103

MATRIX REPRESENTATIONS OF POLYNOMIAL OPERATORS

by P. M. Prenter (*) (Fort Collins, U. S. A.)

SUMMARY. Let H be a separable Hilbert space. Every bounded, n-linear operator L on H n to H ( n = 0 , 1, 2, . . .) is shown to have a unique matrix representation with respect

o~ to each complete orthonormal sequence {~ok} 1 . Conversely, every operator on H n to H pos- sessing a matrix representation is proved to be a bounded, n-linear operator. The foregoing conclusions then apply to polynomial operators P on H to H where

P x = L o + L ~ x - k - L ~ x ~-~- .-. -~-L,~x n

and each L, is a k-linear operator.

1. INTRODUCTION

It is well k n o w n that if H is a real separab le Hi lber t space with a

Schauder bas is {q~i: i = 1, 2, 3, . . .} and if A is a con t inuous l inear opera to r

on H into H, then A has a mat r ix represen ta t ion (a,.j) with respec t to I%}

given by a~ = (A%, %) where ( , ) deno tes inner product . Fur thermore , every

l inear opera to r A hav ing a matr ix represen ta t ion (at~) mus t be a cont inuous

l inear operator . Thus , so lv ing the equat ion A x = y in the Hilber t space H

where A is a bounded linear opera to r is equivalent to so lv ing the infinite

sys t em of l inear equat ions

( 1 . 1 ) y , = i = 1, 2 . . . . j=l

(*) Sponsored by Mathematics Research Center, United States Army, Madison, Wisconsin,

under contract No. D.A.-31-124-ARO-D-462.

1 0 4 P. ~ . P r z ~ r

in l ~. Such systems of equations and their relation to the finite subsystems

N

(1.2) y , - - ~ a,~x~, i - - 1, 2 , . . . , N j- - I

was studied extensively by Hilbert and his followers. Related linear systems

(1.1) and (1.2) also arise in the solution of A x = y via projectional or varia-

tional methods such as least squares and (]alerkin procedures.

The study of solutions of infinite systems of equations and their finite

subsystems is, of course, not restricted to linear equations in a Hilbert space H.

A number of mathematicians have recently been trying to adapt such variational

methods to solving nonlinear equations in Hilbert or Banach spaces. The papers

of Cesari [2] and his student Locker [6], of Urabe [19, 18] and of E. Hopf [4]

are of particular interest. Among the nonlinear operators in a normed linear

space the polynomial operators are perhaps the simplest. Such operators,

through the vehicle of matrix representations can give rise to infinite polynomial

systems o o o o

(1.3) y , - - a,-[- ~ a~l,/, xj, xj, -F- ~ a,i,j,...jk xj, . . . xl, -[- . . . j l j = = l j l j ~ . . . ]k = 1

where i - - -1, 2, . . . . Sufficient conditions for the convergence of direct iterative

techniques for the solution of (1,3) and of its finite subsystems N N

(1.4) y , - a, + a,j,j, xj, xj, + JN x j , . . . j l j= = l j , I2 . .. j N - - 1

and the interrelation of their solutions is treated by Marcus in his papers [7]

and [8] in this journal (1962, 1964). Marcus addressed himself strictly to sy-

stems (1.3) and (1.4). A reformulation of Marcus theory in the language of

polynomial operators is outlined in the paper [14] assuming some characteriza-

tions of these operators which have matrix representations. This same paper

also reviews some other specialized iterative techniques for solving polynomial

operator equations due to Rail [15] and [16], and to McFarland [9]. The purpose

of the present paper is to prove that each continuous polynomial operator P

on a real separable Hilbert space H into H has a matrix representation and

that each polynomial operator endowed with a matrix representation is conti-

nuous. Such theorems generalize to l ~ spaces, 1 ~< p-~< oo in a manner ana-

logous to the linear theory. Importance attaches to such representations for

the same reason importance attaches to the original linear theory of Hilbert.

MATRIX REPRESEI~TATIONS OF POLYNOMIAL OPERATORS 1 0 5

In particular, matrix representations give rise to polynomial sys tems (1.3) and

(1.4) for which some computational algori thms exist. It is also important to

remark that polynomial equations in a normed linear space X to a normed

linear space Y encompass a broad spectrum of applied problems including all

linear equations. Among these are the Riccati differential equation, the Navier-

Stokes equations, the Chandrashekar equat ions of radiative transfer [3]. and a

wide class of Hammerste in integral equations. We start by defining these sim-

ple non linear operators.

