Pacific Journal ofMathematics

NILPOTENT APPROXIMATIONS AND QUASINILPOTENTOPERATORS

CONSTANTIN GELU APOSTOL AND NORBERTO SALINAS

Vol. 61, No. 2 December 1975

PACIFIC JOURNAL OF MATHEMATICS

Vol. 61, No. 2, 1975

NILPOTENT APPROXIMATIONS ANDQUASINILPOTENT OPERATORS

CONSTANTiN APOSTOL AND NORBERTO SALINAS

Let 2έf be a separable, infinite dimensional, complexHubert space and let £f{2t?) denote the algebra of all(bounded, linear) operators on £$?. This paper is concernedwith some aspects of the uniform approximation of an operatorin Sf(£$f) by nilpotent operators.

If T is an operator in £?(3$f) we shall denote by σ(T) thespectrum of T and r(T) the spectral radius of T. We recall thatan operator T in <£f(βέf) is said to be quasinilpotent if σ(T) = {0}or equivalents τ(T) = l i m ^ || Tn\\1/n = 0. In j[5] [7] it wasprovedthat a quasinilpotent operator T in £f(3ίf) is the uniform limitof a sequence {Qk} of nilpotent operators on H (cf. [6, Problem 7]).For each positive integer k let ok be the order of the nilpotentoperator Qk> that is ok is the smallest positive integer such thatQlk = 0. In view of the result mentioned above it is natural toask whether there exists any relationship between the rate ofdecrease of the sequence || T°k\\v°k and the rate of decrease of thesequence | | Γ — Qk\\. Furthermore, it is reasonable to expect thatthere exists a characterization of the set of quasinilpotent operatorsin terms of its nilpotent approximations. These questions seem tobe rather hard and in this paper we present some insights into theseproblems (cf. Corollary 3.4).

In Theorem 3.5 we prove that the distance from an arbitraryoperator T in £f(3ίf) to the set ^Γ{£έ?) of all nilpotent operatorsin Sf(£έf) is at most r(T) and in Theorem 3.1 we give preciseestimates for the nilpotent approximations of T in the case that Tis a biquasitriangular operator and zero is in the essential spectrumof T (see § 3 for the corresponding definitions). Another by-productof our discussion (cf. Proposition 4.4) is that if T is an operator onSίf such that lim i n f ^ Vn \\ Tn \\1/n = 0, then the quasinilpotentoperator T is actually pseudonilpotent (cf. [10, Problem 1]).

We are indebted to D. A. Herrero for his very useful comments,specifically, for his suggestions about the proof of Theorem 3.5 andfor providing us with the example of Remark 4.5.

2. The central lemma, The following lemma is central to ourpurposes.

LEMMA 2.1. Let Te^f(£έf). Then for every a > 0, β > r(T)

327

328 CONSTANTIN APOSTOL AND NORBERTO SALINAS

and every positive integer n there exists Q 6at most n (i.e. Qn = 0) such that

I I (ΓΘO)- + β +aβn

of order

Proof. Let U and V be two isometries in ^(Sίf) such that* = 1 - UU*. We define for l^k^n the subspace ^ C of®£ίf given by ^ = {T*-^ 0 aβ^V^Ux, x e £έ?}. Note thatis closed because it is the image under the isometry ( Q yk-γ jΛ

of the graph of the transpose of Tk~1l(aβk'"1). It is easy to see that^ . Π = {0} if 1 j , k^n, j Φk. This is due to the fact thatthe second components of the elements in ^ and ^£1 are orthogonal.Let ^ = Σ*=i ^^u = {»i + + Vv Vk£ -^k, l^k^n}. Now weprove that ^ is closed. Let {ym} be a sequence in s.t, limm_TO y Λ =0.Since ^ ^ n ^ = {0}, i ^ fc, we can write uniquely ym = Σϊ=iy»,*»where ym>fc = Tk~~ιxm,k ®aβk~ιVk~ιUxm,k1 for some %mΛz2!f, l<*k^n,m = 1, 2, . Since l i m ^ ym = 0, then limm _ Σϊ^ i aβ1*'1 V"1 Uxm, k = 0.Then lim^oo F^" 1 ?/^^ = 0, 1 <L k <L n and hence lim^oo xm,fe = 0,1 ^ k n. Therefore, lim™^ ym>k = 0, 1 & w and hence ^£ isclosed. Now we define Q e T{Sίf 0 Sίf) by

Q = o, Σ* = 1

»—l

Thus, the representing matrix of Q

/0 0 0 ••

* 0 0 ••

0 * 0 ••

0 0 0

on Σ*=i ~i€k is of the form

0 0\0 00 0

* 0/ .

