ON THE SPECTRAL THEORY OF STRONGLY ELLIPTIC DIFFERENTIALOPERATORS*

BY FELIX E. BROWDER

DEPARTMENT OF MATHEMATICS, YALE UNIVERSITY

Communicated by Einar Hille. July 27, 1959

In a recent note in these Proceedings, ' we have discussed existence theorems for ageneral class of boundary-value problems for elliptic partial differential equations,basing our argument upon general a priori estimates for the LI and Holder normsfor the highest order derivatives appearing in the differential operator. We men-tioned at the beginning of that discussion that the techniques which we wereapplying yielded much sharper information for the case of the Dirichlet problemfor a strongly elliptic operator (and indeed, for a whole class of related problemswhich we shall describe in another place). It is the purpose of the present note todevelop some of these sharper results. As before, we make minimal assumptionson the continuity and regularity of the coefficients of our differential operator, andobtain results both in the case of bounded and unbounded domains.2

Let A = EZaj < 2m aa(x)Da be a differential operator of order 2m, (m > 1), whosecoefficients are complex-valued functions defined on an open subset G of the Eucli-dean n-space E' with boundary r. Here, as in our preceding note, for each orderedn-tuple a = (a, . . . , aan) of nonnegative integers, we set D' = [(1/i) (/Ixn) ]" . . .1(1/i) (6/1xn) ]anx a = E 7 I. as. For each n-vector t = (Q1, . . . , in) we seta = ((%,)ai . . . , (Q)"n). The characteristic form a(x, t) of A is defined for x in Gand any real n-vector t by

a(x, t) = EZal = 2maa(x)t'.

The partial differential operator A is said to be uniformly strongly elliptic on Gif there exists a constant c, > 0 such that Ret a(x, t)} 2 cl I t 2m for all x in G andall real n-vectors t. I A will be said to be essentially real if the top-order coefficientsaa for a = 2m are real-valued functions on G.Our basic regularity assumptions on the coefficients of A -and on the domain G

are the following:(I) G is uniformly regular of class C2n. (For the details of the definition of uni-

form regularity of G, we refer to our preceding note.')(II) The coefficients aa ofA are uniformly bounded on G for all a.(III) The top-order coefficients aa of A for a = 2m are uniformly continuous

on G.The formal adjoint A' of A is defined in general as a differential operator with

coefficients which are distributions of order at most 2m - 1, by

A'u = EZal < 2m Da(aa(X)U) = Elal <2mca'(x)Da,

for u in Cc (G), the family of infinitely differentiable functions with compact sup-port in G. The coefficients of A' can be expressed by the Leibniz formula as linearcombinations of derivatives of the a,. ID some, though not all, of our results, weshall utilize the following assumption on the coefficients a,'':

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1424 MATHEMATICS: F. E. BROWDER PROC. N. A. S.

(IV) There exists a constant M > 0 such that I aa' (x) < M for all a, almosteverywhere in G.We are interested in studying the spectrum and the resolvent of the differertial

operator A realized in L2(G) (and more generally in L'(G) for 1 < p < ca) withdomain consisting of functions satisfying null Dirichlet boundary conditions on r ina suitable extended sense.We recall the definition of the Dirichlet problem in its precise terms. We con-

sider the Sobolev space W1m "(G) which, for each p with 1 < p < c, consists of allu in L-(G) for which all the distribution derivatives Dau also lie in LP(G) for Ia | <2m. This linear subset of LP(G) becomes a reflexive Banach space with respectto the norm II8lp2m,P = I Elal . 2mIIDauIILPE()}G/p* Under the assumption of regu-larity (I) upon the domain G, it follows that, if R(G) is the subfamily of W2m' P(G)consisting of the restrictions to G of infinitely differentiable functions with compactsupport in E', then R(G) is dense in W21 P(G), and in addition, the mapping whichassigns to each function u in R (G) the corresponding function Du Jr on r for a fixedj8 with | < 2m, is a bounded linear mapping from the dense subset R(G) ofW21n P(G) into LP(r), where the latter space is taken with respect to the (n - 1)-dimensional volume induced on r by En. If we continue to denote the unique con-tinuous extension of this linear map to a linear mapping from W2m P(G) into LP(r)by DI Ir, we may formulate the following definition:

