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Pacific Journal of

Mathematics

REPRODUCING KERNELS AND OPERATORS WITH A

CYCLIC VECTOR. I

VASHISHTHA NARAYAN SINGH

Vol. 52, No. 2 February 1974

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P A C I F I C J O U R N A L O F M A T H E M A T I C SVol. 52, No. 2, 1974

REPRODUCING KERNELS AND OPERATORS WITH ACYCLIC VECTOR IVAS HIS HTHA N. S INGH

In this paper a study is begun of the complete unitaryinvariant ((1 wT)~1e1 (1 zT)~1e), first considered by Livsicin his paper 'On Spectral Resolution of Linear Nonself Ad-joint Operators' Mat. Sb., 34 (76), 1954, 145-199, of a triple(T, H, e) where T is a bounded linear operator on a Hubertspace H and e is a cyclic vector for T in H, as a reproducingkerne l. One of the important points is the construction ofa subset of the group algebra of the torus closed underpointwise addition and convolution. This obviously willgenerate a ring called the K-ring. A study of this ring willbe done later.Several other theorems and constructions are also given.

Introduction* Let T be a bounded linear operator on a Hubertspace H with a topoJlogically cyclic vector e in H. In this paper wewish to study certain analytic functions associated with the triple(T, H, e) for the sake of the problem of invariant subspaces of T inH. (See also [8] and [15].)The paper is divided into six sections. In 1 we present somefacts about reproducing kerne ls w ith analyticity properties. In 2we consider a triple (T, H, e) of the above type. H can then berepresented as a Hubert space of conjugate analytic functions ae[H]with a reproducing Kernel K. T* on H assumes the form of reverseshift on the Taylor coefficients of functions in ae[H]. (See also [11]or [19].) In 3 we recover (T, H, e) from the reproducing kernel ofae[H] in two ways. The notion of an analytic function of positivedefinite type is introduced and it is shown that only these can ariseas reproducing kernels of ae[H]. These functions are also relatedto invarian t subspaces (4). In 5 a categ ory of triples is constructedand it is connected to the harmonic analysis of the two-torus viathe ana lytic functions of positive definite type. Section 6 consists ofsome examples and counterexamples about the analytic functions ofpositive definite type.The paper is based on the a utho r's dissertation written underthe guidance of Professor John L. Kelley of the University ofCalifornia, Berkeley. The au tho r would like to than k ProfessorKelley for help and advice.1* Reproducing kernels and analytic functions of positivetype. We s ta rt w ith the definition of a reproducing kernel. Le t

56 7

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568 V. N . S I N G HH be a Hubert space of functions on a set X with the inner product( , ) . In Theorem 1.2 we present a few facts of a theory of H dueessentially to E. H. Moore and N. Aronszajn. See [1] and [2].

D E F I N I T I O N 1.1. A reproducing kernel for H is a complex valuedfunction K on X x X such that:

( i ) For all yeX,KyeH where Ky(x) = fo y).(ii) If g e H then for all yeX9 (gf Ky) = g(y), and(iii) The linear span of the set {Ky}yex of functions is dense in H.N o t e that if K is a reproducing kernel then the map g g(y) =

(g, Ky) is a bounded linear functional on H; i.e., evaluation at a pointis continuous. Also if H is a pre- Hilbert space of funtions on a setX with continuous evaluation then the completion H o H can berealized as a space of functions on X by setting h(x) = (fc, i Q for c e l a n d k f f where i ,e 5" is such that (/ , Kx) = f(x) for all / e H.

THEOREM 1.2. Lei H be a pre- Hilbert space of functions on aset X so that the evaluation map f* - * f(x ) is continuous for all xe X,let H be the completion of H. Then the followinghold

( i ) There is a unique reproducing kernel K for H.(ii) If for xeX, xeH, is such that f(x) = (/, x) for all fe Hthen K(x, y) = ( y, x).(iii) / / { jie/ is an orthonormal basis for H, then K(x, y) =(iv) If H is the set of finite formal sums ?= o A, where xt e Xand als are complex numbers, with an inner product (, ) given by

( ?= i ai u = i bjVj) = ?= i f= i dibjKiyj, t), the map ?= i atxt H- ? = I ^i

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REPRODUCING KERNELS AND OPERATORS WITH A CYCLIC VECTOR I 569DEFINITION 1.4. For each s > 0, DS(DS) is the open (closed) discof radius s about 0 in the complex plane . As is the space offunctions defined and conjugate analytic in Ds and As.$ the space offunctions defined on Ds x Ds which are conjugate analytic in thefirst variable and analytic in the second variable. & is the space

of polynomials with complex coefficients. If K is a function of positvetype and is in A 8m 8 for some s then K is an analytic function ofpositive type abbreviated a.f.p.t.

Let Ke A SfS . The following theorem gives an alternate construc-tion of a Hubert space which is equivalent to K being a function ofpositive type.

THEOREM 1.5. Let Ke A 8t8 . Then( i ) K is an a.f.p.t. if 1/A 2 K(z, w)p(z~ )p(w~ )(dz/z) xJ c J c(dwjw) ^ 0 for all p e & where the path of integration C is a simple

contour in Ds such that its inverse winds around Ds once.( i i ) Let (H, (,)) be a pre- Hilbert space of functions analytic ina neighborhood of D}s with the inner product (p, q) 1/47 2J K(z, w)p(w~~1)q(z~1)(dz/ z)(dw/ w). Then the map which takes ac J cmember Ka of H, a a member of Ds, into the rational function1/(1 az) is inner- product preserving.

