Pacific Journal ofMathematics

A NUMERICAL RANGE FOR TWO LINEAR OPERATORS

CHARLES FRANCIS AMELIN

Vol. 48, No. 2 April 1973

PACIFIC JOURNAL OF MATHEMATICSVol. 48, No. 2, 1973

A NUMERICAL RANGE FOR TWO LINEAR OPERATORS

CHARLES F. AMELIN

A numerical range for two closed, linear operators isdefined for the purpose of obtaining some new results onthe stability of index of a Fredholm operator perturbed bya bounded or relatively bounded operator.

()• Introduction* One of the objects of the present paper is tostudy the eigenvalue problem Tx = xAx by means of a two (linear)operator or "bioperative" numerical range. We recall that Toeplitz [30]defined the numerical range for matrices in 1918. Then Wintner[31] in 1930 and Stone [27], [28] in 1930 and 1932 discussed therelationship between the convex hull of the spectrum of a boundedlinear operator on a Hubert space and its numerical range. In 1943, J.Dieudonne [4] and S. M. NikoPskii [24] laid the groundwork whichwould later help to show that the index of a bounded semi-Fredholmoperator is stable under perturbation by a bounded linear operator ofsufficiently small norm. This result was established in 1951 for boundedoperators on a Hubert space by F.V.Atkinson [2] and (independently)I. C. Gohberg [9], [10], and [11]. The following year, M. G. Krein andM. A. KrasnosePskii [20], B. Sz.-Nagy [29], and I. C. Gohberg [12] gen-eralized these results to unbounded closed linear operators. M. G. Kreinand M. A. KrasnosePskii also established the semi-stability of thenullity and deficiency. To understand the foundations and historicaldevelopment of the whole theory, the reader is referred to the com-prehensive article of Gohberg and Krein [13] which appeared in 1957.

Among the many innovations appearing in the 1958 paper of T.Kato [18] was the concept of the "lower bound" (now called the"minimum modulus") of a linear operator A defined on a Banach space.The main reason for defining the minimum modulus of A was to obtainas small a disc about the origin as possible so that ind(Ύ— XA) = ind(A)for all λ outside that disc, A being semi-Fredholm and T beingbounded or relatively bounded with respect to A. In this paper weintroduce a bioperative numerical range which will improve thatresult for Fredholm operators defined on a Hubert space. The im-provement is a consequence of the fact that the bioperative numericalrange for Fredholm A and relatively bounded T defined on a Hubertspace is always contained in the aforementioned disc and that theindex of T — XA remains constant if λ is not in the closure of thatbioperative numerical range.

For another use of the bioperative numerical range, we recall

335

336 CHARLES F. AMELIN

that W Givens [8] showed that co σ(T) = f\{ W(STS"1): S is invertible}if dim(iϊ) < oo. S. Hildebrandt [17] extended this theorem in 1966to bounded linear operators defined on infinite dimensional Hubertspaces H. In 1968, J. G. Stampfli and J. P. Williams [26] definedan "essential" numerical range Wes8(T) for Te^(H). It was shownthat Wess(T) = Π{W(T+ K): K is compact} and co σeS8{T) c Wess(T).According to a result of Stampfli and Williams [26], if T is normalthen coσess(T) = Wβ98(T)f but equality does not hold in general. Inparagraph 2 of this paper we obtain the convex hull of the essentialspectrum of a bounded linear operator defined on a separable Hubertspace in terms of intersections of appropriate bioperative numericalranges.

In paragraph 3, we indicate how to extend some of the resultsof the first paragraph to Banach and Frechet spaces. We give twoexamples in paragraph 4. Any one of the texts Kato [19], Goldberg[14], or Schechter [25] would be a suitable introductory reference tothe elementary definitions and basic theory assumed here.

I would like to thank Robert T. Moore of the University ofWashington for suggesting a problem in generalized spectral theorywhich led to this paper, and Seymour Goldberg and David Lay ofthe University of Maryland for their suggestions.

1* Stability of index theorems in Hubert space*

DEFINITION 1.1. Let H and K be Hubert spaces and T, Ae<Zf(H, K). We define W(T, A) = {(Tx, Ax): \\Ax\\ = 1, x e &(A)Γi&(T)}.

Clearly, if H = if, A is the identity, and Te^(H), then W(T, A)isthe Toeplitz numerical range W(T) = {(Tx, x): \\ x || = 1} If A~ι e ^(H)y

then W(TA^) = W(T, A).