Definition 1.-The domain of Ap, 1 < p <K, i.e., the domain of the differentialoperator A realized in LP(G) under null Dirichlet boundary conditions, is the familyof all u in W2n, P(G) for which Yu Ir = 0 for |,8 I < m. For such u, we setApu =Au, the latter obviously being a well-defined element of LP(G), and Ap is given as alinear transformation from D(A,) c LP(G) into L"(G).Remark 1: The second condition on the vanishing of the extended derivatives

of order < m on the boundary of G is equivalent, as follows from an easy argument,to the following condition which is similar to the generalized Dirichlet null-condi-tion considered in the variational treatment of the Dirichlet problem: For eachpoint x in r, there is a neighborhood N of x in En and a sequence of function {I i}from R(G), each vanishing identically on a neighborhood of N n F, such thatIDI(u - PJ) IILP (N) -. O asj--~ oo forall,3with flB|< m.The basic a priori estimate upon which our discussion is based is the following:

There exists a constant k, > 0, depending only on the constant of uniform ellip-ticity cl, the bounds of (II) and the modulus of continuity in (III) as well as p, suchthat for u in D(A,)

IIUII2m, p < kp{IIAuIILP(G) + IIUIILP(G)}4 (1)

Remark 2: The inequality (1) implies immediately that Ap is a closed operatorfrom LP(G) into LP(G). Indeed, for a single differential operator A (omitting thedependence of kp upon the regularity conditions), it is equivalent by the closedgraph theorem to the fact that Ap is closed.Our objective in the discussion which follows is to prove the following theorems:THEOREM 1.-Let A be an uniformly elliptic partial differential operator of order

2m on a domain G of E', with A and G satisfying the regularity conditions (I), (II),and (III) above. Then:

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VOL. 45, 1959 MATHEMATICS: F. E. BROWDER 1425

(a) Given E > 0, there exists k(e) > 0, such that for all real numbers t and all u inD(A2).

|| (A + MI)uIIL2(G) . {(1 - E)( - k(E) }IIUIIL2(G)-(b) If, in addition, the domain G is bounded, the resolvent set of Ax, for every p with

1 < p < a, eventually contains every ray in the complex plane emanating from theorigin and making an angle of less than 7r/2 with the negative real axis.

(c) If in addition, G being bounded or unbounded, the coefficients of the top orderterms aa ofA (I a = 2m) satisfy an uniform Holder condition on G, then there existsa constant X with 0< X < 1 and a constant k1 > 0 such that for all u in D(A2), ( 02),

I| (A2 + tI)UIIL2(G) . {t - kit" - kl}i IuIL2(G).In this case, if G is bounded, then for g on the spectrum of A p,

Re(h) > -k2(1 + Im(o)|X'),where A' is a constant with 0 < X' < 1.

(d) Suppose that the coefficients of the adjoint operator A' to A satisfy the regularityassumption (IV) above, r locally of class 4m, and that our previous hypotheses areverified. Then, G being bounded or unbounded, the resolvent set of A2 contains a lefthalf-plane Re(r) > k, for some constant k, and on the real axis for t > k,

I I(A2-U)-'I I < Q -k)-1THEOREM 2. Suppose that A and G satisfy condition (I), (II), and (III), and

that G is bounded. Then for t large and negative, (Ap- tI)-1 exists and is a compactlinear transformation in LP(G). In particular, the spectrum of Ap is discrete, theeigenspaces are of finite dimension, and (Ap-DI) -I exists if (A,, -RI) has a trivialnull-space.THEOREM 3. Suppose that A and G satisfy the regularity conditions (I), (II), and

(III), where A is uniformly elliptic and essentially real on G. Then given e > 0,there exists k(e) > 0 such that for r complex with Re(r) 2 0, u E D(A2),

||(A2 + MI)UIIL2(G) 2 { (1 - E) k1 k(e)} IJUIIL2(G).In particular, if G is bounded, the spectrum of Ap for all p with 1 < p < co, lies inafixed half-plane Re(r) > k4.THEOREM 4. Suppose that A is uniformly elliptic and essentially real on a domain

G of E , with A and G satisfying the regularity conditions (I), (II), (III), and (IV),and r1 locally of class 4m. Then:

(a) There exists a constant k5,p > 0 such that for all p with 0 < p < o, A1 =[(2m - 1)/2m], the spectrum of Ap is contained in the set

{I: IIm(r) < k6,s (1 + Re(r)X), Re(h) > -k6}.