Therefore it is clear that Qn = 0. Let Pn be the projection ontokernel F*\ Then

[(T®βPnV) - Σ* = 1

- Γ

= Tnxn φ ΓP^/S"F" UX + g α ^ P , - 1) V"

= Tnxn φ 0 .

NILPOTENTS APPROXIMATIONS AND QUASINILPOTENT 329

Since

X. =aβ'1-1

1

— /v/O71"1

~ aβn

We conclude that || [ ( T 0 βPnV) - Q] \^t || ^ || Tn \\/aβ-1. Fromnow on given a subspace gf of £ίf, P^ will denote the projection from

onto <%f. Then

II(ΓΘO)

/S

Thus in order to complete the proof it suffices to show thatP^tf- II 5 α. Notice that ^ x = {^0^: (y, x) + a(z, Ux) = 0

for all x e £ίf}. Hence ^^ί 1 = {( —α *7*z) 0 «, 2 e <^}. Thereforelip p . II _ II p I I II < κyII -^.^φo-^^r^ II — II - ^ ^ φ o l^r^ II = Ui

Following [9] we shall denote by Re(T) the reducing essentialspectrum of the operator Γ, that is Re(T) is the set of complexnumbers λ such that there exists a projection P in £?(£ίf) of infiniterank and nullity such that (T — X)P and (T* — λ)P are compactoperators.

THEOREM 2.2. Lβέ T be in £?(<%?) such that Re(T) Φ 0 andsuppose that 0 e i2e(Γ). Tftίm /or ever?/ α: > 0, /9 > r(Γ), 7 > 1 απώever?/ positive integer n there exists Q e ^V(Sίf) of order at most nsuch that

(*> Q l l ^ T\\+ βaβn~

Proof. From [9, Theorem 4.6] it follows that for every ε > 0there exists a unitary transformation Uε: <§ίf'—• Sίf'0 Sίf such that|| UtTΌ* - ( Γ © 0 ) | | < ε. On the other hand, from Lemma 2.1 wededuce that there exists a nilpotent operator Q1 on Sίf 0 έ%f oforder at most n such that || Γ0O - Q ' || ^ α || T \\ + /S + || Tn WKaβ^1).Letting now Q = ?7e*Q;?7£ we observe that || Γ - Q || - || USTU?-Q' \\ s + | | ( T 0 O ) - Q'\\ ^ ε + α | | Γ || + β + \\ Tn WKaβ"'1). The proof ofthe theorem is completed by choosing ε small enough.

3* Nilpotent approximation of biquasitriangular operators*In what follows, given an operator T in ^(Sίf) we denote by E{T)

330 CONSTANTIN APOSTOL AND NORBERTO SALINAS

the essential spectrum of T, i.e. E{T) = {X e σ(T): T - λ is notFredholm}. Also Et(T) and Er(T) will denote the left essential spec-trum and the right essential spectrum of the operator T, that isEt(T) = {X e E(T): Γ - λ is not semi-Fredholm with dim [null (T-λ)]< 00}and Er(T) = [^(ϊ7*)]*.

Following [6] we say that an operator T on £{f is quasitriangularif there exists an increasing sequence {Pn} of finite rank projectionsin £?(£ίf) tending strongly to the identity operator such thatlim^oo ||(1 — Pn)TPn\\ = 0 and we say that T is biquasitriangular ifboth T and T* are quasitriangular [2], In [3] it was shown thatT is biquasitriangular if and only if the index of T — λ is zero forevery complex number λ such that T — λ is semi-Fredholm. Inparticular, if T is biquasitrianglar E(T) = Et(T) Π Er(T).

In the following theorem we give some precise estimates for thenilpotent approximation of a biquasitriangular operator.