Proof. Consider l/4 r2 K{z, w)p(z~ )p(w~1)(dz/ z)(dwlw) as alimit of R i e m a n n sums l/ 4 2 ?,i= K(ZJ, dP^i P^T^j+i ~ z$lz) x(z i+ 1 zjzj. Since K is a function of positive type it is clear t h a t 1/47 2(J) (J) K(z, w)p(z~1)p(w~1)(dz/ z)(dw/ w) is positive. Let now alf a?, , an

be any complex numbers and let au , an e Ds. Consider the rationalfunction / given by f{z) = * (l - a^Y1. Then ?, i= i ai :3 K(ajf az) =l/ 4 r2f f K(z, w)f(z^f(w- ){dz/ z)(dw/ w) - lim l/4 2 f K(z,w)x

J C1 J C2 - * 00 J C1J C2p(z~ )p(w~1)(dz/ z)(dw/ w) ^ 0 where p% is a sequence of polynoimialsconverging to / uniformly on compact subsets of Djs and Cl9 C2 arechosen suitably. Thus (i) is proved the proof of (ii) follows fromth e observation t h a t the n o rm in the space of rational functions ofth e function z H^ ? = I *(l ~~ ^^Y1 is ?, i= i ^jK(ajf a) which is thesame as the norm of ?= i ^% Ka. in H.

2. The kernel fun ct ion of cyclic triple* Consider now a H u b e r tspace H and a bounded linear operator T on H. We construct certa infunct ions of positive type associated with T and H. Some factsfrom the functional calculus of T are used. (See [12].) If is a

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570 V. N . S I N G Hcomplex valued function analytic in the neighborhood of the spectrum ( T ) of T define ( ).as follows.

D E F I N I T I O N 2.1. Let C be a contour lying in the domain ofanalyticity of b which winds around each point of (T) once. Thenb (T ) = l/2r f b(z)(zl - T)~ dz. It is known that bH * b(T) is a lineara n d multiplicative homomorphism of the space of functions analyticin a neighborhood of (T) into the space of bounded linear operatorson H. (See [12], page 199.)The following result is probably well-known and will be usedlater.

THEOREM 2.2. Let T be a bounded linear operator on a Hubertspace H and suppose that there is an analytic function b definedand nonzero on a connected neighborhood of the spectrum o(T) ofT such that b{T) 0. Then there is a nonzero polynomial p suchthat p(T) = 0.

We leave the proof of this fact to the reader. See for example[ 2 0 ] .

N ow we make the following definition.D E F I N I T I O N 2.3. For r > 0. J5r is the space of functions analyticin a neighborhood of the closed disc Dr with topology given as theinductive limit topology of the spaces A( U) of the functions analytic

in s neighborhood U of Dr. (See [14], page 219, problem D.)I t is known that Br is a Montel space. (See [14], page 196,problem F(e).)F or r > 0, define Ar as follows.D E F I N I T I O N 2.4. Ar is the space consisting of functions whichare complex conjugates of the functions in Ar.Let E be a linear space with a locally convex topology r and let

E' be its dual. Then the strong topology for Er is the topology ofuniform convergence on r- bounded sets.

The following result will be important in the sequel.LEMMA 2.5. Let r > 0. Then AUr and Br are strong duals for

the pairing given by [/, g] I f(z)g(z~1)dz/ z for fe Br and g e A llrwhere C is the contour tv->(r + )eu, 0 ^ t ^ 2 , for some small depending on f and g.

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R E P R O D U C I N G K E R N E L S AND OPERATORS WITH A CYCLIC VECTOR I 571Proof. We present an outline of the proof and refer to [14] for

more information. Since Br is a Montel space it is reflexive ([14],20 F(a)) and hence it suffices to show that th e dual of 2?r, withstrong topology is A llrf for the pairing above.To show this, let for each z e D1/ r, hz be th emember of B r givenby hz( ) = 1/(1 z) for e Dr. For anycontinuous linear functionalg on Br le t %{g) be defined by iQ(g)(z) = g(hz). Then %{g) is a well-defined function and is a member of A1/ r. Moreover, g(f) = l/ 2 i

/ fi^ioidX^dz/ z = l/ 2 i[f9 %{g)\, where C is a contour as in th ecstatement of the proposition being so small that it is contained inth e domain of holomorphy of / . Conversely any member of A1/ rdefines a continuous linear functional on B r by the above formulaand thus A lr and B'r are algebraically isomorphic.To complete the proof it remains to show that the topology ofu .c . c . on A llr coincides with th e strong topology of B'r. To provethis first observe that each bounded set of B r is contained in abounded set of r. for some r ' > r ([14], 17G (6) (iii)). Hence itfollows that B'r with strong topology is metrizable ([14], 18.4). Sinceeach bounded subset of Ar, is uniformly bounded on every compactsubset of r,, a Cauchy sequence in th e u.c.c. topology of ~A^ r is alsoa Cauchy sequence in its strong topology. That a Cauchy sequencein the strong topology of B'r is also a Cauchy sequence in its u.c.c.topology follows from the observation that for any compact subsetK of Dljr the family of functions {hz}ze is bounded in Br and alsoth e fact that for any h e A1/rf [hz, h] = 2 ih(z) where z e K.Consider now an operator T on a Hubert space H and let e e H.

D E F I N I T I O N 2.6. A Hubert triple is a triple (T.H.e) where H isa Hubert space. T a bounded linear operator on H and e a memberof H* A Hubert triple is a cyclic triple if the orbit of e (the linearspan of th e set {T*e}T=o is dense in H. If r = || || th emap e\ Br- *His given by e(b) = b(T)e.The map e depends on T, H, and e. Note that e(Br) is densein H iff ( ,H f e) is cyclic.The following is a consequence of Theorem 2.2.

THEOREM 2.7. If(T,H, e) is cyclic then His infinite dimensionalif and only if e is injective.Proof I f H is f inite d i m e n s i o n a l t h e n it follows f rom the Caley-

H a m i l t o n t h e o r e m of l i n e a r a l g e b r a t h a t p(T) = 0 for some pe^( [4 ] , page 320) an d t h u s e is not i n j e c t i ve . S in ce ( , H, e) is cyclict h e c o n v e r s e is t h e s t a t e m e n t of T h e o r e m 2.2.

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572 V. JS . S I N G H

Now we establish commutativity of certain maps and constructa reproducing kernel which is an a.f.p.t . The reason for this roundabout construction will be clear from Theorem 4.3.