(b) For all r with Re (I) < 0, r > k6,Jf(A2 - 'I)-11 . {Ir -k6I 1x2 - k6}-1,

where X2 = [(em - 1)/(92m + 1) ].We shall prove Theorems 1, 2, and 3 in their general form, but Theorem 4 only

for p = 2. The proof of the latter result for general p will be given in another

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1426 MATHEMATICS: F. E. BROWDER PRoc. N. A. S.

place. For these proofs, we shall make use of Theorems 5 and 6 below, which wehave previously announced and whose proofs we sketch below. Some of the de-tails of the proof of Theorems 5(b) were given in ref. 1 (essentially the completeproof for p = 2, or for G bounded) and we present only that portion of the argumentfor general p and G which was omitted there for lack of space.THEOREM 5.-(a) Suppose that A is uniformly elliptic on G with A and G satisfying

the regularity conditions (I), (II), and (III). Let N, be the null space of Ap. ThenNp < Np, if p < Pi equality holding if G is bounded.

(b) Suppose that A is uniformly elliptic on G with A and G satisfying the regularityconditions (I), (II), (III), and (IV), r locally of class 4m. Let 1 < p, q, (p'- + q-I =1), and let Ap and A' be the realizations of A and A' under null Dirichlet boundaryconditions in LP(G) and L (G), respectively. Then (A,)* = A .

Proof of Theorem 5(a): This follows from the sharper formulation of the a prioriestimates as given in our preceding note. i

Proof of 5(b): Since under the hypotheses of 5(b), A, and A' are closed oper-ators and the operators A and A' are interchangeable, it suffices to consider thecase q > 2. It suffices, moreover, to show that if u lies in the domain of (Ap)*,then u lies locally in W21n q on a neighborhood of each point of G u r. We proceedin two stages: (1) We show that u lies locally in W2m, 2; (2) If u lies locally inW2n 2 and if A'u lies in LV(G), then if u satisfies the null Dirichlet conditions in theW2M 2 sense locally at each point of r, then u lies locally in W~M q.For the proof of (1), let G1 be a bounded uniformly regular subdomain of G having

a portion F' of its boundary rI contained in r. We define two operators A2, e and2, 1 in L2(GI) as follows:

D(A2, 0) = {U: u E WM 2(GI); DurIr = 0 for < m; Du Iri-r- Ofor <3j< 2m},

A2, IOU= Au, for u in D(A2, o);

D(A',,) = {u: u lies in W2mn 2 locally in a neighborhood of everypoint of G1 u F', A'u lies in L2(G,); Du Ir' = 0 for < ;<

A2I = A'u, for u in D(A',,).The operators A2, o and 4, 1 are closed operators in L2(G1), as follows from simplevariants of our a priori estimates. Moreover, the restriction of our given functionu from D(Ap) to GU clearly lies in the domain of A4 O. It therefore suffices to showthat A2* = A4 1. By the procedure of our preceding note, we may assumewithout loss of generality that the coefficients ofA and A' are infinitely differentiablefunctions on G,. It suffices by the argument of our preceding note to show thatA, oA2, o = Al, 1A2, o and that (A4, 1)*A' 1 = A2, oA2, I.The argument for the second equality is the simpler, since if v lies in the domain

of (A4, )*A4 1, then A4, v lies in the domain of the operator (A, 1) * and, since theoperator A2' (taken for the domain G1) is contained in A, 1, (A. 1) * c (A /)* - A2.But if A4'v = vl lies in D(A2) and if in addition (v,, A4, w) = (f, w) for some f inL2(GU) and all w in D(A, 1), the same equality persists if we set vl = 0 outside GUand consider all w in D(A'). It follows immediately that vl satisfies the nullCauchy boundary conditions on r, - r', and hence that v lies in D(A2, ,4, 1).