THEOREM 3.1. Let T in ^f(J^) be a biquasitriangular operatorand suppose that QeE(T). Then for every a > 0, β > r(T), 7 > 1and every positive integer n there exists a nilpotent operator Q in

of order at most An such that (*) is valid.

Proof. From [11, Proposition 3.2] it follows that for every ε > 0there exists a unitary transformation V,\S^^^f@^f®3ίf@S^and a compact operator Kε in £f(£έf) with \\Kt\\ < e such that

, 0 0 0 \

where Lί5 e £f(£έf), 1 ^ 3 < i ^ 4, and for each 1 ^ 3 ^ 4, S, is anoperator in £?{£έf) such that Sy is unitarily equivalent to M3 φ N3-where Mό is a block diagonal operator on 3$f with E{M5)<^E{T)and JVy is a normal operator in ^{3ίf) such that E(Nj) = E(T).Since E(Nj) = Λ#(JVy) [9, Theorem 3.10], 0 e ί?(Γ) and Be(Md 0 iV» =Λ#(Λfy) U Re(Nj) [9, Lemma 4.10], it follows that 0 6 J2.(Sy), 1 ^ i ^ 4.Now we chose ε > 0 small enough so that r(T — i Q < β. SinceII SJ II ^ II (Γ - Kε)

k II, 1 ^ i ^ 4, A? = 1, 2, . . , we also have r ( ^ ) < /S,1 ^ 3 ^ 4. Given a fixed number δ with 1 < δ < 7, from Theorem2.2 we can find nilpotent operators Q3 in ^f(β^) of order at mostn such that || Sd - Q3 \\ ^ δ(a \\ Sf \\ + β + \\ Sj \\/aβn-% 1 ^ j £ 4.Now let Q be the operator in ^f(βέf) defined by

NILPOTENTS APPROXIMATIONS AND QUASINILPOTENT 331

Qx 0 0 0 \

I/21 Q2 0 0VtQVt= / γ

L/31 -^32 **3 "

\ ί/41 Z/42 x/43 Q4

It is obvious that Q is a nilpotent operator of order at most inand we see that

|| T - if. - Q || = || Vε(T - if. - Q)Ff || = max || S, - Q/||

^ δfα || Γ - K || + β + lKΓ- g«)*lh .

Therefore,

T - Kt || + β

and hence the theorem follows by choosing ε small enough.

COROLLARY 3.2. Let T bea biquasitrίangular operator insuch that 0 e E(T). Then the distance dist (T, ΛT(2ί?)) ^ r(Γ).

Proof. From Theorem 3.1 we deduce that for an arbitraryε > 0 (taking a = ε, β = r(T) + ε and 7 = 1 + e) there exists anilpotent operator Q in £f{^f) such that

ε){ε || Γ|| + r(T)~r

Since l i m , ^ || Tn \\1/n = r(Γ), it follows that for w sufficiently large

[|| Tn \\Vn/(r(T) + ε)]n < ε2 and hence

|| T - Q \\< (1 + ε)[ε || Γ || + (r(T) + ε)(l + ε)] .

Since ε is arbitrary the proof of the corollary is complete.

The following corollary expresses the fact that if T is a quasi-nilpotent operator on Sff and | |Γ*| | 1 / n doesn't decrease very fast,then there exists a sequence {Qn} in .yK(β^) where each Qn hasorder oΛ, n — 1, 2, such that the rate of decrease of the sequence||T°"|!1/0% is the same as the rate of decrease of the sequenceI I T - Q . H .

COROLLARY 3.3. Let T in ^f(β^) be quasίnilpotent and let

332 CONSTANTIN APOSTOL AND NORBERTO SALINAS

δ > 1. Then there exists a sequence {Qn} in ^(Sίf) such that Qin = 0and for some c > 0, || T - Qn\\ < c(\\ T\\ δ~« + || Tn |Γ) .

Proof. It is a direct consequence of Theorem 3.1 where a = δ~n,

COROLLARY 3.4. Let T in £f(£ίf) be quasinίlpotent. Then there

exists a sequence {Qn} in ^V^^^f) such that Q4

n

n = 0 and \\T — Qn\\ <Z

3(1 + \\T\\)(n\\Tn\\y/n+ί.