2. POLYNOMIAL OPERATORS

Let X and Y be linear spaces over the field of real (complex) numbers.

For each n = l, 2, . . . , let X n denote the direct product of X with itself n

times. That is

X n ----- {(x~, x2 . . . . . xn): x~ ~ X, i ----- 1, 2 . . . . , nl.

An n-linear operator L on X ~ into Y is a function L(x~, xo . . . . , xn) which is

linear and homogeneous in each of its arguments separately. Tha t is, fixing

each coordinate x i of the n-tuple (xt, x2, . . . . xn) except the k th one obtains a

linear operator in the variable xk on the space X into the space Y. A O-linear

operator Lo on X is a constant function on X into Y. We shall identify a

0-linear operator L0 with its range so that Lox = Lo for all x E X. In the

event L is n-linear, n ----- I, 2, . . . , and xt = x~ = . . . = Xn = x we adopt the

notation

L(x t , x~, . . . , x n ) = L x n.

For each k---~0, 1, 2, . . . let Lk be a k-linear operator on X to Y. The

operator P on X to Y given by

P x = Lo + L i x + L~x 2 + . . . + Lnx n

is called a polynomial operator of degree n on X. The polynomial operators

are in general non-linear and are among the most simple of the non-linear

operators. This non-l ineari ty is obvious in the case of polynomials of degree

zero and is s imply illustrated when n = 2 by the quadratic P x = L2x 2 through

the equation

P (x + y) = L~ (x -q- y, x q- y) ----- L~ x 2 q- L~ x y + L~ y x -q- L~ y2.

1 0 6 P. ~ . ~,R~.~T~R

Clearly P(x + y) = P x + P y iff L~xy = - - L~yx. Polynomial operators are a

direct generalization of ordinary polynomials in n real (complex) variables to a

linear space setting. Although a comprehensive theory of these operators is not

yet available, some appealing analogies to the theory of ordinary polynomials

in n real (complex) variables have been observed. In particular, L. B. Rall [15]

has investigated quadratic equations (polynomials of degree 2) and found a

quadratic formula in Banach space; the Weierstrass Theorem has been proved

[l l , 13] for polynomial operators on a separable Hilbert space; and J. R.

Phillips [10] has drawn upon the theory of compact, self-adjoint operators on

a separable Hilbert space H to obtain eigenfunction expansions for a certain

class of self-adjoint quadratic operators (2 nd degree) on H. The interested reader

is referred to the bibliography for a list of that work done on polynomial

operators that is familiar to the author.

Let ~n(X, Y) denote the family of n-linear operators from X to Y and

let ~n(X, Y) denote the family of polynomials of degree n from X to Y.

Clearly, each of the families ~n(X, Y) and ~ ( X , Y) is a linear space with

addition and scalar multiplication defined by:

(2.1) (S + T)(z) .~ S z + Tz, (aS) (z) -~ a (Sz),

for each z E X ~ and each a in the scalar field of X when S, TEZ~,(X, Y) and

(2.2) (S + T) (x) ----- S x + Tx, (a S) (x) = a (Sx),

for each x E X when S, TE ~ ( X , Y).

Examples of polynomials abound, Many of the equations of elasticity

theory are of this type, the Chandrasekhar equation of radiative transfer [3] is

quadratic, and all of the examples worked on in the papers of the Cesari

school [2, 6] are quadratics or cubics. We give several examples here and

leave the rest to the readers experience and imagination.

E x a m p l e 1. In the case of ordinary differential equations, the famous

Riccati equation

(2.2) dy a t + a(t)y + b(t)y 2 = c(t), y(0) = c,

is quadratic. Assuming the coefficient functions a, b, c to be continuous, the

operator P defined by

P(y) ---- + Lty + L0

M A T R I X R E P R E S E N T A T I O N S O F P O L Y N O M I A L O P E R A T O R S 107

where L~(y, z ) = b ( t ) y ( t ) z ( t ) , L ~ y = - a ( t ) y - [ - y ' , and L0 = - - c ( t ) may be

regarded as a quadrat ic operator from the space C 1 [0, a] into the space C[0, a].