LEMMA 1.2. W(T9 A) is a convex set.

Proof. We may assume that there exist two distinct points\, λ2e W(T, A) since the lemma is trivial otherwise. I.e., thereexist xu xze&(A) f] £&(T) such that (Txu Axt) = \ \\ Axt || = 1 and(Tx2, Ax2) = λ2, || Aα?2|| = 1. Since &r(A) Π &(T) is a subspace of Hwe may consider the binary forms d: C x C —> C, i = 1, 2 definedby

Cλ(alf a2) = (T(a,x, + a2x2) , A(aίx1 + a2x2))

C2(alf a2) = (A(a1xι + a2x2) , A(a,x, + a2x2)) .

We must show that CΊ assumes every value on the line segmentjoining \ and λ2 while C2 = 1. If we let

A NUMERICAL RANGE FOR TWO LINEAR OPERATORS 337

Q \ n _ _ _C(aly a2) = —i — = aγax + α12αxα:2 + a2ιaxa2

where

(\ - λ2)α12 = (Γα;2, Ax,) - λ2(Aa;2, AaJ

and (λL — λ2)α21 = (Txu Ax2) — \2(Axu Ax2), then it suffices to show thatthe form C takes on every real value from 0 to 1, while C2 = 1.Choose β of modulus 1 so that (α12 — a2ι)β is real and

ΈLe(Axl9 Ax2)β ^ 0 .

If a, = u and a2 = βv where u and v are real variables, then C =2ύ2 + r^v and C2 = u2 + 2suv + v2 where r — βa12 + βa2ι is real and

0. Solving for C2(u, βv) = 1 we obtain v = — su + V l + (s2 — l ) u 2

which is real valued for ue[ — l, + 1] Since v is now a function

of w, we can let CQ(u) = C(%, #y) = ^62(l — rs) + ru Vl + (s2 — 1)%2.For ^ e [0, 1], C0(u) is evidently a continuous real valued function withC0(0) = 0, C0(l) = 1 so that C0(u) assumes all values between 0 and1.

This proof was adapted from the proof given by Stone [28], p. 131.Certain other facts about the bioperative numerical range are easyto verify; e.g., W(aT + βS, A) aaW(T, A) + βW(S, A) where weset Sf{aT + βS) = &(T) Π &(S). For another fact, let T, A e &(H)and ϊ7*, A* denote the Hubert space ad joints of T and A. ThenW(T, A) lies on the line y = x tan(^0) if A*T = eUΘ«T*A. In particular,if T and A are self-adjoint and commute, then W(T, A) is real.

In what follows if A e ^(H, K), we let PA denote the projectionPA: H —> N(A)L onto the orthogonal complement of the nullspace ofA, and we denote the class of semi-Fredholm operators with finitenullity by Φ+. We recall that the minimum modulus of A e CSP{H, K),is the greatest number 7 which satisfies the inequality

7 dist (x, N(A)) ^ || As ||

for every xe&(A). Thus

7(A) = inf j }lAxj}τ, . xeD(A)}i dist (x, N(A)) )

and

(where 0/0 = cc). If xeH and x = xx + n e N(A)1 0 N(A), then

338 CHARLES F. AMELIN

| | a ? | | 2 = | | α ? 1 | | I + \\n\\2 a n d άist(x,N(A)) = \\xί\\ = \\PAx\\. I n o t h e rwords, if H is a Hubert space and Ae^(H, K), then

Ί{A) =

THEOREM 1.3. Let T,Ae <έ?(H, K) with &(T)n&(A). IfAeΦ+and X $ W(TPA,A), T-xAeΦ+, then ind(Γ- λA) = ind(A), nul(Γ- λA)^nul(A), and def (Γ - XA) ^ def (A).

