Estimates of operator polynomials in symmetric spaces. Functional calculus for absolute contraction operators

ESTIMATES OF OPERATOR POLYNOMIALS IN SYMMETRIC SPACES.

FUNCTIONAL CALCULUS FOR ABSOLUTE CONTRACTION OPERATORS

V. V. Peller

I. Formulation of the Problem. Von Neumann Inequality. One of the most effective methods in the nonclassical spectral theory of operators is the construction of a rich functional calculus for a given class of operators. The most difficult part of the construction of the maximally possible calculus ~-+ ~ (T) is to obtain sharp estimates for the norms of the operator polynomials ~ (T). Perhaps the most well-known estimate of this type for Hil- bert spaces is the Von Neumann inequality [i]

Icy (T)i < max {Icy (r~) I: I ~ I < 1} for an arbitrary contraction operator T (i.e., a linear operator whose norm IT1 does not ex- ceed one) and for an arbitrary polynomial ~ (we denote the set of all polynomials by ~A). On the basis of this inequality, a functional calculus for contraction operators in a Hilbert space is constructed (see [2]). There are many different proofs of the Von Neumann inequality [1-4].

An analog of this inequality for a Banach space X consists in the explicit computation of the norm:

[llq~Jl[xt-'sup{tq~(T) l: I r lx< i } , where T is a linear operator in X and IT IX is its norm. It is obvious that

max {1r I~ I <- i l < Ill ~ IIIx < Y,~>o[ ~(n)I.

Here, and everywhere in the sequel, the symbol @ (n) denotes the n-th Taylor (Fourier) coef- ficient of the function ~.

Case of the Space LP. Let I:~ be the space of all the analytic functions in the disk

D {~C: I ~ I<l} with the finite norm II][[~=~>0[f(n)[p<q-o~. An analytic function g in

D is called a p-multlplier if

/~_ z~ ~ / g ~ z~.

The space of all p'multipliers is denoted oy Mp, and the norm [glp of an element g, g ~_ My, is called the norm of the corresponding multiplication operator in the space l~. It is obvious that I~ Iv I~ (8)It~ for each polynomial ~, where S is a shift operator in the sequence space

/P:S(Xo, Zl . . . . ) = ( 0 , Xo, x l , . . . ) .

About l 0 y e a r s ago Ma t saev ( s e e [5.]) advanced t h e h y p o t h e s i s t h a t f o r i n f i n i t e - d i m e n - s i o n a l LP-spaces the norm Ill III,: is a p-multipller, i.e.,

I r (T)It: < I ~ I,, r ~ .~A, (1)

for an arbitrary contraction operator T in the space LP.

The author has proved (see [4, 6, 7]) that inequality (i) is valid for absolute contraction operators of the space L p, i.e., for operators T such that max (] TIL,, [T[LO~)<i, and for certain other contraction operators. For isometric operators T this inequality has been proved by Kitover (see [4]).

Symmetric Spaces, A Banach space of measurable functions on segment [0, i] is said to be symmetric [8] if it has the following properties:

= V.A. Steklov Mathematics Institute, Leningrad Branch, Academy of Sciences of the USSR. Translated from Matematicheskie Zametki, Vol. 25, No. 6, pp. 899-912, June, 1979. Original article submitted May 18, 1977.

464 0001-4346/79/2556"0464507.50 �9 1979 Plenum Publishing Corporation

i) If f~E and g is a measurable function such that Ig~) I~ [f~) I for almost all z, ~ [0, i], then g ~ E and HgllE<l l f l I~.

2) If Ifl and IgJ are equimeasurable functions and f~E, then g~E and IIfI[E =llgIIE.

If E is a symmetric space, then L~E CL I (see [8]). The LP-spaces and the Orlich, the Lorentz, and the Marcinkiewicz spaces are examples of symmetric spaces.

Let E be a symmetric space. We can assume that the functions of the space E are defined on the circle T = {~t:t~[0,1]}. A numerical sequence c = {~}~z is called a multiplier of the space E if for an arbitrary function f E E the sequenc$ {f(~ ~}~z is the sequence of the Fourier coefficients of a function from E. We will denote the space of all multipliers of the space E by M(E), and we define the norm ITc~E) of a multiplier c as the norm of the corresponding operator Qc in the space QJ (n) = cJ (n), n ~ Z.