D E F I N I T I O N 2.8. Multiplication by z, Mz: Br- +Br is the operatorgiven by M,f(Q = / ( ) for e Domain (/ ).The following lemma is an easy consequence of the functional

calculus.L E M M A 2.9. e and Mz are continuous linear maps such that .( ) = e and T- e = e- M z.D E F I N I T I O N 2.10. :H>H' is defined by ( x ) (y) = (y, x) for allx, yH.H' is conjugate linearly isomorphic to H via the map and

.T* = T' where T* is the H ubert space adjoint of Ton H and Tis the Banach space dual of T. Also from T- e = e- M z we obtain e T = M- 'e where 'e, M z, etc. are Banach space duals of e, Mz,etc . We know from Lemma 2.5 that B'r can be identified with A lfrvia the map i0.

D E F I N I T I O N 2.11. The map ae: H~> A1}r is the composition of themaps , 'e, i0 and as given in the diagramH H > Br Aur A 1 / r

where takes a function into its complex conjugate.F r o m Lemma 2.9 and the discussion following it we see that thereis a map S*: A / r* Al!r so that ae- T* = S* :e where ae is as above.

T H E O R E M 2.12. Let (T, H, e) be a Hilbert triple with \\T\\ = r.Then S*ae = ae T* where ae is such that ae(x)(z) = (x, (1 zT e)for all z e DUr and all x e H, and S*f(z) = f(z) f(0)/z. Moreover, aeis injective iff (T, H, e) is cyclic and in that case} if dim H = oo,the range of ae is dense in A1/r.

Proof. We show that ae is such that ae{x){z) = (x, (1 zT)~ e)for all z e DSr and S*f(z) = f(z) f(0)/z, because if this were so thena . (T*x ) ( z ) = ( * ?f (1 - zT)~ e) = (x, (1 - zT)~ Te) = S*(a.{x))(z) andhence jg*- g = ,- *, Now - 0i00 (x)(z) = '0 ( x )(h z) = (x)( e(hz)) =((1 - zT)- \ x) = (a? (1 - zTY'e).Now we prove the rest of the theorem. Since ae(x) - 0 iff (x, (1 z T ) ~ e) = 0 for all zeDl!r it follows that ae(x) = 0 iff (x, p(T)e) = 0for all pe&>. If e is cyclic then this holds iff x = 0. The statemen t

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R E P R O D U C I N G K E R N E L S AND OPERATORS WITH A CYCLIC VECTOR I 573t h a t if e is cyclic and if dim H = oo then the range of ae is dense inA !r is a mere dualisation of the statement in Theorem 2.2.N o t e that in general ae is injective on the orbit of e and zeroon its orthogonal complement. If e is cyclic then we can assign ani nner product ( ) on ae[H] by (ax), ae(y))*e = (a? y). This innerproduct makes T* on Hunitarily equivalent to S* on ae[H]. Moreover,and this is the point of the construction, ae[H] has a reproducingkernel.

THEOREM 2.13. Let (T,H,e) be cyclic. Then for each zeDllrthe evaluation map ae(x) f+ ae(x)(z) defined on ae[H] is continuousand ae(x)(z) = (a e( x ) , z) where ez = ae{(l z T ) ~ e) . Consequently ae[H]has a reproducing kernel K given explicitly as K(z, w) = ((1 wT)~ le, (1 - zTYxe) = U - o e, T me)zmw\

Proof. Since ae(x)(z) = (x, (1 - zT)fty = {ae{x\ ae((l -for all ae(x) e ae[H]f we see that the evaluation at z is explicitly givenby inner product with a member of ae[H] and is hence continuous.F u r t h e r m o r e , the particular member of ae[H] corresponding to eval-u a t i o n a t z is ae((l zTY1) and hence z = ae((l zT xe). Now inview of Theorem 1.2 it follows that ae[H) has a reproducing kernelgiven explicitly as K(z, w) = ( ( ( l - wTY e\ ae((l - zT e))*9 = ((1 -lo 1 ) - 1 ^ (1 - zTYe).D E F I N I T I O N 2.14. The kernel function of a cyclic triple (I7, , e)

is the reproducing kernel of ae[H]. If ( , Z, e) is just a Huberttriple, not necessarily cyclic, then the kernel function for (T, H, e) isdefined to be the kernel function for ( o, 0(e), e) where Q(e) is theclosed orbit of e and T o is the restriction of T to 0(e).

Two cyclic triples (T , Ji, ex) and (T 2, H2, e2) are defined to beunitarily equivalent, (2\ , fl , ex) (J 2, 2, e2), iff there exists a unitarymap C7: fli ^ H2 with U-T, = T 2- U and ^( O - e2.Let (T, , e) be a cyclic triple with kernel function K. Considerth e Hubert space ae[H] with the inner product ( )ae obtained from( , H, e). We can construct ae[H] from Kin the manner of Theorem1.2, since K is the reproducing kernel for ae[H]. Since ae(e) = Ko

we have the following as a corollary to the preceding.COROLLARY 2.15. Let ( , H, e) be a cyclic triple. Then the kernel

function K for {T, H, e) is a completeunitary invariant for ( , H9 e),i.e., (T*, H, e) is unitarily equivalent via ae to (S*, ae[H], KQ).The above corollary can also be deduced directly. See for

example [15].

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574 V. N . S I N G H3* The triple of an A *F *P * \ In this section we study the

kernel function of a triple more closely. F irst we mention someproperties of a kernel function. Note that if K is the kernelfunction of (T, H, e) where || || = r then K can be written as ^o

1^, T

me)z

mw

nfor z, w e D1/r. The boundedness of T is reflectedin a special property of K. The infinite dimensionality of H is also

reflected in another property of K. We describe these properties.First the following definitions.

D E F I N I T I O N 3.1. The map S?\ A S tS > A S y8 is given by f(a)(z, w) =[a(z, w) a(z, 0) (0,w) + (0,0)]/ (zw) for z, we Ds. A m e m b e r Kof As,s which is an a.f.p.t. is an analytic function of positive definitetype (abbreviated a.f.p.d.t.) iff there is a positive real r so thatr2K f{K) is also an a.f . p . t . We write (K) for the least such r.Ke A S fS is a degenerate a.f . p . t . iff K is an f . p . t . and there is ap o l y n o m ia l p so t h a t I K(z, w)p(z~ 1)p(w~ 1) ( d z / z ) ( dw/w) = (p, p) = 0,J c J cwhere C is such that its inverse contains Dls and is preferably acircle with center 0.