Proof. It is an immediate consequence of Theorem 3.1 taking|| T\\))VΛ+ι and 7 = 3/2.

In the next theorem we shall need the following terminology:If X is a compact subset of the plane we denote by X the comple-ment of the unbounded component of the complement of X. If ε 0we denote by X£ the set Xε — {λ e C: inf^x I λ — μ | ε}. We notice thatif ε sup ; e x I λ I, then Xe = (X)ε = (Xε)" and 0 e Xε.

THEOREM 3.5. Let T be in ^f(^f). Then the distancedist(T, ^r{£e?))^r{T).

Proof. Let ε > 0 and let Σ = <*(T)\E{T\. If Σ Φ 0 , then^ == {λi, , λw} where each λyis an isolated eigenvalue of σ(T) suchthat T — Xj is Fredholm, 1 i ^ m. Then the spectral idempotentEΣ corresponding to Σ> associated with T, has finite rank. Let

= range JE7Σ, ^— ά?ΓL and let P be the projection from Sίf ontoIt readily follows that σ(T\^) = Σ and σ ( P Γ | ^ ) = σ(Γ)\Σ.

Furthermore, ^[(PΓ) | T] = (Γ) and ^ [ P T | ^T] = ( Γ ) . Nowwe recall that a normal operator M in ^f(β^) is called diagonalizableif J ^ has an orthonormal basis consisting of eigenvectors of M andM is said to have uniform infinite multiplicity if every eigenvalueof M has infinite multiplicity. Notice that in this case M is unitarilyequivalent t o l © l and that σ(M) = E(M). From [3, Theorem 2.2]there exists a unitary transformation Uε: £ —> ^ θ ££" and acompact operator Kε on . ^ whose norm is so small that ||iΓe[| < ε,

- Kε]c:E(T)2ε and such that we also have

where M is a diagonalizable normal operator of uniform infinitemultiplicity in £f(£έ?) such that σ(M) = Et[PT | ^ # ] . Applying nowthe same reasoning to the operator S we further deduce that there

NILPOTENT APPROXIMATIONS AND QUASINILPOTENT 333

M0

0

AC

0

BD

N

exist a unitary transformation F e ' . ^ ^ ^ θ ^ θ 3lf and acompact operator Lε in £f(^f) with norm so small that | |L 6 | | < 2ε,

σ[PT I ,^~ £fε\ c J5(Γ)Sβ and such that

Tε = Vε[PT 1 ^ - J2ffi] V? =

where M is as before, N is a diagonalizable normal operator of infi-nite uniform multiplicity in S^(^f), A, B, C, D are in J^{Sίf) and

σ(N) = Er{C) = Er(£ ^j(=.Er[PT \ Λ\. Now let δ - r(Λf), η - r(iV)

and notice that Ύ] <^ δ. From the remark preceding the present

theorem we see that σ(M)δ — σ(M)δi σ(N)η — σ(N)v c σ(M)δ and

Oeσ(N)v. Employing a minor variation of [3, Proposition 1.11] it

follows that there exist two normal operators M' and Nr in £^{^f)

such that σ(M') - E(M') = σ(M)δ+3ε = E7T)5+3: - E(T)δ+3ε, σ(N') =E(N') = σ(N)v and || M - M' \\ = δ + 3e, || JV - iVr || = η. Let Γ,f bethe 3 x 3 operator matrix defined by

rpr

We claim that Eι(T'e) Π Er(T's) =- σ(Tε) - σ(M'). In order to prove

this assertion we first observe that, since M' acts on an invariant

subspace of T'e, it is obvious that σ(M') = Et(M') c Et{T[). Also, if

X$Er(T'β), then X$Er(9 ^,) and hence X £ Er(N') = σ(N'). Since

Et(C) c 4^(C) and Er(C) = σ(iV) c σ(iV;) = σjw'), it follows that

X ί E(C)( = Er(C) U S,(C)). Therefore λ g J ? ^ ^ , ) , and since λ g JB7r(Γβ')