Clearly P(y)= 0 is the Riccati equation (2.2).

Example 2. The Hammerstein Integral Equations provide us with a second

example. Let K(s, tr t2, t3) be a square integrable function of four variables

for which

Then

/o'fo'/o'/o' [k(s, t i , t~, t3)[=dti dt~ d t 3 ds < oo.

r l ~'1 1

C(x,, x~, x3)=Jo -t0 -f'0 k(s, t , , t~, t3) x( t , )x( t~)x( t3)dt , d t2d t 3

is a 3-linear operator on (L2[O, 1]) 3 to L~[O, 1]. Let yEL2[O, 1] and let ;~ be a

scalar. The equation

(2.3) C x 8 - - Zx : y

is a cubic Fredholm integral equation on Le[O, 1] into L2[O, 1].

Example 3. The Navier-Stokes equations provide us with a polynomial

partial differential equat ion which is quadratic. Here the problem is to find

a velocity field u = u(x, t) and a pressure field p = p(x, t) which, for

x-----(xi, x~, . . . , xn)E A (A a bounded simply connected region in C") and

for t > 0 satisfy the differential equat ions

(2.4) u~ -Jr- u �9 grad u = - - grad p -~- A u, div u = O,

where A is the Laplacian, div is the divergence, and u(x, t) satisfies the initial

condit ions

u (x, 0) ---- Uo (x), x ~ A

and the boundary condit ion

Letting L~(u, v) = u �9 grad v,

quadrat ic equation in u.

u(x, t) = 0, x E boundary of A.

L i u = u t - A u , it is apparent that (2.4) is a

1 0 8 e . ~ . ~,REr~rE~

3. BOUNDED POLYNOMIAL OPERATORS

In the event X and Y carry topologies, one can speak of cont inuous

n-linear operators and of cont inuous polynomial operators. To this end let X

and Y be normed, linear spaces and let X ~ carry the product topology. A basic

open set U(x) about z E X ~ is a direct product ~ S ( ~ , x~) of ~ neighborhoods t = I

of the coordinates x~ of z=(x~, x2, . . . , x,) where S (% x~) = {y~EX: [[x~-y~ll < ~t}.

We define an e neighborhood U(~, z) o f z E X ~ to be the direct product f IS(e,x~). 1=1

Clearly z ' = (x~, x~, . . . , x'~)EU(e, z) iff max {]!x~--x~]l: i = 1 , 2 . . . . , n} < ~.

Furthermore X ~ is a linear space with a metric d where for each z=(x~ , x2,..., x,)

and z ~ = (xl, x~ . . . . , x ~ ) i n X ~

z + z ' = (x, + + x; . . . . . x. + x'.),

a z = (ax , , axe , . . . , ax.) ,

d(z , z') = max {[Ixi - - x~[[: i : 1, 2 . . . . . n}.

The metric d induces a norm I] ]]max on X" through the equation [Iz[]max = d(z, 6).

Here 0 is the identity in X ~ (that is, 0----(x~, x2 . . . . , x.) where x, = x2~ ... = x , = O).

It can be shown that the topology generated on X ~ by the norm II [[max is

identical to the product topology on X" and that X ~ together with this topology

is a topological linear space. In the event X is a Banach space, it fol lows

that X" is also a Banach space with respect to the norm II [Ima~.

Let T be an n-linear operator on X ~ to Y. T is said to be bounded if

there exists a positive constant M such that

Ir r ( x l , . . . , x,)ll <MFIx, ll Itx ll . . . Hx~

for all (xl, x2 . . . . . x,,) ~ X". In the event T is 0-linear, we say T is bounded

if H T x I I ~ M [ I x [ I ~ for all x E X . Thus, every zero linear operator is

a lways bounded. One can then quite easily prove

Theorem 3.1. For each n = 0, 1, 2 . . . . . l'et T be an n-linear operator. Then

a) T is bounded iff T is continuous,

b) T is bounded iff T is continuous at 6,

c) T is continuous iff T is continuous at ().