Proof. Since Γ, A e i f (#, ϋΓ) and 3f{T)z>3f(A) there exist non-negative constants α, 6 such that || Tx || tS α || x || + & || Ax || for every

In particular, 11 TPAx \\ £ a \\ PAx \\ + 611 Ax \\ for everyWe recall that Kato [18] has proven that if

then {T-ζA)PA is semi-Fredholm and ind ((T - ξA)PA) = ind (A).But if λ£ W(TPA, A), we can find a δ > 0 such that (*) δ \\Ax\\ ^|| (Γ - xA)PAx || for every α? e &{A) = ^ ( Γ - λA) = ^ ( ( Γ - λA)P^).(Apply the Schwarz inequality and the fact that \\Ax\\ — 1, to theinequality d \(TPAx, Ax) — λ|). The starred inequality implies that(T- XA)PA is closed. By the same inequality nul((Γ- XA)PA) ^ nul(A)so that the dimensions nul((Γ — XA)PA) and nul(A) are equal by thedefinition of PA. (Notice A = APA). Furthermore, since H is a Hubertspace, 7(A) equals Ύ(A\NU)±) and thus 7(A)||PA^|| ^ ||Aa?|| so that therange of (T—XA)PA is closed (by the starred inequality); i.e., (T—XA)PA

is semi-Fredholm with finite nullity for every λg W(TPA, A). SinceW(TPA, A) is a convex, subset of the complex plane, its complement isconnected. But this implies by the asserted result of Kato et aL, that

i n d ( ( Γ - XA)PA) = ind (A)

for every λg W(TPA, A) because φ(X) = ind ((Γ - XA)PA) is a constantfunction on connected subsets of the plane for which (Γ — XA)PA issemi-Fredholm. Γ — λA must have nullity smaller than that ofnul (A) = nul (PA), because (T — XA)PA is one-to-one on N(A)1 andequals T — λA on N(A)L. In fact, suppose there exists an orthonormalset {Xii i = 1, , n + l } c ^ ( Γ - λA) with n = nul(A), ( Γ - λA)^ = 0.Write α?i = j?< + qieN(A)λ φiV(A). Since ϊ7 — λA is one-to-one onN(A)1, qι Φ 0 for every i. At most n of the g are linearly inde-pendent so qn+ι = Σ?=i /Si?*- Since

(Γ - XA)Pi = - ( Γ - λA)?4, (Γ -

and thus pw + 1 = Σii/Si^i by the linearity of the one-to-one map

A NUMERICAL RANGE FOR TWO LINEAR OPERATORS 339

T — xA. The resulting fact, xn+ί = Σ"=ι β&i ί s a contradiction son u l ( Γ - XA) ^nul(A).

Further,

(T - XA)2f{A) = (Γ - λA)ΛΓ(A) + (Γ - λA)(iV(A)' n ^(A))

is closed being the sum of two closed subspaces one of which is finitedimensional. Since T — XA is thereby semi-Fredholm for everyX$W{TPA,A),

ind(Γ - XA) = ind(T - xA) + inA(PA) = ind((T- XA)PA) = ind (A) .

But then def (Γ - XA) ^ def (A).

By the Schwarz inequality

I (TPAx, Ax) I ίS || TPAx \\ ^ a \\ PAx || + b \\ Ax \\ -?~ + b

if || Ax || = 1. We recall that Kato [18] proved that if

| λ | > + b

and A is semi-Fredholm, then ind (T — λA) = ind (A). Since we haveshown that W{TPA, A) c {z: \z\<> (a/y(A)) + 6}, Theorem 1.3 alwaysgives results at least as good as those previous for AeΦ+ definedon a Hubert space. The examples in paragraph 4 demonstrate thatW(TPA, A) sometimes gives better results. We note that the stabilityof index result ind (T — λA) = ind (A) of Theorem 1.3 remains validfor λ ? Π {W(TPA + K, A): K is compact} but the semi-stability ofthe nullity and deficiency need no longer hold.

For the case of semi-Fredholm A with finite deficiency it is pos-sible to define a "dual" bioperative numerical range

W(T*, A*)- = {(A*x, T*x): || A*x || = 1, a e ^(A*) n

and prove similar results using W(T*PA*, A*)~ in place of W(TPA, A)being careful to hypothesize &r(T*)z>&(A*). (The small bar denotescomplex conjugation.) If Γ, A e &(H, K) and A is Fredholm it iseasy to verify the stability results for λ g W(TPA, A) Γi W{T*PA*, A*)~.This may be of interest since both W(TPA, A) and W(T*PA*y A*)~ arecontained in {z: \ z \ (|| Γ||/7(A))}.

For simplicity suppose now that T and A are bounded,

r(Γ, A) = sup {| λ |: T - λA is not Fredholm}

and I W\(T9A) = sup{|λ|: λe W(T, A)} .