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VOL. 45, 1959 MATHEMATICS: F. E. BROWDER 1427

For the first equality, we observe that if v lies in D(A2, O A2, a), then v lies inD(A2, o) and (A2, ov, A2, Ow) = (f, w) for some f in L2(G,) and all w in D(A2, o). Inother words, v is a solution of the variational boundary-value problem correspondingto the coercive form (A2, ov, A2, A) on the subspace D(A2, C) on W21n, 2(G,), (thecoerciveness being a consequence of the a priori estimate in W~M 2(G,)). By theregularity theory for solutions of variational boundary-value problems, v lies inW4, 2(G1) and therefore A2, ov lies in D(A', 1), since it is a function v, from W2m 2(G,)satisfying the equality (v,, A2, Ow) = (A'v,, w) for all w in D(A2, 0).Remark 3: It is a direct consequence of Theorem 5(a) that if N(A2 - rI) is

trivial, then so is N(AP- rI) for every p with i < p < 2 (if we apply 5(a) to theoperator A - rI instead of our original operator A). If G is bounded and we ob-tain a restriction on r for (A2 - rI) to have a nontrivial null-space which dependsonly on the regularity constants of (I), (II), and (III), then approximating A bydifferential operator A (k) uniformly on G, where the A (*) have infinitely differentiablecoefficients on the closure of G, we see that each A (k) - t I will have a trivial null-space for the excluded region in D. By the variational theory of the Dirichlet prob-lem for strongly elliptic operators with regular coefficients, (A2(k) -rI)-I will existfor all sufficiently large k. If the norms of the operators (A2(k) -rI)-I are uni-formly bounded for large k (as they will be if inequalities of the form Theorem 1 (a),1(c), or Theorem 3 have been established), it follows immediately that (A2 -CI)-1 exists on all of L2(G). For f in LP(G), however, if u = (A2 -DI)-1f, u liesin W~M 2(G) and for p'1 > 2-1 - 2m-n-1, it follows that u and Au both lie inLP(G). By the sharper form of our a priori estimates, u must lie in W~M P(G). Itfollows that for such p, (Ap -DI)-I exists. If we continue the boot-strap proce-dure of which the preceding remarks are the first step, we see finally that (A p-CI)-I exists for all such r for 1 < p < c.

Proof of Theorem 1 (a): Under the hypothesis (I) that G is uniformly regular ofclass C2m, given any 6 > 0, there exists a family of functions {14,} in C' (Es), eachhaving support of diameter less than 6, such that >j [#j(x) ]2 = 1 on G, and withthe additional property that for any two indices a and f3 with a , | 8 < 2m,E>j D¶aj(X) D$&j(x) < ka, (6) < oo, where ka, 9(6) < k', a6a -la1.

Let t be a real number, t > 0, u any element of D(A2). Then, (designating byIJvj| the L2-norm of v, unless otherwise indicated),

1 (A2 + (I)UII2 = (Au + (u, Au + tu) = lIAu 112 + 42I|U112 + 2(Re(Au, u). (2)By the a priori estimate (1) above, IlAu 112 > COIIU11m, 2- klIu 112, with constants

co and k > 0 which are independent of u.Choose a > 0, and a corresponding partition of unity {4,'jl as above. We have:

(Au, u) = EJEIal < 2m (aa(x)DG(1j2u), U). (3)

Choosing a point xj in the support of Oj and in G, for a | = 2m,

(aa(x)Da(#ju), u) = aa(xj) (Da(Oju), )PjU) +({aa(x) - aa(xj)}Da(#ju), u) + EZst,IXI<2mC0, x(aa(xj)D(6Pju)DXu, 0,tju). (4)

Letw(b) = EZaj=2m sup { a. (x) - aa(y) }. Summing (4) over all a withIx-yj <.

a | = 2m and combining the result with (3), we obtain,

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1428 MATHEMATICS: F. E. BROWDER PROC. N. A. S.