we deduce that X g Er(M') — σ{M'). Now we shall prove the other

inclusion. To this end we note that σ(jί ?r)cα(Γ £ )c£(T) 3 . , and

hence σί ^ j c E(T)Mε = σ(M;). In particular, it follows that

σ(C) c (τ(M'). Thus, if λ g σ(M'), then λ g cr(C) U (JSΓ') and hence

The last observation establishes our claim. Let 7 = dist (ΓJ,^ ^ θ ^ ) ) . Our first conclusion is that dist (PT \ l!Lβ|| + dist(Γβ, ^ί^(^f ® ^f ® ^f)) ^ 2ε + 7 + max(| |M~ Jlί'||,II J\Γ — 2V"). Since |[ ikf' — M\\ = δ + 3ε ^ r(Γ) + 3ε and ||ΛΓ'- iV|| -

334 CONSTANTIN APOSTOL AND NORBERTO SALINAS

η ^ δ, we deduce that dist (PT \ Λ\^£)) ^ 7 + r(T) + 5ε. Usingthe fact that ^ is finite dimensional it is elementary to check thatdist (T I j r ^ ( J H ) ^ r(Γ I ^ H ^ r(Γ) and hence we conclude that

dist (T, Λ\3έf)) £ max [dist (Tdist (PT I ^ < > T ( ^ ) ) ] ^ 7 + r(Γ) + 5ε .

Since 0 e JK(TJ) and we have shown that T' is a biquasitriangularoperator we may conclude from Corollary 3.2 that dist (Γ, .yK(£έ?)) <£r(Γ;) + r(Γ) + 5ε <; 2r(Γ) + 8ε. However, we are interested in asharper estimate. At this point we make use of the fact that T[enjoys the further property that σ(T[) is simple connected and coin-cides with E(T[) and therefore from [4, Proposition 1.6], 7 = 0. Thus,dist(T, ^ r ( < ^ ) ) ^ r(T) + 5ε and since ε is arbitrary the proof ofthe theorem is complete.

4* Concluding remarks* In this section we pose some problemsconcerning the nilpotent approximation of quasinilpotent operators.We motivate these problems by presenting some pertinent obser-vations.

PROPOSITION 4.1. Let T be in ^f\^f) and suppose that thereexists a sequence {Qn} in ^V(<%?) such that Q°n

n = 0, n = 1, 2, andlim inf^TO || T ~ Qn \\1/0^ = 0. Then T is quasinilpotent.

Proof. Since T°» = T°- - Q°« = Σi=i T°^j(T - Q^Qt1 then forn sufficiently large there exists c>0 such that || T°* \\1/On ^\\T-Qn \\Vo".[ΣJ"i II T \\°»-j || Qn \\*-ψ** ^ c ]| T - Qn |Γ° . This completes the proofof the proposition.

PROBLEM 1. Is the converse of Proposition 4.1 valid? (We expecta negative answer.)

REMARK 4.2. In a different circle of ideas we note that minormodifications of the arguments given in this paper show that if wereplace ^f(£έ?) by the Calkin algebra over £ίf (i.e., the quotientalgebra of £f(£ίf) by the ideal of compact operators on S(f\ thenthe conclusion of Theorem 3.5 still holds. Thus it is natural to ask:

PROBLEM 2. If J ^ is a C*-algebra and Λ* is the set of nilpotentelements of J Γ when does it follow that the distance dist (A,r(A), for every A e

If s*f is a finite type I von Neumann algebra, it is easy to see

NILPOTENT APPROXIMATIONS AND QUASINILPOTENT 335

that the above question has an affirmative answer. On the otherhand, it is worth noting that, by a well known fact in the theoryof C*-algebras [8], any noncommutative C*-algebra contains nonzeronilpotent elements. However, if & is a semi-simple Banach algebra,then there exists no nilpotent element different from zero. In par-ticular, if & is a semi-simple commutative Banach algebra such thatthe Gelf and transformation is not an isometry, then the correspondingversion of the above question has a negative answer. This is thereason that in the preceding problem we imposed the condition that

be a C*-algebra.The following terminology was introduced in [10].