MATRIX REPRESENTATIONS OF POLYNOMIAL OPERATORS 1 0 ~

We say that a polynomial operator P

P x -~ Lo -Jr- L~x ~ L2x 2 -Jr- . . . -q- Lnx n

of degree n is bounded iff each of its component k'linear operators Lo, L~, . . . , Ln

is bounded. It follows that Theorem 3.1 also applies to polynomial operators T.

Clearly unbounded polynomials exist. For example, any unbounded linear ope-

rator is an unbounded polynomial of degree one.

If T is a bounded, n-linear operator we define the norm [IT]I of T through

the equation

(3.1) I lZl l - - inf{M: HT(x,, x~, ..., xn)]l ~<Mtlx, ll'lLx211" ... "llxoll for all (xl, x2 .... ,x,)EX"}.

It is then simple to prove

Theorem 3.2. Let T be a bounded n-linear operator. Then

[kTll = sup l iT(x , , x2, . . . , x,)ll llx~l]<l

= sup liT(x,, x2, . . . , xn) ll i = 1 , 2, . . . , n. l i x t l [ = l

Let ~ n ( X , Y) denote the family of bounded, n-linear operators from

X n to Y together with addition and scalar multiplication defined by (2.1)

and (2.2). In the event Y is a Banach space, one can easily show that

c't3~(X, Y) together with the norm defined by (3.1) is itself a Banach space.

An analogous statement holds for the space of bounded, polynomial operators.

A function f from a topological space Z into the real numbers R (complex

numbers C) is said to be bounded at a point Xo if there exists an open

neighborhood U(xo) and a positive constant K such that IIf(x)ll ~ < K for all

x E U(Xo). One can easily prove

Theorem 3.3. Let T be an n-linear operator. Then for all n ~ 1

a) T is bounded at z E X n iff T is bounded at O,

b) T is bounded at z E X ~ iff T is a bounded operator.

4. MATRIX REPRESENTATIONS OF POLYNOMIALS

Let A be an operator on a separable Hilbert space H and let /q0i}7 be a

complete, orthonormal basis for H. Then A is said to have a matrix represen-

tation (aij) with respect to lq0i}7 if for each x E H

A x = ~ dt~i i = l

1 ] 0 P . M . PRENTEn

where

and

co

di = ~ at i x i y=l

x = ~ x ~ % . j=l

It is well known that every continuous, linear operator has a matrix represen-

tation and that any linear operator having a matrix representation is continuous.

Now let A be an operator on H a to H where k ~ l . Then A is said to

have a matr ix representation (a:,6,i~ .... i~) with respect to {%}~=, if, for each k

elements x, y , . . . , z in H,

oo

A ( x , y, . . . , z ) = ~-~djr j = l

where

dj = E {aj, il,i2 ..... & xik y&-i . . . Zl, : i t , i~, . . . , i~ = 1, 2 . . . . I

and

X ~- ~'~ Xt ~ l

o o

y = ~ - ~ y , %

o o

Z ~ EZi~i. i=l

Then it is a simple matter to prove:

Theorem 4.1. Let H be a separable Hilbert space with a complete, orthonormal o~

basis {cpil~= 1 . Then

a) Every continuous k-l inear operator A, k = 1, 2 . . . . . has a unique matr ix

representation (aj, il,i~ ..... ik), with respect to {q0i}~ 1 where j , is, is, . . . , ik = 1, 2, ... and

b) ( a j , 6 , 6 . . . . . ik) = (A ~i~ ~ik_~ . . . ~i,, ~j) where (,) denotes inner product.

MATRIX REPRESENTATIONS OF POLYNOMIAL OPERATORS 111

P r o o f . Let x ~, x~, . . . , x k be any k elements of H. Let

--" L ( xi, ~/) ~i X N j = l

for each i = 1, 2 , . . . , k and let

Then

a j , i , , i , . . . . . ik = ( A ~ i , ~ i k _ 1 . . . ~ i l , ( ~ j ) .