340 CHARLES F. AMELIN

We have seen that if A is Fredholm, r(Γ, A) ^\W\ (TPA, A). If Tis not Fredholm, then 1 ^ r(A — T, A) so we have just proven thefollowing: If \W\(A- TPA, A) < 1 and A is Fredholm, then T isFredholm.

Theorem 1.3 holds if we replace W(TPA, A) by W(T, A) sinceW(TPA, A) c W(T, A). But W(T, A) may be too large. In fact, letA,Te^(H), A be Fredholm, and T(N(A)) ς£ R(A)λ. Then thereexists an neN(A) and an a eff such that (Tn, Ax) Φ 0, ||Aa?|| =]| A(x + an) || = 1 for every α e C and

(Γ(a? + an), A(x + an)) = (Tx, Ax) + a(Trc, Ax) ,

so that TF(T, A) — C. On the other hand, one can verify that thecondition T(N(A)) c R{A)] or the condition that N(A) reduces bothT and A each guarantee that W(T, A) = W(TPΛ, A).

If A e &(H, K), then H = N(A) 0 N(A)\ K = Έ(A) © R(A)L

and A is a one-to-one operator from NiA)1 onto R(A). A+, thegeneralized inverse of A, has domain &{A+) — R(A) 0 R(A)L and isdefined to be zero on R(A)1 and the inverse of the one-to-one operatorinduced by A on R(A); i.e., A+Ax = P ^ , a? e ^ ( A ) . (α? e &r(A) c ^ ( ϊ 7 ) )Since

(TPAx, Ax) = {TA+Ax, Ax) ,

the following reduction of the bioperative numerical range to theToeplitz numerical range is immediate: W(TPA, A) = W(TA+ \A{H)).

2. The convex hull of the essential spectrum* For simplicitywe assume in this paragraph that all operators are bounded. Thereare many possible definitions of the essential spectrum of T; e.g.,the semi-Fredholm spectrum σSF(T), the Fredholm spectrum σF(T),the Weyl spectrum ω(T), or the Browder spectrum. But it is easyto see that the convex hull is the same for any one of the variousessential spectra and the essential radius ress(T) — sup {| λ |: λe σess(T)}is uniquely defined.

PROPOSITION 2.1. Let H be a separable Hubert space andTe &(H). Then co σess(T) c f| {W(ATP,AP): AeΦ+,P a projectionwith nul(P) < oo} c WeS8(T).

Proof. If λg Π W(ATP,AP), then λg W(ATP,AP) for someAeΦ+, some projection P with finite nullity; i.e., there exists a8 > 0 such that

δ || APx || ^ I (ATPx, APx) - λ |^\\A(T-X)Px\\

A NUMERICAL RANGE FOR TWO LINEAR OPERATORS 341

Thus (T - λ)P and therefore T - XeΦ+ which yields

σSF(T) c Π {W{ATP, AP): AeΦ+,P a projection with nul(P) < 00}

and the first inclusion.For the second inclusion we let PJΆ = ~sp {e{: i Ξ> n) where

{e,: ΐ = 1, 2, •} is an orthonormal basis for H. If λ 6 ΠW(TPn,Pn),then there exists yn such that || Pnyn |] = 1 and | {TPnyni Pnyn) - λ \<ljn.If xn = Pnyn, then ^ ^ 0 , ||α;Λ | | = l and (Txn, xn)~+X so thatλe Wess(T) by Theorem 5.1 of Pillmore, Stampfli, and Williams [6].But fi {W(ATP, AP): AeΦ+,P a projection with

nul ( P ) < 00} c f "

THEOREM 2.2. Lei H be a separable Hilbert space and TThen ress(T) = inf {|| ATA+/iP(//) ||: A e Φ+, P a projection withnul(P)< co, PAP = P).

Proof. By the remark made at the end of paragraph 1 and thefact that A+AP = PAP = P, it is clear that Proposition 2.1 and theSchwarz inequality imply that ress(T) ^ inf {|| ATA+/ΛPiH) ||: A e Φ+,Pa projection with nul (P) < ©o, P = P^P}. We now prove the reverseinequality; i.e., if ress(T) < 1, then for any ε > 0, there exists anAeΦ+, A: H-+12

+(H) and a projection P: ϋ->£Γ with nul (P) < co,PAP = P such that sup{|| ATPx ||: ||APa?H = 1} g 1 + ε.