DEFINITION. Let T e £?{&?). We say that T is pseudonilpotentif for every ε > 0 there exists a finite orthogonal family of projec-tions Pl9 , Pn in Sf{Sίf) such that Σ*=i P* = l and || Σ*ai-P<ΪΉ H<ε

The set of all pseudonilpotent operators in Sf^Sίf) will be denotedby

In [10], it was observed that Ψ(Sέf) = Ψ(gέf)* and it was shown

that ^ ( ^ r ) <z.ΦΨ(3έf) c * 3 F ( 5 F ) and that there are pseudonilpotentoperators which are not quasinilpotent.

LEMMA 4.3. Let Γ 6 ^ ( ^ ) and suppose that there exists asequence {Qn} in ^K(^f) such that Q°n

n = 0, n = 1, 2, andlim inf oo Vo~ \\ T — Qn \\ = 0. Then T is pseudonilpotent.

Proof. Given ε > 0, let n be sufficiently large so thatVol || T - Qn ||" < ε. For each 1 ^ j ^ on let P3 be the projectiononto kernel Qί kernel Qt1. Also let x e £έf. Then

On °n

3=1 i=j=

VII

°n

Σ

1

Therefore || Σ £ i ΣίΞy PiTPj II ^ Von \\ T - Qn\\ < ε, as desired.

PROPOSITION 4.4. Let T be in ^f(J^) such that

lim inf V7 n \\Tn \\υn = 0 .

T is pseudonilpotent and quasinilpotent.

336 CONSTANTIN APOSTOL AND NORBERTO SALINAS

Proof. It is an immediate consequence of Corollary 3.3 andLemma 4.3.

REMARK 4.5. Let {en} be an orthonormal basis for £%f and let{an} be a sequence of complex numbers. We recall that an operatorT in £f{£έf) is called a unilateral weighted shift with weights an,n = 1, 2, if Ten = anen+1. Now we let T in £f(£έf) be a unilateralweighted shift with weights ao = ax = = anι = 1, α.1+1 = =α%2 = 1/2, , αnjfc_1+1 = ank = I/A?, , where {%*} is a strictly in-creasing sequence of natural numbers. Given an arbitrary slowlyincreasing sequence {cn} of real numbers cn > 1, tending to infinite,it is not difficult to define the sequence {nk} inductively so thatlim^oo cnjk = oo. It follows that for k > 1, if nk^ < n <; nk, thencn\\Tn\\Vn > cWJb-1|| Γ MΓ7** > o%h_Jc-\ Since Γ is a quasinilpotentcompact operator, from [10, Theorem 3.4], TzΨ{<%f) but

This shows that the sufficient condition of Proposition 4.4 (or anyreasonable relaxation of it) is not necessary. Moreover, T (x) 1^provides a similar example with a noncompact quasinilpotent andpseudonilpotent operator.

Finally, we pose two questions concerning pseudonilpotent oper-ators which were already asked in [10].

PROBLEM 3. Is every quasinilpotent operator pseudonilpotent?

REMARK 4.6. From results of [3] and [4] it follows that everyoperator in J*f(<&?) has a nontrivial invariant subspace if and only

if every operator in ^V{^f) has the same property.

PROBLEM 4. Does every pseudonilpotent operator have a non-trivial invariant subspace?

REFERENCES

1. C. Apostol, On the norm-closure of nilpotents, Rev. Roum. Math. Pures et Appl.,19 (1974), 277-282.2. C. Apostol and C. Foias, On the distance to the biquasitriangular operators, AnnalesUniv. Bucharest, 1974.3. C. Apostol, C. Foias and D. Voiculescu, Some results on non-quasitriangular oper-ators, IV. Rev. Roum. Math. Pures et Appl. 18 (1973), 487-514.4. , On the norm closure of nilpotents, II, Rev. Roum. Math. Pures et Appl.,19 (1974), 549-577.5. C. Apostol and D. Voiculescu, On a problem of Halmos, Rev. Roum. Math. Pureset Appl., 19 (1974), 283-284.

NILPOTENT APPROXIMATIONS AND QUASINILPOTENT 337

6. P. R. Halmos, Ten problems in Hubert space, Bull. Amer. Math. Soc, 76 (1970),887-933.7. D. A. Herrero, Toward a spectral characterization of the set of norm limits ofnilpotent operators, Indiana Univ. Math. J., 24 (1974/75), 847-864.8. I. Kaplansky, Rings of Operators, W. A. Benjamin, Inc., New York-Amsterdam, 1968.9. N. Salinas, Reducing essential eigenvalues, Duke Math. J., 40 (1973), 561-580.10. , Subnormal limits of nilpotent operators, Acta Sci. Math. (Szeged), 37(1975), 117-124.11. D. Voiculescu, Norm-limits of algebraic operators, Rev. Roum. Math. Pures etAppl., 19 (1974), 371-376.