1 2 k A x~v x N . . . x N ----

N

E i~ , i~, . . . , i~ --- I

x ~. x ~ x k A q~,, q~i~ qoik. l l i~ " " " i k " " "

It follows that

1 2 k q ) j ) = (A x N X,v . . . XN,

N

E i l , i 2 , . . . . i k --- 1

, X 1 X z X k a j , ik , i k _ l , ... i~ i~ & " ' " ik "

Since A is continuous

and

1 2 k _ _ A x ~ x ~ . x k lim Ax Nx N . . . x N .. N-.-~.oo

N

(Ax i x 2 . . , x k, %) = lim N - , o o t ~ , i ~ , . . . . & ' - I

X 1. X z X k a j , i k , i k - 1 , . . . , i~ ta i~ " " " i k

o o

E il , i2, . . . . ik -~ I

a j , & & _ ~ i~ x 1 x 2 x k. . . . . . . i l i~ ~ " " I k ~

converges for all x ~, x ~ , . . . , x* in H. Letting d j - - ( A x ~ x ~ . . , x k, %) we see

that A has a matrix representation with respect to Iq0,t~. Uniqueness follows

simply from the uniqueness of (Aq0~q~. . . q0/k, %). This completes the proof of

the theorem.

Every continuous k-linear operator has a matrix representation. Does every

infinite matrix (aj, gl,~.~ ..... ~k) represent a continuous k-linear operator on H" to H

or even a non-continuous k-linear operator on H k to H ? The answer to this

question is clearly " n o " since counter-examples are easily constructed.

A somewhat more subtle conjecture which does have an affirmative answer

asks whether every operator on H k to H, k >~ 1, having a matrix representation

is a continuous, k-linear operator. Well known proofs of this theorem in the

case k-----1 are based on either Landau's Theorem combined with the closed

112 P.M. PRENTER

graph theorem, or on properties of the adjoint of a linear operator, or on

convexity arguments. It is a simple matter to adapt the theory for the linear

case to the k-linear case to prove

Theorem 4.2. Let H be a complete, separable Hilbert space. Every operator

L on H k to H possessing a matrix representation (aj, i,,i~ ..... i~) is a continuous

k-linear operator.

We shall prove this theorem through a sequence of lemmas using mathe-

matical induction. First recall that if ~ a k bk converges for all (bk)E l", p ~ 1, k : l

then Landau ' s Theorem states that (ak)E l q where 1 - I - 1 ~ 1. If L has a P q

matrix representation (aj, i~,i, ..... i~), then for each x = (x ~, x ~ . . . . , x k) E H k

( 1 ) L x = j = l

n If x ~ = ~ x ~ t , n = - l , 2, . . . , k then

i : 1

(2) o o

jCx)= a j , . . . . . x2 x k ik l k - I " ' " il "

i l , i2 . . . . , i k ~ l

It is well known [17] that Theorem 4.2 is true when n : 1. We now make the

induction hypothesis that Theorem 4.2 is true when n ~ k - 1.

With this assumption we are able to prove

Lemma 4.3. Let (a],i l , i~ . . . . . ik) be a matrix representation of an operator L

f rom H k to t t where L x = ~ ~j(x)cpj and each Qj is given by equation (2). j : l

Then each Qj is a continuous k-linear functional.

Proof. It is obvious that each flj is k-linear. For fixed j let

be given by

c = (ci~: ii = l, 2 . . . . )

~ j a j , i] , ir . . . . " x I x 2 x k - 1 , t k ik t k _ l " " " i2 "

i2,i3 . . . . . lk----I

It is clear from Landau 's theorem that c ~ l 2 since Xcilx i , converges for all

(xi,) E 12 . We now use the induction hypothesis. Since for each j -~-1 , 2, . . . ,

Y, aj , il,i~ . . . . . i k X 1. x 2 x k-~ converges for all (x ~, x ~ x k-~) E H k-i it ~k i k ~ l " ' " i2 ' " ' " '

i s , . . . , l k

MATRIX H]~PR~,SENTA.TION'$ O~ POLYI~O~IAL OPERATORS 1 ] 3

fol lows that, for each j , (aj, g,,~, ..... i~) is a matrix representation of a cont inuous

(k - -1 ) - l i nea r operator Aj. Furthermore, c ~ A/(x ~, x ~ . . . . . xk-~). N o w simply

note that

~j(x ~, x~, . . . , x ~ ) = (c, x ~)

-----(Aj(x t, x~, . . . , xk-t), xk).