According to a theorem of Stampfli (cf Lancaster [21]) for anyTe&(H), there exists a compact operator K such that σ(T + K) —ω(T). If ε > 0 is given, there exists a projection P: £Γ—>iJ withnul(P)<oo such that || iΓP || < e/Σ?=01| (Γ + iΓ)% ||2 since ff is aHilbert space and the spectral radius of T + K, r(T + K) < 1. (Weomit the trivial case Γ = —K). If A: £Γ —> Z2

f(ίί) i s defined by Ax —(Px,P(T+ K)x, . . . , P ( Γ + jBΓJ ίc, •••) then

It is clear that AeΦ+ and P^P = P. Also A(T + K) = U*A whereU* is the backward shift. Thus

\\ATPx\\ ^ ||AXPa;|| + || UΐAPx\\

^\\A\\\\KP\\\\Px\\ + \\

Taking the sup over ||APα;|| = 1 yields the desired result.

THEOREM 2.3. Let Te.^'(H) where H is a separable Hilbertspace. Then

342 CHARLES F. AMELIN

co σess(T) = Γl {W(ATP, AP): AeΦ+, P a projection with nul(P) < 00}

= Π {W(ATP, AP): AeΦ+, P a projection with nul(P) <

PΛP - P)

Proof. In view of Proposition 2.1, we need only show

Π {W(ATP, AP): AeΦ+, P a projection with nul(P) < 00,

The proof of this inclusion is based on the fact that any closed,convex subset of the plane is the intersection of all open discs con-taining it. Let D be an open disc containing co σess(T) with centerλ and radius r. Then the essential spectrum of (l/r)(Γ — λl) lies inthe open unit disc so there exists an A e Φ4 and a projection P offinite nullity such that sup || (l/r)A(T - Xl)Px ||: || APx || = 1} < 1.Thus I WI ((l/r)A(T - λ/)P, AP) < 1 or 11 W \ (ATP, AP) - λ | < r; i.e.,W(ATP, AP) is contained in D.

This proof is similar to one due to J.P. Williams in showing theGivens-Hildebrandt result coσ(T) = Π{W(STS^): S is invertible};cf. Fillmore [5], p. 22, To show the analogy between the Givens-Hildebrandt result and our result we can rewrite the result of the

theorem as coσess(T) = Π {W(ATA+/AP(H)): AeΦ+,P a projection offinite nullity}.

3* Extensions to Banach and Frechet spaces* Theorem 1.3 canbe extended to a certain extent to Banach or Frechet spaces by theuse of Lumer [22] semi-inner products or "states", cf. Bonsall andDuncan [3]. E.g., if X and Y are Banach spaces and T,Ae &(X, Y),by the Hahn Banach theorem there exists a choice function ξ: X—>X*such that <&,£(&)> = ||α?|| and ||£(α?)|| = 1. The map ζ(x) = ||a?||£(a?)then defines a semi-inner product [ , ]ζ on X x X by

[xl9 x2]ζ = <&!, ζ(x2)) .

It is easily verified that this semi-inner product satisfies most of theusual properties of an inner product except for conjugate linearity;cf. Lumer [22], Giles [7]. Since our numerical range theory doesnot need conjugate linearity we can define

It can then be verified that Theorem 1.3 holds for W{TPA, A, ζ) inplace of W(TPA9 A) for any ζ (where PA is the continuous projection

A NUMERICAL RANGE FOR TWO LINEAR OPERATORS 343

taking X onto M, the closed complement of N(A) in X). For aBanach space, however, equality 7(A) = Ύ(A/M) is not valid in generaland therefore the remarks made immediately after Theorem 1.3 arenot quite relevant to this case. Much the same can be said aboutoperators T, AeL(X, Y) for (X, τ) and (Y,o) Freehet spaces withtopologies τ and υ generated by the families of semi-norms Γ and A,respectively. According to a lemma of R. T. Moore [23], for everype A, there exists a choice function ζP: Y—> Y* such that

(a) p(yy =<y,ζ,(y)>

for every y e Y and

(b) KK.

For every pe A select a ζp and define Λ1 = {ζp: pe A). By usingtechniques developed in Amelin [1], it can be shown that Theorem 1.3holds with W(TPA, A) replaced by

W(TPA, A, Λx) = co {[TPAx, Ax)ζp: ζP e Λl9 p(Ax) = 1} .