Received June 12, 1975. During the preparation of this article the second authorwas partially supported by a grant of the University of Kansas General ResearchFund.

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Pacific Journal of MathematicsVol. 61, No. 2 December, 1975

Graham Donald Allen, Francis Joseph Narcowich and James Patrick Williams, Anoperator version of a theorem of Kolmogorov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

Joel Hilary Anderson and Ciprian Foias, Properties which normal operators share withnormal derivations and related operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

Constantin Gelu Apostol and Norberto Salinas, Nilpotent approximations andquasinilpotent operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

James M. Briggs, Jr., Finitely generated ideals in regular F-algebras . . . . . . . . . . . . . . . . 339Frank Benjamin Cannonito and Ronald Wallace Gatterdam, The word problem and

power problem in 1-relator groups are primitive recursive . . . . . . . . . . . . . . . . . . . . . . 351Clifton Earle Corzatt, Permutation polynomials over the rational numbers . . . . . . . . . . . . 361L. S. Dube, An inversion of the S2 transform for generalized functions . . . . . . . . . . . . . . . 383William Richard Emerson, Averaging strongly subadditive set functions in unimodular

amenable groups. I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391Barry J. Gardner, Semi-simple radical classes of algebras and attainability of

identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401Irving Leonard Glicksberg, Removable discontinuities of A-holomorphic functions . . . . 417Fred Halpern, Transfer theorems for topological structures . . . . . . . . . . . . . . . . . . . . . . . . . . 427H. B. Hamilton, T. E. Nordahl and Takayuki Tamura, Commutative cancellative

semigroups without idempotents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441Melvin Hochster, An obstruction to lifting cyclic modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 457Alistair H. Lachlan, Theories with a finite number of models in an uncountable power

are categorical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465Kjeld Laursen, Continuity of linear maps from C∗-algebras . . . . . . . . . . . . . . . . . . . . . . . . . 483Tsai Sheng Liu, Oscillation of even order differential equations with deviating

arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493Jorge Martinez, Doubling chains, singular elements and hyper-Z l-groups . . . . . . . . . . . . 503Mehdi Radjabalipour and Heydar Radjavi, On the geometry of numerical ranges . . . . . . 507Thomas I. Seidman, The solution of singular equations, I. Linear equations in Hilbert

space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513R. James Tomkins, Properties of martingale-like sequences . . . . . . . . . . . . . . . . . . . . . . . . . 521Alfons Van Daele, A Radon Nikodým theorem for weights on von Neumann

algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527Kenneth S. Williams, On Euler’s criterion for quintic nonresidues . . . . . . . . . . . . . . . . . . . 543Manfred Wischnewsky, On linear representations of affine groups. I . . . . . . . . . . . . . . . . . 551Scott Andrew Wolpert, Noncompleteness of the Weil-Petersson metric for Teichmüller

space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573Volker Wrobel, Some generalizations of Schauder’s theorem in locally convex

spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579Birge Huisgen-Zimmermann, Endomorphism rings of self-generators . . . . . . . . . . . . . . . . 587Kelly Denis McKennon, Corrections to: “Multipliers of type (p, p)”; “Multipliers of

type (p, p) and multipliers of the group L p-algebras”; “Multipliers and thegroup L p-algebras” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603

Andrew M. W. Glass, W. Charles (Wilbur) Holland Jr. and Stephen H. McCleary,Correction to: “a∗-closures to completely distributive lattice-orderedgroups” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606

Zvi Arad and George Isaac Glauberman, Correction to: “A characteristic subgroup ofa group of odd order” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607

Roger W. Barnard and John Lawson Lewis, Correction to: “Subordination theoremsfor some classes of starlike functions” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607

David Westreich, Corrections to: “Bifurcation of operator equations with unboundedlinearized part” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608

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