient for L-boundedness or Lcompactness of B are obtained for the case p and the case in which the coefficients of L are arbitrarily large.These theorems rely heavily on investigations by Brown and Hinton [3] on sufficient conditions for interpolation inequalities.Examples of each theorem are presented and contrasted for the situation in which the coefficient in T is an exponential function.
The final theorem, Theorem 3.1, deals exclusively with the case p 1. Sufficient, integral average conditions are given for T-boundedness of B.

INTEGRAL AVERAGE CONDITIONS FOR EULER-LIKE OPERATORS
In this section we consider operators whose coefficients are powers of a fixed function s times a weight function w and a bounded function.In the simplest case, i.e., w(t) s(t) 1, Theorem 1.2 gives Theorem VI.8.1 of [4].For tx 0, w(t) 1, and s(t) t, the sufficiency condition of part (ii) of Theorem 1.2 yields Corollary VI.8.4 of [4] for perturbations of the Euler operator.Since we do not require w(t) or a 0, we refer to the unperturbed operator of Theorem 1.2 as Euler- like.

k-I
Let a be the maximal operator with action given by a 17 bj D By the sufficiency 1--0 argument above, A is T-bounded.Thus since B is T-bounded, Minkowski's inequality implies that A n-! + B is T-bounded.Note that (A + B)y W/, , bj yO), y D (T).With and , defined l=k as above (see (1.9)), define h(t) O(t) ., t>a.
Then h C'(R) and h (k) -= on[0,1].For each r > a, define (1.13) hr(t where u ds(r)" (A + B)h,= W p for r < < r+s(r), and (1.15) By Lemma A, we obtain for a constant C, g. (r) r) where the last inequality follows from the relative boundedness of A + B with respect to T. By calculations like those used in deriving (1.11) and (1.12), we obtain for r >_ a, Ilh, il" <_ c, w(r) s(r) (1.17) and where C and C are constants independent of r.Thus (1.6) implies that (1.1) holds forj k and any d (0, 1/(2N0)).This establishes necessity of(1.1).

By
We show that B N converges to B in the space of on (N, .o). bounded operators on D(T) with the T-norm.First note that T is closed.To see this, let f, -f and Tf.--.g in /.,(a, .,,).Let J be a compact subinterval of [a, *,,) and restrict the functions f, f,, and g to J. Define Tj /.(J) ---) /.(J) to be the maximal operator corresponding to z on J.
Clearly, f. --4 f in /,(J) and f.D (T).Since Tf, (Tf.)l,,Tf. g in L(J).By Theorems VI.3.1 and IV.1.7 in Goldberg [4], T is closed.Therefore, f D(T) and Tjf g.Thus, f D (T) and Tf g.Hence T is dosed.By the argument used in proving sufficiency in (i), Theorem A applies to the interval I [N, .o)with the same choices for the weights, f, and e 0. By (1.3) and (1.4), for 0 < e < S(e) < C sup {w(t)-'s(t) with a similar estimate for S:(e), 0 < e < 6, where C is a constant independent of N and e.It follows from Theorem A that for all y e D(T), I' b, Y"'I" <--I W IY[ p + I? P '(' where Cj is independent of y and N (but depends on e).Use of (1.23) in (1.19) gives IIy->1 _< Z c, sup gj., (t)   Ily]l ,=o ,,,.-, (1.24) for all y D(T) such that y 0. By (1.2), the term on the right side approaches 0 as N ---> ,,*.
Therefore, Bu ---> B in the space of bounded operators on D(T) with the T-norm.
Next, we show that each Bu is T-compact.Let {ft} be a T-bounded sequence, say IIf, L -< ' for all I.We will show that {fJ'}, 0 < j < n-1, is uniformly bounded on [a, N].
Necessity.Suppose B is T-compact.First we show that (1.2) holds for j 0. We proceed by a contradiction argument.Suppose that for any d; 0, l/(2N0) ), there exists e > 0 and a sequence {r}7__i of positive numbers such that r --> and > e, > 1.
To prove Theorem 1.2, we will use the following lemmas.

1-e
Since a is C-compact, {ayn} contains a convergent subsequence.Since {Yn} was an arbitrary (C + D)-bounded sequence, A is (C + D)-compact. 1 LEM1VIA 1.2.Let B, L and T be the operators in Theorems 1.1 and 1.2.Then: B is L-bounded if and only if B is T-bounded.Further, the relative bound for B with respect to L is 0 if and only if the relative bound for B with respect to T is O. (ii) B is L-compact if and only if B is T-compact.PROOF.Consider the differential expression l-'r W Ip p, tl, D'.Its I--0 coefficients satisfy the perturbation conditions (1.1) since for e I and 0 < < n 1, p, < (constant).s(t) la, w s "+') s(t) --, a, (0 < < n-1) e L"(1).Hence by Theorem 1.1(i), (i) Suppose B isL-bounded.Then B is (-,/L-bounded since L"(1).Another an application of Lemma 1.1 (withA =B, C ml L, and D T-__1 L) shows thatB isT- bounded.
Next, suppose B is T-bounded.By Lemma 1.1 (with A B, C T, and D is The statementaboutzerorelative bounds also follows from Lemma 1.1.(ii) This part is proved in a similar manner using Lemma 1.1 (ii).I PROOF OF THEOREM 1.2.
(i) Sufficiency.Suppose (1.42)holds for 0 _< j _< n land some t$ 0, 1/(2N0)).By Theorem 1.1(i), B is T-bounded with relative bound 0. Hence Lemma 1.3 implies that B is Lbounded with relative bound 0. The result D(//v) D(L) follows by the same argument used in showing that D(T/v) D (T) in the proof of Theorem 1.1.
(ii) Sufficiency.S.lppose (1.43) holds for 0 j P-and some /0, No ) Then by Theorem 1.1, B is T-compact and hence L-compact by Lemma 1.2.The invariance of the essential spectrum and Fredholm index of L under perturbations by B follow as in the proof of Theorem 1.1.
Necessity.Suppose B is L-compact.Then B is T-compact by Lemma 1.2.By Theorem 1.1, there exists t$ 0, 1/(2N0) suchthat bj (0 _< j _< n 1) satisfy(l.43). 1 REMARK.Theorems 1.1 and 1.2 apply to operators T and L with coefficients eventually bounded above by the corresponding coefficients of an Euler operator.To see this, note that the hypothesis s'(t)l -< No a.e. on I implies that there exists a positive constant C such that s(t) < C for all sufficiently large.Now, by definition of P, and W and the hypothesis that a, (0 < < n) L"(1), we have la,(t)l ,(t) s(t)' < C, t' (1.44) for all sufficiently large, where C, are constants independent of and 0 < < and L-compact if and tel onlyif lim g.,(t) 0 if=0, 1) forsome d (0, l/(2N0)).
Next we prove a corollary of Theorem 1.2 in which an nth order perturbation B of L is considered.The perturbation is such that the coefficients of the highest-order terms in L and L + B obey the same hypotheses.Before stating the corollary, we prove a lemma concerning the domains of the single-term operator T and multi-term operator L.
Then R(A/-L)is closed and L(a, *,,) R(it/-L) M, where M N(I-I.:).Since L*y ity has at most n L(a, oo) solutions, M is finite-dimensional.
-/)[,/.,O)]bl, } the same estimate holds for S(e) up to a multiplicative constant.By Lemma A, for and E