I t follows from Theorem 1.5 that if 6a' is a function of positivet y p e so is ^{a) 9 s i nc e ^{a){z, w)p(z~ 1)p(w~ ) ( d z / z ) ( dw/w) = (^ , w)p ( z~ 1)p1(w~ 1) ( d z / z ) ( dw/w) , w h e r e p1 is the polynomia l suchJ c J ct h a t px(z) = zp{z). As shown by the following result the kernelfunction of a Hubert triple is always an a.f.p.d.t.

PROPOSITION 3.2. If K is the kernel function for the triple(T, Hy e) then K is an a.f.p.d.t. and S^{K) is the kernel functionfor the triple ( , H, Te). Moreover, if ( , H, e) is cyclic, then p{K) =|| || and K is nondegenerate iff imH = oo.

We omit the straightforward proof of this proposition.Recall that if (T, H, e) is a cyclic triple then we have seen in

2 (Theorem 2.15) that ( *, H, e) is unitarily equivalent via ae to(S*, ae[H], Ko) and that ae[H] with ( ) is a space of functions withcontinuous evaluations at points, thus the space ae[H] and in factth e triple (T*, H, e) and hence ( , H, e), is determined by the kernelfunction K. We want to describe explicitly in two different wayscorresponding to the two Theorems 1.2 and 1.5 of 1, the construc-tion of (T, H, e) from K.

First it follows from ae- T* = S*- ^ and the fact that a* = att h a t ae- T = S-ae where S is the adjoint of S* relative to the innerproduct ( )ae of ae[H]. We construct ae[H] in terms of the kernelK of (T, H, e). We make the construction for an arbitrary a.f .p.t .

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R E P R O D U C I N G K E R N E L S AND OPERATORS WITH A CYCLIC VECTOR I 575Ke As,s. A formal for S is given in Proposition 3.4. F irst a definition.

D E F I N I T I O N 3.3. For K an a.f . p . t . , Ke A r>r> let H be the com-pletion of the space of functions which are finite linear combinationsY^aiKai for at e C and at e Dr with respect to the inner product( * atK*v ; bjK .)k = iti a&Ki j, a,).We have the following result .PROPOSITION 3.4. Let KeAr,r be an a.f.p.t. ThenH cAr. IfK is the kernel function for a cyclic triple (T, H, e) then ae[H] = H and the adjoint S of S*: H H is given by S(KW) = Kw KJwfor w 0 and S(KQ) = dK/ dw \w=0.Proof. The fact that H consists of functions is known from

1. Let fe H and let fn = i &?%

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576 V. N . S I N G H

Ci is a contour such that its inverse is rectifiable in the complexplane, contains Dr and has winding number 1 about Dr and is con-tained in the domain of holomorphy of bt. If multiplication by z, M zis bounded on Br relative to ( ) then Ml is its bounded extensionto H .

H is well- defined despite the arbitrariness of the contours andr. This is so because of Cauchy's theorem. Also note tha t H doesn o t consist of functions in general (see counterexample 1, 6).PROPOSITION 3.8. Let K be an a.f.p.t. e A Ur ir. Then( i ) M z on Br has bounded extension Msz to H iff K is ana.f.p.d.t.(ii) If K is an a.f.p.d.t. then K is the kernel function of(iii) If K is the kernel function of a cyclic triple (T, H, e) then

(M ZA, HKf 1) is unitarily equivalent to (T, H9 e) under the map pv-p(T)e for all polynomials p.Proof.( i ) K is an a.f.p.d.t. iff there is t ^ 0 so that ?K- S^(K) is of

positive type. However, I S^{K)(z, w)a(z~1)a(w~1)(dz/ z)(dw/ w) =f J K(z, wXM^iz- 'XM^Xw- 'Jd Xdw/ w), a e Br. Hence t2K-is a function of positive definite type iff t ^ ||ikC ||, Br being densein Hx. To prove(ii) note that if e Dllr then the function R : z > (1 ^)"1 e Brand {Rh R) = l/ 4 2 f f K(z, w)l/ ((l - Xw'1)^ - z- ))(dzlz){dwlw)J c J c2where Cf C2 are contours specified as in Definition 3.7. I t followsfrom the Cauchy formula (see [13], page 26) that (R , R) = K(X, )and thus (ii) is proved.

The proof of (iii) is straightforward and is left to the reader.COROLLARY 3.9. A function K is the kernel function for some

cyclic triple iff K is an a.f.p.d.t.4* F urther properties of an A *F *P * \D E F I N I T I O N 4.1. The linear map C": H > H is given on K'a$ byC\Ka)(z) = (1 az)~ and the map J: H -H is given on polynomials

by J(p) ~ l/ 2 i I Kvp(w~1)dw/ w for a simple contour C which liesin the domain of analyticity of the function w Kw and whoseinverse contains T r and has winding number 1 about it.

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R E P R O D U C I N G K E R N E L S AND OPERATORS WITH A CYCLIC VECTOR I 577To be sure we should refer to vector valued integration but here

we mean that J(p) is th e function so that J(p)(z) = l/ 2 i K(Z, W) XJ cp{w~~ )dwlw. Clearly J(p) is independent of the contour C so long it

has the properties given in th e definition.We have the following proposition.PROPOSITION 4.2. / / ( ,H, e) is a cyclic triple then ae and Care unitary and the diagram

is commutative. Moreover, C'~ = J for any a a.f.p.t. K and C" isunitary for any a.f.p.t.

Proof. We know that ae is unitary. That C is unitary for anya.f.p.t. K is a consequence of Theorem 1.5. Since ae[H] = H weknow that th e diagram

l s

is commutative. To prove that the rest of the diagram is alsocommutative it suffices to prove that C'- S = M z. C on elementsof the type Ka of H . However, this is true since C'- S(Ka)(z) = C".

Ka~~K(z) = ( 1 ~ g g ) " 1 ~ 1 = z(l - az)- 1 - M,- C'(K)(z) .a aNow we prove that Cf~ = J for any a.f.p.t. K. To do so it

suffices to consider elements of the form C\Ka) of H . For suchanelementJ-C'(Ka) = - 1- r Kw(l - aw~ ){dwlw) = - ?v f - ^- dw = Ka .