Thus , using Schwartz 's inequali ty

[~/(x ' , x 2 . . . . , x*)l = [(Aj(x I, x ~ , . . . , xk-t), xk) l

~< I I&(x ~, x ~- . . . . , x~-~)l[ - tlx~ll

~< [I A/I1" II x ~ I1" II x ~l l ' '" II x~ II.

Thus f~j is cont inuous as was to be proved.

Lemma 4.4. Let

L x = ~ Q j ( x ) % . j ~ l

I f each ~/ is continuous, then L is closed.

Proof. Let ( x , ) = (x~, x~, . . . , x~) be a sequence in H k which converges

to y : ( y ~ , y~, . . . , yk) in H k. Suppose L xn converges to z in H. We must

prove that L y = z. But (Lx , , %)-----~/(x,) which converges to ~ ( y ) since ~ is

continuous. Since (Lxn, %)-~ (z, %), it fol lows that ~(y)~-- - (z , %) for each

j = I, 2, . . . . But then o o

[[z - - tyll~ = Y'~ [(z - - t y , %)12 j = l

= E I(z, r,) - (Z y, r,) l 2 J

= ~ I~(Y) - - (Ly, %)1 ~ = 0. J

Thus L y = z and L is closed.

Now let L be given as in Lemma 4.4. Observe that if T x ~ L x where

x E H, then it fol lows from the induction hypotheses that T x E q3~_~IH, HI, the

family of bounded, ( k - - 1 ) - l i n e a r operators from H ~-' to H. We then have

Lemma 4.5. Let L be an operator from H ~ to H having a matrix represen-

tation (aj, g,i~ ..... ~k). I f L is closed and T: H ~ q3k_~[H, HI is given by T x = L x

where x E H, then T is closed. 8 - R e n d . C i r c . b l a l e m . P a l e r m o - Serie n - Tomo XXI - Anno 1972

114 v.u. W,~NTES

Proof. Let (x~) be a sequence in H which converges to x in H and suppose

Tx~ converges to A in c'Bk_~[H, H]. We must prove that T x - ~ A. For each

Y __ (y~, y~, . . . , yk_~) in H k-~, (Tx,)(y) converges pointwise to A y. However,

L is closed. Thus for all y in H k-~(x,,, y~, y2, . . . , yk-~) converges to

(x, y~, y 2 , . . . , yk-~), (Txo) (y ) - -L(x~ , y~, y 2 , . . . , yk-~) converges to Ay and thus by the closedness of L

L(x, y', / , . . . , / - ' ) = (rx)(y) = Ay

for all y EH*-'. Thus ( A - T x) is a ( / r 1)-linear operator which vanishes at

all y E H k-'. Thus A - - T x which was to be proved.

Lemma 4.6. Let T: H-~ cB,_,[H, H] linearly. If T is closed then T is continuous.

Proof. The proof is a simple adaptation of a canonical proof for the linear

case (see Taylor, p. 180). Let graph T - -{ (x , T x): x EHI. This set is closed

in the product topology on H X c~,_~[H, H]. Observe that graph T is a sub-

space when addition on H X Q3,_,[H, H] is defined by

(x, A ) + ( y , B ) - - ( x + y , A + B )

where x, y EH and A, BEc~,_,[H, H]. Let H �9 cB,_~[H, H] be the space

H X c~,_~[H, H] topologized by the norm

}l(x, B)II = Itx II + 11 B !1

where B E c~k_~[H , H]. This topology is equivalent to the product topology on

H X c~k_,[H, H]. Since H and 93,_,[H, H] are complete, H �9 c~k_~[H, H] is

complete. Observe that graph T is a closed subspace of a complete space and

thus is itself complete. Define a function A from graph T to H by

Then

A (x, 7"x) ---- x.

]1 A (x, Tx) ll = 11 x 11 ~ I1 x 11 + [i T x II = II (x, Tx) ll;

so that A is bounded. Furthermore, A is linear since

A ((x, T x ) + (y, r y ) ) = A (x + y, Y(x + y))

- - A(x + y, r x + Ty)

,-- x + y - - A (x, Tx) + A (y, Ty).

MATRIX REPRESENTATIONS OF POLYNOMIAL OPERATORS 115

Thus A is a bounded linear transformation on graph T onto H. Also A is I - - 1 .