4* Examples*

(a) Let T,Ae^(ll) be defined as T - KP(0), A = UP(0)P(ΐ)where CΠs the bilateral shift, and if et = {(ξn) \ ί< = l, £, = 0, ί=£j}9 then

= P(0)P(l), so TPA = A and

A) - {|| Ax ||2: || Aα; || = 1} = {1}

Thus if λ Φ 1, ind (Γ — λA) = ind (A) = 0, We note that previousresults stated that for | λ | > || T ||/7(A) = 1, ind (T - λA) = ind (A).

(b) Cf Goldberg and Schubert [15] (G - S). Let T = Dn andA = T — λ (for λ G Λ and w odd) be the maximal operators onU [0, co); i.e., 3f(T) = ^ ( A ) - {/: / € L2 [0, oo), /<—1> absolutely con-tinuous on [0, co), ϊ 7 / G L2 [0, oo)}# By Goldberg [14] Ch. 6, A isFredholm with nul(A) g w, def (A) = 0. By Theorem 9 of G - S, τ(A) =min {| (iσ)n - λ|: σ e R} = | λ |. Since

by Kato's result, if | ζ \ > (α/7(A)) + 6 = 2, then

ind (T - f A) - ind(A) .

On the other hand,

344 CHARLES F. AMELIN

W(TPA, A) = {(TPAf, Af): \\ Af \\ = 1}

= ((Γ - X)PAf, Af) + X(PAf, Af)

= 1 + \{PJ, Af) .

But 1 (PAf, Af) 1 <ί [I PAf H ^ 11 A/H/7(A) = 1/1 λ I; i.e., λ(P4/, A/) c{2: I2I ^ 1}. Thus W(TPA, A) is contained in a circle of radius 1centered at (1,0).

REFERENCES

1. C. F. Amelin, Norm and metric topologies, Fredholm theory, and generalizednumerical range for locally convex space operators. Ph. D. thesis, University ofCalifornia, Berkeley (1972).2. F. V. Atkinson, The normal solubility of linear equations in normed spaces, Mat.Sb. N. S., 28 (70), 3-14 (1951).3. F. F. Bonsall and J. Duncan, Numerical Ranges of Operators on Normed Spaces andof Elements of Normed Algebras, Lond. Math. Soc. Lecture Note Ser. (2) Cambridge Univ.Press (1971).4. J. Dieudonne, Sur les homomorphismes d'espaces normes, Bull. Sci. Math., (2) 67(1943), 72-84.5. P. A. Fillmore, Notes on Operator Theory, Van Nostrand Reinhold Co., New York1970.6. P. A. Fillmore, J. G. Stampfli, and J. P. Williams, On the essential numerical range,the essential spectrum and a problem of Halmos, Technical report, University ofIndiana.7. J. R. Giles, Classes of semi-inner product spaces, Trans. Amer. Math. Soc, 129(1967), 436-446.8. W. Givens, Field of values of a matrix, Proc. Amer. Math. Soc, 3 (1952), 206-209.9. I. C. Gohberg, On linear equations in Hilbert space, Dokl. Akad. Nauk SSSR(N.S ),76 (1951), 9-12.10. , On linear equations in normed spaces, Dokl. Akad. Nauk SSSR (N.S.),76 (1951), 477-480.11. , On linear operators depending analytically on a parameter, Dokl. Akad.Nauk SSSR (N.S.), 78 (1951), 629-632.12. , On the index of an unbounded operator, Mat. Sb. N. S., 33 (75) (1953),193-198.13. I. C. Gohberg and M. G. Krein, Fundamental theorems on deficiency numbers,root numbers, and indices of linear operators, Usp. Mat. Nauk., 12 (1957), 43-118.14. S. Goldberg, Unbounded Linear Operators: Theory and Applications, McGraw-Hill, New York 1966.15. S. Goldberg and C. Schubert, Some applications of the theory of unboundedoperators to ordinary differential equations, J. Math. Anal, and Appl., 19 (1967), 78-92.16. P. R. Halmos, A Hilbert Space Problem Book, Van Nostrand, Princeton 1967.17. S. Hildebrandt, Uber den numerischen Wertebereich eines operators, Math. Ann.,163 (1966), 230-247.18. T. Kato, Perturbation theory for nullity, deficiency and other quantities of linearoperators, J. d'Analyse Math., 6 (1958), 273-322.19. , Perturbation Theory for Linear Operators, Springer-Verlag, New York1966.20. M. G. Krein and M. A. KrasnoseΓskii, Stability of the index of an unbounded oper-ator, Mat. Sb. N. S., 30 (72) (1952), 219-224.