2 % J c 2z% J c w aNow le t H be th e closure of Br under an inner product. Thenwe have th eTHEOREM 4.3. H = H for some a.f.p.t. Ke A lr, i / r iff H* = H .Proof. I f H H then t h e inclusion of B r in H is continuous

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578 V. N . S I N G Hwhere Br is taken with its u.c.c. topology. Hence the dual of H canbe identified with H as in Proposition 4.2. Conversely if if* = H for some a.f.p.t. K in A1}r, 1/ r, then the dual can be identified withH is the same as H c Br via the map C".

The following two theorems give characteristics of an a.f.p.d.t.supplementing Definition 3.1 and furnishing a connection between ana.f.p.d.t. and an invariant subspace.

THEOREM 4.4. Let Ke Ar,r be an a.f.p.t. Then K is an a.f.p.d.t.if and only if H is S*- invariant.The above theorem is an immediate consequence, of the factt h a t the inclusion of H in Ar is continuous, and the closed grapht h e o r e m . We leave the proof to the reader.Let ( , H, e) be a cyclic triple. We prove another proposition

which relates invariant subspaces of T in H to functions of positivedefinite type arising from the kernel function of (T, H9 e) and theprojection corresponding to the invariant subspace.

THEOREM 4.5. Let {T, Hy e) be a cyclic triple. Then P is anorthogonal projection so that P{H) is invariant under T if and onlyif the function K' defined by K'(z, w) = ((I - P)(l - wT)'ef(I - P) x(1 zT)~ e) is an a.f.p.d.t.

We have to show that P(H) is invariant under T if and only ifth e operator (I P)x\- *(I P)Tx is bounded. We prove this factas a consequence of a more general lemma.

LEMMA 4.6. Let ( , H, e) be a Hilbert triple and H a firstcountable Hausdorff linear topological space, T a continuous linearoperator defined on H and e a member of H. Let H? = {xe H; thereis a sequence {pi{T)e}i>x and {Pi(f)e}i*0 where each PieP,i =1,2, } Then H? is a closed invariant subspace for T.

Proof. Hz is clearly a linear subspace of H invariant under T.The whole difficulty lies in showing that it is closed. This is alsoquite straightforward and we do it as follows. Let {pT(T)e}i bea sequence converging to xm and let also the {xm}m converge to x.Also let {p ( ) }i converge to 0 for each m. Let B(x, 2~n)(B(x, 2~n))be the open (closed) ball of radius 2~n with center at x and let {n^be the sequence for which {xm}m B(x, 2~ni) - B(x, 2"W"1) 0 . Sucha sequence {n^ t oo exists since {xm}m * x. For each nt choose anxn.e B(x, 2~w0 B(x, 2~%~1). Suppose also that {[/.}* is a countableneighborhood basis at 0 for H. For each nt choose a p*f%i) so thatG Un. and p*f%i)(T)e is in B(x, z~ni) - B(x, 2- ni '1) . I t is possible

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R E P R O D U C I N G K E R N E L S AND O P E R A T O R S W I T H A CYCLIC VECTOR I 579to do so since {pp(T)e}j> xn. and {^( jeJy^O for each % bysupposition.

Now the sequence {p]i{n.){T)e}%i has 0 as a cluster point in H sothere is a proper infinite subsequence of this sequence which tendst o 0. Since {p*fni)(T)e}n{ - +x in that subsequence if we replece T ande by T and e it tends to x. Thus H? is closed.

The proof of this lemma can perhaps be simplified by construc-t ion of a bounded intertwining operator from H to H whose kernelcoincides with the subspace given by the limits of {Pi(T)e}i as above.

Now we complete the proof of Theorem 4.5 as follows.Proof of Theorem 4.5. Suppose that P(H) is invariant for

T. We k n o w t h a t \\p \\B , = \\(I - P)p{T)e\\ H. Thus \\Mzp\\H , =| - P)Tp{T)e \\ H = || (I - P)TPp{T)e + ( I - P)T(I - P)p(T)e \\H =P)T(I- P)p(T)e \\H ^ || T(I - P)p(T)e \\ H ^ M\\ (I- P)p(T)e\\ H =M\\p\\H ,, where f is the norm of T, and the suffix indicates thespace in which norm is to be taken. I t follows that Mz is boundedon H , and hence K' is an a.f.p.d.t.I n order to prove the converse we use Lemma 4.6 and put for

H, , e HKfJ Ml and 1 respectively.We conclude by characterizing interlacing maps of cyclic triplest o another triple in terms of kernel functions as a preparation for 5.T H E O R E M 4.7. Let (T lf Hlf ex) be a cyclic triple and let (T 2, H2, e2)be another triple and 12the unique map given by 12{p{T^e^) = p( T 2)e2for all p e ^ . Then 12 extends to a unique bounded linear operator iz' Hx + H2 iff there exists a real number r so that r2K K2 is ana.f.p.t. where Kx and K2 are the kernel functions of the triples(T lf Hu ex) and (T2, H2, e2) respectively.5* A category of triples and harnomic analysis* Let & beth e class of all triples (T, H, e) so that T is a proper contraction.T h en ^ becomes a category if we define morphism between triples

(T lf Hu e and (T 2, H 2, e2) to be a bounded linear map l2\ I ' - + 2 sot h a t 12{e^) = e2 and T 2- 12 = ^ T^ We discuss some elementary factsabout this category. See [16] for terminology.Let .Hi and H2 be two H ubert spaces with linear operators 2\

and T 2. Then H, H2 and T 0 H2 on J?i H2 are defined as usual.(See [17].) I t is well-known that if T.'.H.^H, and T 2.H 2- *HZ arebounded linear operators then T 0 T 2 is also bounded and infact

We summarize some properties of this category ^ in the follow-ing lemma.