(Au, u) = EZI:al>2maa(Xj) (D 6hu), #jU) + R1(u) + R2(U), (5)where

|R1(u) < ki E2m- 1 6-2mIIUI1r 211U11 (6)

|R2(U) < k2W(6)IlIUI2m, 211U IIBy the Fourier transform argument of Van Hove and Gdrding, however,

ReI EJEZIaI =2m aa(xj) (Da(#ju) X Oju)} > 0 for all u in D(A2) (since all such u satisfythe null Dirichlet boundary conditions in the variational sense). Hence, it followsfrom (2), (3), (5), and (6) that

JJ(Au) + tU)112 > COIIU1IJm,2 - kIlU112 + 021IUI12 -tk2W(6)11UII2m, 2IUIl- ki EZ2m1- 1 6-2mIlU IrI 21|U I. (7)

It follows from an elementary analytic argument that there exists a constant k3such that for all u in WIM, 2(G) on 0< r < 2m,

U11ur, 2 < k31uIIj/2m2.llU~j(2m-r)/2m (8)Using (8), we see that

(kjiar-2m11ulUr, 21U11 < {k 6'r-2m 11u11r/2M" UI(4m - r)/2m <

4*2m llu ll2m, 2 + 2 llul12 + k45 - 4mem/(2m7) llu 12. (9)

Under the hypotheses of Theorem 5(a), w(b) -- 0 as 6 -a 0. Let e > 0 be given,and choose 6 so small that w(6) < (cOE) /2k2-1. Then

tk2W(S)IIUII2m, 2111U11 < E6a2lU 112 + k22[w(b)]2 (4e)-lIUl|m <

E24u 112 + - |I|U|lI2m 2. (10)

Summing (9) over r from 0 to 2m - 1, and combining the result with (7) and (10),we obtain,

II (Au + (U)112 > 12 II|m, 2 + ((1 - )02- K(E, 6))IuI21 (11)where K(e, 6) is bounded by a rational function of E and 6. In particular, droppingthe (2m)-norm of u and taking the square root of both sides,

|| (Au + (u) || > { (1 - e)t - Ki(e) I I|u11, (12)and 1 (a) is proved.

For the proof of Theorem 1(c), we remark that if the coefficients of A of highestorder are Holder-continuous, the function w(b) is bounded by a power of 6. In thatcase, Ki(e) may be taken in the form k5E-' for some positive M. We have, therefore,

|| (Au + {u) || > { (1 - E)t -k5-} lull.Choosing e = '/2 for r < 2, e = t-'/(,u + 1) for >> 2, we have the inequality of

Theorem 1(c) withX = 1/(r + 1).The further conclusions of (b) and (c) of Theorem 1 for r real follow from Remark

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VOL. 45, 1959 MATHEMATICS: F. E. BROWDER 1429

3 and Theorem 5(a). The conclusions on the location of the spectrum follow fromthe first resolvent equation for closed operators.We shall obtain Theorem 1(d) by means of the following result which is of in-

terest in its own right:THEOREM 6. Suppose that A is an uniformly elliptic differential operator on G,

with the boundary r of G locally of class C46, and A and G satisfy the regularityand boundedness conditions (J), (JI), (III), and (IV) above. Then:

(a) There exist constants c > 0, k > 0, such that for all u in D(A2),

Re(Au, u) > CIII|U2, 2 - kilUI1122(b) If A is formally self-adjoint, i.e., A = A', then A2 is a Hermitian operator in