Thus A -~ exists and is continuous since graph T and H are both complete.

It follows that if the sequence (x~) in H converges to x in H, then A-~(x~)

converges to A-~x. That is (x~, Tx~)converges to (x, T x) in H �9 ~k_~[H, HI.

But then T x~ converges to T x since

II(x., Txo) - - (x, r x ) ll = llx - - x.ll + tl T xo - - T~II.

Thus T is continuous as was to be proved.

We can now prove

Theorem 4.2. Let H be a complete, separable Hilbert space. Every operator

L on H k to H possessing a matrix representation (ai, gx,~ ..... ~k) is a continuous

k-linear operator.

Proof. The theorem is Clearly true when n - - I . Assume the theorem is

true when n - - k - 1 (induction hypothesis) and let n - - k . Define T from H

to c~3k_~[H, H] by T x = L x. Invoking Lemmas 4.3 through 4.6 we see that

L is closed which implies T is closed which implies T is continuous. Let

x = ( x ~, x ~-, . . . , x k) E H k. Then

II t. (x', x ~, . . . , x')II = l l ( r x ~) (x ~, x' , . . . , x ~) II

~< II r x' IJ Jl x = tl H x' lJ . . . I[ x~ll

II T II it x' il I1 x = l J . . . [I x~Jt

and L is continuous. This completes the proof of the theorem.

We have given a proof of Theorem 4.2 by arguments involving closed

graph theory. There is a second and somewhat more complicated proof involving

n-convexity notions for n-linear functionals. We shall however, omit this proof.

The preceding theory pertains to n-linear operators where n ~>~ 1. When

n - - 0 we can define a matrix representation of a O-linear operator. Let L0 be a

constant function on H to H. Then there exists a fixed y = L o E H such that

LoX -- y for all x E H. But y ~ 2 y ~ . The one way matrix (y~" i - - 1, 2, . . . ) i = I

is said to be a matrix representation of the operator Lo. It is clear that to each

one way matrix (c~)t=~ for which ,2 I c~]2( oo there corresponds a 0-1inear operator t=1

L 0 defined by Lo x ~ X c~ t for all x E H . t-'-I

1 1 6 P .M. P,,~m'rm..

5. FINAL REMARKS

We have proved that every continuous n-linear operator (and hence every

continuous polynomial operator) has a unique matrix representation with respect

to each complete, orthonormal system t~,l~ in a separable Hilbert space and

conversely. However, not every matrix (aj, il,~ ..... /,), j, it, i2, . . . , i~ = 1, 2 . . . . ,

where c o

A (x', x', . . . , x")-~ ~ dj%, ./=I

and

Z " " " ] " ' ~

il, i2,..., tn

x + ~ + = x j ?j j = l

corresponds to an n-linear operator on H" to H. For example, let n = 2 and

define aj, k,t ~ j " 8kt where 8k~ is the Kronecher delta. Then for each j

If x ~ %

and

a j k l

[i ~176 j 0

0 j

A x = [i ~176 oO oo

Ax~-~- ~ n?. ~H. n = l

Necessary conditions upon the terms of the matrix (aj, il,i~ ..... i.) to guarantee it

to be an operator on H" to H are rather easily come by. However, such

conditions are usually not sufficient conditions. In the event n = 1, Schur's

Lemma [11] gives a very stringent set of sufficient conditions on (ajk) to gua-

rantee it to be a matrix representation of a linear operator. Can Sehur's Lemma

MATRIX REPRESENTATIONS OF POLYNOMIAL OPERATORS | 1 7

be extended to the cases n =>/2? Also conditions exist which guarantee that

the matrix (aij) represents a compact linear operator on H to H. Certainly,

these conditions can also be extended to n ~ 2.

All of the foregoing theory reduces to the case H=12(n) where n = 1, 2, . . . .

A good part of it should carry over to the spaces H = IP(n) where p > 1 or

p = 1. At least, one can certainly ask the same questions in any separable

Banach space.

Acknowledgement. The author wishes to thank Professor J. B. Rosser for

pointing out a serious error in the initial draft of this paper. Any remaining

errors are certainly the sole responsibility of the author.

Fort Collins (Colorado), January 1971.

1 1 8 P . M . ~,RSNrEn

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