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580 V. N . S I N G HL E M M A 5.1. The following hold in the category ^:( i ) A morphism l2: (T lf Hu et) + (T 2, H2, e2) is an epimorphismiff uiHi) is dense in H 2.(ii) If (T, H, e) is cyclic then there is at most one morphism

from (T , H, e) to any other object (Tu

Hl9

et) of (T 2, H 2y e2) is a monomorphism. To prove theconverse le t 12 be a monomorphism. T h e n we have to show t h a tKer ( 12) = {0}. I t is easy to see t h a t Ker ( 12) is a closed subspace of H xinvariant u n d e r 2\ . Consider th e triple (Tx 0 T u H, 0 K e r ( & 2) , e1 0 0).Define t h e morphisms / and g:(T L T u H x 0 Ker ( 12) , ex 0) - *(T lf Hu ej by sett in g / = p and g = n + n where pt is the projec-t i o n on th e first coordinate and n is the identity mapping of (T 19 H19 eji n t o ( \ , Hu e,) and n: (T lf Ker ( 12) , 0) ~> ( 2 \ , Hu 0) is the injection ofKer ( i2) i n t o . . Thus g(x &y) = x + y. We see now t h a t t h emorphisms 12g and 12f are equal. However f g. Thus l2 is nota monomorphism if Ker ( 2) {0}.T h e category ^ also admits the usual tensor p r o d u c t o p e r a t i o n(x). iZ H2 and its 2- completion H x (g) H2, T x (x) T 2 and its extensionTi T t are defined as usual. I t is well- known t h a t if T and T 2 arebounded linear operaters t h e n so is \ T 2 and in fact || T x 21| ^l l || || TJI . (See [6].)

LEMMA 5.2. The operation (x) is a product in &.We leave the proof of this statement to the reader. I t is easy

to check that (x) is not a sum in

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R E P R O D U C I N G KERN ELS AND OPERATORS WITH A CYCLIC VECTOR I 581invariant subspace let it be given by the range of an orthogonalprojection P. Let ( , H, e) be written as ( , {p(T)e \ pe &*}, e) con-sider also the triple (M {&}P, 1) where the inner product on { }P isgiven by (p, q) = ((I - P)p(T)e, (I - P)q(T)e). Then (JIC WU 1)is a cyclic triple (Theorem 4.5). Now the desired morphism *. ( , {p(T)e\pe^}, e) -> (MA , { }P, 1) is given by 2{p{T)e) = p.I t is not clear whether the category & also has an initial object.However, if we adjoin to ^ the triple (M z, H\ 1) where H2 is theHardy space of analytic functions defined on D1 whose boundaryvalues are square integrable on the unit circle, and obtain anothercategory &" then W certainly has an initial object; namely thetriple (M z, H2, 1). This will be given as a corollary in the latterpart of this paper.

We now make the following definition and show that the kernelfunction map k defined as below connects ^ and the harmonicanalysis of the two- torus.

D E F I N I T I O N 5.5. The map k named the kernel function map is themap which assigns to a member (T y H, e) of c^ its kernel function.

Now we prove the following properties of k related to theoperations 0 and (x) of ^

T H E O R E M 5.6. If K and K2 are the kernel functions for (Tf Z , ejand ( 2, H2, e2) respectively, then the kernel functions for (T x0 T2, Hx 0H i, &i 0 e2) and ( \ () T2j H^ H2, e (x)e2) are K + K2 and Kx *K2 respec-tively where K^K2 (z, w) = * I"" K(ze~ f we-^)K2{eid, e**) (d / 2 ) (d f/ 2 )J o Jo

and where d j2 and d / 2 denote the normalized Haar measure ojthe circle.

Proof. Let Cm,n and C2m>n be the coefficients of zmwn in a Tay l o rexpans ion of the k e r n e l f u n c t i o n s of the t r i p l e s ( T 0T 2, H1a n d (2\ (g) T 2, Hx (x) H2, e1 (x) e2) respectively.T h e n ,Ci,n = ((T,0 T 2y( ei 0 e2\ {T x 0 T 2r{e 0 e2))H

= (2\* 0 T 2\e, 0 e2) , T? 0 T^e, e2))Hl ff2= ( IX, T?e)Hl + ( a e2, T?e2)H2and

c , n = ((T, ( 2 (ei (g) 2), ( , (g) 2yn(ei = ( (g) ^ (x) 2), T (x) 2m( i 2))

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582 V. N . S I N G Hand hence m ,%= 0 C1mtnz~ mwn = K^z, w) + K2(z, w). We also observet h a t since 2\ and T 2 sue proper contractionsK and K2e A 8iS for somes > 1 and hence

C2m,nz~

mw

n= * P ( i, T?e1) z

me

im w

ne-

in+m,n=0 Jo JO Jra,w= 0

x (T?e2, T 2me2)e- im ein*d / 2 df/ 2m,n=0- P K(ze~ i , w

jo JoD E F I N I T I O N 5.7. The equivalence relation ^ defined on ^ is

such t h a t ( , H, e) ~ (T u H l9ej iff there is a unitary map : 0(e)~>0( O so t h a t (e)= and = 7^^.

Now let us consider the category ^ with the equivalence relation~ and denote this new object by | . Then the operations 0 and (x>are defined on %_ and we have th e following corollary.

THEOREM 5.8. The class % is somorphic to a subsetof the groupalgebra of the torus and so it is a cancellative abelian semigroupunder the operation 0 . I t also has no divisors of zero under theoperation(x).

We remark t h a t th e set of a.f.p.t.'s is closed under pointwisemultiplication. However, th e set of a.f.p.d.t.'s is not closed underpointwise multiplication. (Seecounterexample 3.)

6* Some examples an d counterexamples*EXAMPLES.1. Consider (M z, H\ 1). Then {M z, H\ 1) is a cyclic triple andits kernel function K is given by

K(z, w) = ( zm)zmwn = (1 - zw)'1.m,n=0

If q is an inner function t h e n k(M z, H 2, 1) = k(M z, H\ q).2. The triple (F, U [0,1], 1), where V is the Volterra operatoron th e Hubert space of square in tegrable real- valued functions ont h e closed interval [0, 1]with Lebesgue measure, is also a cyclic t riple.I t s kernel function K is given by

K(z, w) = (V n( l ) , Vm(l))z~mwn0

I (2 + w)dx = t-Z

( + w)xo Z + W

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R E P R O D U C I N G K E R N E L S AND OPERATORS WITH A CYCLIC VECTOR I 5833. The kernel function of the triple (M a, C, 1) is the functionaz)~\l aw)~ . This is a special case of an example to be

given in the latter part of this paper.Counterexamples.1. This counterexample shows th at in general H is not a spaceof functions. We claim that the following holds.Let (Mz, HKi 1) be a canonical triple with

K{z, W )= A e2{~ en+1)znwn .Let M(X) denote the space of functions (measurable or otherwise)for a Borel field (X A, ) finite valued except for a set of measurezero, and let it be given the topology of pointwise convergence.Then the identity map i of the polynomials contained in H to thepolynomials contained in M(X) if injective is not continuous. This isseen as follows.