L2(G), which by (a) has its spectrum bounded from below on the real axis.Proof of Theorem 6: Consider the differential operator H = '/2(A + A'). H is

formally self-adjoint, and the hypotheses (I), (II), (III), and (IV) on A imply thecorresponding facts for H, which in addition is uniformly elliptic on G since itstop-order coefficients are the real parts of the corresponding coefficients of A.Hence by Theorem 5(a) for p = 2, if H2 is the realization of H in K2(G) under nullDirichlet boundary conditions on the boundary of G (in the sense which we havedefined above), then H2 is a Hermitian operator, i.e., H2* = H2. On the other hand,by Theorem 1(a) which we have already established, II(H2 + {I)uII . hull for tsufficiently large and positive. Since H2 is closed, it follows from this latter factthat for such I, the range of (H2 + {I) is closed in L2(G). Since (H2 + (I)* = H2 +{I, the orthogonal complement of this range is the null-space of H2 + aI, which bythe inequality is trivial. Hence (H2 + SI) 1 exists on all of L2(G) for r large andpositive, and H2 must be bounded from below (in the sense of the ordering of Hermit-ian operators) by -kI, i.e. (H2U, U) > -k(u, u) for all u in D(H2). But D(H2) =D(A2), and Re(A2u, u) = (H2U, u), which imply that

Re(A2u, u) > -k(u, u).

Consider finally A - e(- 1) 'An = A,, where e is a small positive constant and Ais the ordinary Laplace operator in En. For e sufficiently small, A, satisfies all thehypotheses on A and hence for u in D(A2) = D(A,,2), Re(Au, u) 2 - k(u, u).But, (Afu, u) = (Au, u) - E(-1)m(Amu, u), while (- 1)m(Amu, u) 2 C l|ul1,2.Therefore,

Re(Au, u) = Re(A Eu, u) + E(- 1)m(Amu, u) 2 C'EhU|lr, 2- k luhl2, and all parts ofthe theorems have been proved.

Proof of Theorem 1(d): We again consider

l1(A2 + $J)Ull2 = (Au, Au) + 2(U, u) + 2tRe(Au, u).

By theorem 6(a), Re(Au, u) 2 -k(u, u). Hence for t . 0,

hl(A2 + j)l11 2(> 2 - 2kt - k)llu112 > (- k6)2 - k721u 112.Using the fact that (a2 - b2)12 > a - b if a > b > 0, we obtain I1(A2 + tI)U >-k6 - k7) IuII. By the same argument, I|(A2 + tI)uII > ( -k6 - k7) ull.

The first inequality says that for t > k6 + k7, the range of A2 + (I is closed. Itsorthogonal complement is the null-space of (A2 + (I) | = A2' + (I, which by the

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1430 MATHEMATICS: F. E. BROWDER PROC. N. A. S.

second inequality is trivial under the same restriction on t. Hence, (A2 + {I)1exists for t > k8 = k6 + k7, and satisfies the inequality

II(A2 + (')-'11 < ( - k8)-1.The assertion that the spectrum lies in a right half-plane follows once more from

the first resolvent equation for closed operators.Proof of Theor-m 3: We proceed as before, except that we now consider 11(A2 +

UI)ulj for Re(r) > O,u in D(A2).II(A2 + D)U 12 = I1A2UI12 + |D 121|U||2 + 2 Re {D (Au, u)}.

Carrying through the same decomposition as before and using the fact that ifA is essentially real, the top-order coefficients aa are real-valued functions on G, wefind

Re f(Au, u) > Re(r) ZJEicri=2mraa(xj) (Da61(ju), ftju) - l¢j|{Ri(u) + R2(u)J,where Rj(u) and R2(u) satisfy the inequalities (6) above. If Re(t) 2 0, the first termis nonnegative, and the remainder of the proof is identical with that of Theorem 1 (a)if one substitutes for t in the latter.

Proof of Theorem 3: We have already observed that (A2 -{I) -1 exists for t largeand negative if G is bounded, and by Remark 3, so does (A, - I) -. But if(A, - I)-1 exists, it must be compact for G bounded and regular, since for u =

(Ap -{I)-1V, IIUII2m, p <cp. v| and on a bounded regular domain sets bounded in(2m, p)-norm are pre-compact in L'(G). The remainder of the assertions followfrom the compactness of (Ap-t) -I1 for any single value of t.