A NUMERICAL RANGE FOR TWO LINEAR OPERATORS 345

21. J. S. Lancaster, Lifting from the Calkin algebra, Ph. D. thesis, Indiana University(1972).22. G. Lumer, Semi-inner product spaces, Trans. Amer. Math. Soc, 100 (1961), 29-43.23. R. T. Moore, Ad joints, numerical ranges and spectra of operators on locally convexspaces, Bull. Amer. Math. Soc, 75 (1969), 85-90.24. S. M. NikoΓskii, Linear equations in linear normed spaces, Izv. Akad. Nauk SSSR.Ser. Mat., 7 (1943), 147-166.25. Martin Schechter, Principles of Functional Analysis, Academic Press, New York1971.26- J. G. Stampfli and J. P. Williams, Growth conditions and the numerical range ina Banach algebra, Tδhoku Math. J., 20 (1968), 417-424.27. M. H. Stone, Hausdorff's theorems concerning hermitian forms, Bull. Amer. Math.Soc, 36 (1930), 259-261.28. , Linear transformations in Hiΐbert space, Amer. Math. Soc Colloq. Publ.,15 (1932).29. B. Sz.-Nagy, On the stability of the index of unbounded linear transformations,Acta Math. Acad. Sci. Hungar, 3 (1952), 49-52.30. 0. Toeplitz, Das algebraische analogon zu einem satze von Fejer, Math Zeit., 2(1918), 187-197.31. Wintner, Spektraltheorie der unendlichen matrizen, Leipzig (1930).

Received June 23, 1972 and in revised form February 28, 1973.

UNIVERSITY OF MARYLAND

PACIFIC JOURNAL OF MATHEMATICS

EDITORS

RICHARD ARENS (Managing Editor) J. DUGUNDJI*

University of California Department of MathematicsLos Angeles, California 90024 University of Southern California

Los Angeles, California 90007

R. A. BEAUMONT D. GILBARG AND J. MILGRAM

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Copyright © 1973 byPacific Journal of Mathematics

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Pacific Journal of MathematicsVol. 48, No. 2 April, 1973

Mir Maswood Ali, Content of the frustum of a simplex . . . . . . . . . . . . . . . . . . . . 313Mieczyslaw Altman, Contractors, approximate identities and factorization

in Banach algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323Charles Francis Amelin, A numerical range for two linear operators . . . . . . . 335John Robert Baxter and Rafael Van Severen Chacon, Nonlinear functionals

on C([0, 1]× [0, 1]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347Stephen Dale Bronn, Cotorsion theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355Peter A. Fowler, Capacity theory in Banach spaces . . . . . . . . . . . . . . . . . . . . . . . 365Jerome A. Goldstein, Groups of isometries on Orlicz spaces . . . . . . . . . . . . . . 387Kenneth R. Goodearl, Idealizers and nonsingular rings . . . . . . . . . . . . . . . . . . . 395Robert L. Griess, Jr., Automorphisms of extra special groups and

nonvanishing degree 2 cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403Paul M. Krajkiewicz, The Picard theorem for multianalytic functions . . . . . . 423Peter A. McCoy, Value distribution of linear combinations of axisymmetric

harmonic polynomials and their derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 441A. P. Morse and Donald Chesley Pfaff, Separative relations for

measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451Albert David Polimeni, Groups in which Aut(G) is transitive on the

isomorphism classes of G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473Aribindi Satyanarayan Rao, Matrix summability of a class of derived

Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481Thomas Jay Sanders, Shape groups and products . . . . . . . . . . . . . . . . . . . . . . . . 485Ruth Silverman, Decomposition of plane convex sets. II. Sets associated

with a width function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497Richard Snay, Decompositions of E3 into points and countably many

flexible dendrites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503John Griggs Thompson, Nonsolvable finite groups all of whose local

subgroups are solvable, IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511Robert E. Waterman, Invariant subspaces, similarity and isometric

equivalence of certain commuting operators in L p . . . . . . . . . . . . . . . . . . . 593James Chin-Sze Wong, An ergodic property of locally compact amenable

semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615Julius Martin Zelmanowitz, Orders in simple Artinian rings are strongly

equivalent to matrix rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621

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