First the space M(X) is complete. Second the sequence of func-tions {fn}n where fn(z) een~ xzn is a complete orthonormal sequence inH . Hence the sequence of polynomials {p^ neN given by pn{z) = ?= o 2~V~V converges to a member h of H . Now let X be anysubset of the complex plane . Since i is injective X {0}. I tsuffices to show that the sequence pn(z) n does not converge for anyz e 0, z 0. Let z = rei0. Then | pn(z) - pn^ (z ) | = e %~\r/2)n =ee%~ 1+wlogr/ 2, which ten ds to oo if r 0. H ence the sequence {pn(z)}nis divergent unless z = 0.

2. If (7, H, e) is a cyclic triple then the subspace {p(T)e \pcan be given the structure of an abelian ring if we definep(T)e q(T)e = pq(T)e. However, this operation is not continuousand does not extend to all of H. Take for ( , H9 e) the triple(M z, H2, 1). Consider the convergent sequence {pn}n given by pn(z) = m= i zm/ ms/ . The limit is of course in H2. However, a simple calcu-lation shows that p\ does not converge in H2. We leave the calcu-lation to the reader. (To see how it relates to (x) not being a sumin f take for j\: (T, Hy e)->{T T, H H,ee) the map f(T)eh->/ ( )e(x)eand for j2: ( , H, e) - > ( , H(g)H, e (x) ) the map flr( )e- >e (x) (T)e and let (T3, i/"3, s) be the triple ( , fl", e) with 013 and ^28 asth e identity map (f{T)e\- f(T)e) then there is no map from ^:( () , (g) , e 0 e) -> (7, iJ, e) so that ^o^ = z and ^o^2 = 2Z .)

3. We show now that every a.f.p.t. is not an a.f . p . d . t . as wellas the pointwise product of two a.f.p.d.t .'s is not necessarily ana.f.p.d.t . The kernel function of the triple (1, C, 1) is the function

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584 V. N . S I N G HK(z, w) (1 z)'1^ w)~\ This is an a.f . p . d . t . Its square thefunction Kx(z, w) = (1 z)~2(l w)~ 2 is obviously an a.f.p.t. But itis not an a.f.p.d.t. as H is one dimensional and the a.f.p.d.t. cor-responding to a triple (M a, 0, ) is the function \l az)~ (l aw)~\

REFERENCES1. N . Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc, 68 (1950),337-404.2. S. Bergman, The Kernel Function and Conformed Mapping, American MathematicalSociety Surveys N umber V, N ew York, 1950.3. A. Beurling, On two problems concerning linear transformations in Hilbert space,Acta. Math. , 8 1 (1949), 239- 255.4. G. Birkhof and S. MacLane, A Survey of Modern Algebra, Revised edition,M a cm i l l a n Co., New York, 1963.5. S. Bochner and W. T. Mart in, Several Complex Variables, Fourth printing, PrincetonUniversity Press, 1964.6. J. Dixmier, Les Algebres dy Operateurs danse Le Espace Hilbertian, (Algebres de VonNeumann) , Gauthier- Villaries, 55 Quai des Grandes Augustins, 1957.7. N . Dunford and J. Schwartz, Linear Operators, Part II, Interscience, Wiley andSons, New York, Third printing, March, 1967.8. I . C . Gohberg and M. G. Krein, Theory and Applications of Volterra Operators inHilbert Space, Translation from the Russian by A. Feinstein , Amer. Math . Soc, Provi-de nce , R. I., 1970.9. P. R. Halmos, Finite Dimensional Vector Spaces, Second edition, Van N ostrand ,P r i n c e t o n , N . J. , 1958.10. , Introduction to Hilbert Space, Second edition, Van Nostrand, Princeton,N . J., 1957.1 1 . H. Helson, Lectures on Invariant Subspaces, Academic Press, New York, 1964.12. E. Hille and R. S. Ph illips, Functional Analysis and Semi- groups, Revised edition,Amer. Math . Soc. Colloq. Publ., Vol. 31, Amer. Math . Soc, Providence, R. I., 1957.1 3. L. Hormander, An Introduction to Complex Analysis in Several Variables, VanN o st ra n d , Princeton , N . J., 1967.14. J. L. Kelley and I . Namioka, Linear Topological Spaces, Van Nostrand, Princeton,N . J., 1963.15. M. S. Livsic, On Spectral Resolution of Linear Nonself Adjoint Operators, Mat.Sb., 34 (76), (1954), 145-199. English tran slat ion Amer. Math. Soc. Translations (2), 5(1957), 67- 114.16. S. MacLane, Homology, Academic Press, New York, 1963.17. M. A. N aimark, Normed Rings, P. N oordhoff N . V., Groningen, N etherlands, 1964.18. R. R. Phelps, Lectures on Choquets Theorem, Van Nostrand Mathematical Studies7, Van N ostrand, Princeton, N . J., 1966.19. G. C. Rota, On models for linear operators, Comm. Pure Appl. Math., 13 (1960),469- 472.20. V. N. Singh, Reproducing Kernels and Operators with a Cyclic Vector, Thesis,University of California, Berkeley, 1969.Received Febru ary 27, 1973.I N D I A N STATISTICAL INSTITUTE

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PACIFIC JOURNAL OF MATHEMATICSE D I T O R S

RICHARD ARENS (Managing Editor) J. DUGUNDJIUniversity of California Department of MathematicsLos Angeles, California 90024 Un iver sity of Southern CaliforniaLos Angeles, California 90007R. A. B E A U M O N T D. G I L B A R G AND J. M I L G R A MUniversity of Washington Stanford UniversitySea ttle. Wa shington 98105 Stan ford . California 94305