Proof of Theorem 4: Let H = '/2(A + A'). H is formally self-adjoint, and ifA is essentially real, A = H + R, where R is of order < 2m. Let R2 be R restrictedto the domain of A2, H2 and A2 as before. Suppose that R(¢) > 0. Then for u inD(A2),

II(A2 + MU|11 >.|(H2 + UI) - |IR2UII, (13)1I(H2 + UI)U 12 = IIH2U 12 + | 12I1U112 - 2Re(r) (H2U, U) >

I 1 12 - kg ¢ - kio} 1IUI12 + CI1UI12m,2 (14)IIRUII < kiiIIUll2mi, 2 < k12I1uII (2m,21)/2I|UII1/2 (15)

Combining these inequalities, we obtain

II(A2 + DI)UII > (j¢j - k6l1lX2 - k6)IjUII- (16)

As before, we see that the range of (A2 + U) is closed for k- sufficiently largein the half-plane Re(v) > 0. The parallel inequality for (A2' + RI) implies thatthe range of (A2 + UI) is all of L2(G), while (16) implies for such r that (A2 + UI)-'exists and is a bounded operator. The assertions about the position of the spec-trum follow from the inequality (16) and the first resolvent equation.

* The preparation of this paper was partially supported by the National Science Foundationunder NSF Research Grant G8236.

1 These PROCEEDINGS, 45, 365-372 (1959).2 Some of these results (and in particular Theorems 1(d) and 6) were previously announced by

the writer in Comptes Rendus Acad. Sci., 246, 526-528, 1363-1365 (1958).

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VOL. 45, 1959 MATHEMATICS: G. T. WHYBURN 1431

3 If n = 2, we require the additional condition that for each point x of the boundary, n the unitnormal vector to r at x, t any unit tangent vector, the polynomial a(x, t + Xn) in the complexvariable X should have exactly m roots in the upper-half X-plane (with multiplicities). This con-dition is superfluous if A is essentially real.4A sharper form of the a priori estimate was given in reference 1 which we apply in the dis-

cussion of Theorem 5 and Remark 3, namely that if u lies in W2m, P locally on a neighborhood inG of each point of G u r and satisfies the null Dirichlet conditions at each point of r, and, inaddition, if u and Au lie in LP(G), then u lies in D(Ap) and (1) holds.

MAPPING NORMS*

BY G. T. WHYBURN

UNIVERSITY OF VIRGINIA

Communicated July 20, 1959

1. Introduction.-If X and Y are metric spaces and f(X) = Y is a mapping, weunderstand by the norm N(f) of f the least upper bound of the diameter of pointinverses under f, so that

N(f) = l.u.b. {I [f-'(y) ] }, for y eY.

This number has entered in various ways into numerous studies of metric andtopological properties of mappings and of the spaces on which the mappings act.We shall be concerned with certain cases of the following general question: Fora given metric space X and class T of mappings, under what conditions does there exista positive constant d(X, T) such that any mapping of the class T operating on X as itsdomain is necessarily of norm > d(X, T)? For example, if X is a circle of radius rand T is the class of nontopological open mappings, it is readily shown that we maytake d(X, T) = rx/3; and, of course, w = z3 is a mapping of this class on the circleJzj = r of norm exactly r3.Our results have to do largely with the cases in which the spaces X are simple

closed curves and the mappings are nontopological open mappings. Application ismade to the case where X is an arbitrary 2-dimensional manifold. Proofs for themain theorems are only sketched in this paper and are complete only for the casesof circles, although the statements are made in their full generality. Detailedproofs will follow in a later article.

2. Altitude Norms. If J is a simple closed curve and a, b, and c are any threepoints on J we denote by 5(a, b, c) the smallest of the distances p(a, bc), p(b, ac),p(c, ab), where bc is the arc of J not containing a, and so on. We then definethe altitude a(J) of J as the least upper bound of the aggregate [6(a, b, c) ] for alltriples a, b, c of points on J. It readily follows that actually a(J) is the maximumnumber in this aggregate. Note also that a circle of radius r has altitude r-/3 inthis sense.

2.1. THEOREM. If J is a simple closed curve, any nontopological open mappingfonJhasnorm > a(J).

Let B = f(J). The case in which B is a simple closed curve is a direct consequenceof the following two statements which we verify.

i. f is topological on any arc a ofJ for which

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