ASSOCIATE EDITORSE. P. BECKENBACH B. H. NEUMANN F. W O LF K. YOSHIDA

SUPPORTING INSTITUTIONSUNIVERSITY OF BRITISH COLUMBIA UNIV ERSITY OF SOUTHERN CALIFORNIACALIFORNIA INSTITUTE OF TECHNOLOGY STANFOR D UNIVERSITYUNIVERSITY OF CALIFORNIA UNIVERSITY OF TOKYOMONTANA STATE UNIVERSITY UNIVERSITY OF UTAHUNIVERSITY OF NEVADA WASHINGTON STATE UNIVERSITYNEW MEXICO STATE UNIV ERSITY UNIV ERSITY OF WASHINGTONOREGON STATE UNIVERSITY * * *UNIVERSITY OF OREGON AM ERICAN MATHEMATICAL SOCIETYOSAKA UNIVER SITY NAV AL WEAPON S CENTERThe Supporting Institutions listed above contribute to the cost of publication of th is Journal ,but they are not owners or publishers and have no responsibility for its content or policies.Mathematical papers intended for publication in the Pacific Journal of Mathematics shouldbe in typed form or offset-reproduced, (not dittoed ), double spaced with large ma rgins . Under-line Greek letters in red, German in green, and scr ipt in blue. The first paragraph or two mustbe capable of being used separately as a synopsis of the enti re paper. Items of the bibliographyshould not be cited there unless absolutely necessary, in which case they must be identified byauthor and Journal , ra ther than by item num ber. Man uscripts, in duplicate if possible, may besent to any one of the four edi tor s. Please classify according to the scheme of Math. Rev. Indexto Vol. 39. All other communications to the editor s should be addressed to the managing edi tor ,or Elaine Barth, University of California, Los Angeles, California, 90024.100 reprints are provided free for each article, only if page charges have been substantiallypaid Add itional copies may be obtained at cost in multip les of 50.The Pacific of Journal Mathematics is issued monthly as of January 1966. Regular sub-scription ra te : $72.00 a year (6 Vols., 12 issues). Special ra te : $36.00 a year to individualmembers of support ing inst i tu t ions.Subscriptions, orders for back numbers, and changes of address should be sent to PacificJournal of Mathematics, 103 Highland Boulevard, Berkeley, California, 94708.

PUBLISHED BY PACIFIC JOURNAL OF MATHEMATICS, A NON-PROFIT CORPORATIONPrinted at Kokusai Bunken Insatsusha (International Academic Printing Co., Ltd.) , 270,3-chome Totsuka-cho. Shinjuku-ku, Tokyo 160, Japan .Copyright 1973 by Pacific Journal of MathematicsManufactured and first issued in Japan

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Pacific Journal of MathematicsVol. 52, No. 2 February, 1974

Harm Bart, Spectral properties of locally holomorphic vector-valued functions . . . . . 321

J. Adrian (John) Bondy and Robert Louis Hemminger, Reconstructing infinite

graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

Bryan Edmund Cain and Richard J. Tondra, Biholomorphic approximation of planar

domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341Richard Carey and Joel David Pincus, Eigenvalues of seminormal operators,

examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

Tyrone Duncan, Absolute continuity for abstract Wiener spaces . . . . . . . . . . . . . . . . . . . 359

Joe Wayne Fisher and Louis Halle Rowen, An embedding of semiprime

P.I.-rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

Andrew S. Geue, Precompact and collectively semi-precompact sets of

semi-precompact continuous linear operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

Charles Lemuel Hagopian, Locally homeomorphic connected plane continua . . . . . 403

Darald Joe Hartfiel, A study of convex sets of stochastic matrices induced by

probability vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405

Yasunori Ishibashi, Some remarks on high order derivations . . . . . . . . . . . . . . . . . . . . . . 419

Donald Gordon James, Orthogonal groups of dyadic unimodular quadratic forms.

II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

Geoffrey Thomas Jones, Projective pseudo-complemented semilattices . . . . . . . . . . . . 443

Darrell Conley Kent, Kelly Denis McKennon, G. Richardson and M. Schroder,

Continuous convergence in C(X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457

J. J. Koliha, Some convergence theorems in Banach algebras . . . . . . . . . . . . . . . . . . . . . 467Tsang Hai Kuo, Projections in the spaces of bounded linear operations . . . . . . . . . . . . 475

George Berry Leeman, Jr., A local estimate for typically real functions . . . . . . . . . . . . . 481

Andrew Guy Markoe, A characterization of normal analytic spaces by the

homological codimension of the structure sheaf. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

Kunio Murasugi, On the divisibility of knot groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491

John Phillips, Perturbations of type I von Neumann algebras . . . . . . . . . . . . . . . . . . . . . 505

Billy E. Rhoades, Commutants of some quasi-Hausdorff matrices . . . . . . . . . . . . . . . . . 513

David W. Roeder, Category theory applied to Pontryagin duality . . . . . . . . . . . . . . . . . . 519

Maxwell Alexander Rosenlicht, The nonminimality of the differential closure . . . . . . 529

Peter Michael Rosenthal, On an inversion theorem for the general Mehler-Fock

transform pair. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539

Alan Saleski, Stopping times for Bernoulli automorphisms . . . . . . . . . . . . . . . . . . . . . . . 547

John Herman Scheuneman, Fundamental groups of compact complete locally affine

complex surfaces. II. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553

Vashishtha Narayan Singh, Reproducing kernels and operators with a cyclic vector.

I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567

Peggy Strait, On the maximum and minimum of partial sums of randomvariables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585

J. L. Brenner, Maximal ideals in the near ring of polynomials modulo 2 . . . . . . . . . . . 595

Ernst Gabor Straus, Remark on the preceding paper: Ideals in near rings of

polynomials over a field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601

Masamichi Takesaki, Faithful states on a C-algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605

R. Michael Tanner, Some content maximizing properties of the regular simplex . . . . . 611

A d B h W A l f th P l Wi th f t i

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