The main result of See. 2 is an estimate, analogous to inequality (i), for I~(T) IE in a separable symmetric space E. However, if E~LP, then we cannot reduce the whole thing to the multiplication operators (multipliers) in a suitable space of analytic functions. There- fore, it is necessary to use (possibly less convenient for the description of the calculus) the multipliers of the Fourier coefficients of Qc. Namely, if T is an absolute contraction operator (in this case T is a contraction operator in the space LP by the Mityagin interpolation theorem [9]) and ~ ( ~ A , then

~ (T) ~s < I ~ [~ or T[ {~ (an)}-~z TYM(~), (2)

where ~ = ~ and 0 is an irrational number (the correctness of this definition, i.e., its independence from the choice of ~, is contained in the proof of Theorem i). It is obvious that I~ IE = ~ (To)~E, where the operator T O is defined by the equation

(roO(e ~mt) = f (#m(t+~

If E = LP, then]~ [E = ]~ I~ ; and for an arbitrary symmetric space E

sup {yr [: ~ D ) < Ir IE<E->~]~ (n ) l-

Functional Calculus for Absolute Contraction Operators. Let T be an absolute contraction operator in LP(,~, ~,~). Then inequality (i) is valid for the operator T. It is very important to describe operators T for which this inequality is valid within its natural bounds, i.e., for which the calculus ~--~ (T), ~ ~@A, can be extended to a contractive representation of the algebra Mp. If p = 2 (i.e., T is a contraction operator in the Hil- bert space and Mp = M2 = H~), then the above-mentioned description is a consequence of the Nagy--Foias theory [2]: These are those contraction operators T that do not have unitary components.

In Sec. 3 of this note we will find conditions on the operator T that are sufficient for such a "completely nonunitary character" of the operator T in the space LP, i.e., for the extension of the calculus from~A to Mp. Analogous statements will be obtained for symmetric spaces also.

2. Estimates for Operator Polynomials in Symmetric Spaces. The main aim of this sec- tion is tO prove the following theorem.

THEOREM i. Let T be an absolute contraction operator in a separable symmetric space E. Then for each polynomial

I~ (r)I~ < I~ I~-

The method of proof is based on the approximation of an arbitrary absolute contraction operator T by the operators for which the desired inequality holds.

We require the following facts about separable symmetric spaces [I0]: a) The adjoint space E* can be identified with the space of measurable functions g on the segment [0, i] such that

The duality between E and E* is realized by the form

YN (I, g) = /~E, g ~ E * :

465

b) The set of continuous functions C[0, i] is dense in the space E. c) if [ ~ E, then |im }I/y~IIE ~--0, where Xe denotes the characteristic function of a measurable set e and lel let-,,o denotes its Lebesgue measure.

Definition. A net of operators (Ta}=~A in a space E is said to converge to an operator T in the pw-topology if

lira (r~f , g) = (r~l , g) O~

for all /~E, g ~ E * and k > / O .

Remark. If a net of operators {Ta}~A pw-converges to an operator T, then [~ (T)]E lira [ ~(T~) IT for each polynomial T (see [7, Lamina i]).

Let {6r)r"_ I and {Ar}r~_-, be partitions of the segment [0, i] with binary rational end points such that JAr I = ]St I, 1<r< n, {%r}~=, be the set of complex numbers with modulus at most one, and let ~ be a mapping of the segment [0, i] onto itself such that ~I6r is an order- preserving linear mapping of the segment dr onto the segment Ar. Let us define an operator R by the equation

(RI ) (z) = ~rl ( r (x)) ff x ~ 6r. (3)

We denote the set of all such operators by ~ . If ] ~rl ---- 1, i ~ r ~ n , in the definition of the operator R, then we will say that R ~ ~,.

LEMMA i. If T is an absolute contraction operator, then T ~ ~pw.

In the case E = LP this statement is contained in the proof of Theorem 5 in [7]; moreover, this proof can be carried over verbatim to the case of a separable symmetric space if the statements a) and b) and the Mityagin interpolation theorem [9] are used.

LEMMA 2. Let R~$ and ~ A . Then

I r (n) I . < s u p {p (q) I . : Q ~ :~,}.

Proof. Let the operator R be defined by Eq. (3). Let us consider the polydisk

D" = { ( i x , , . . . , i~ , , )~C" : I t~ I < l . . . . . l~,, I < t } .

Let ~ =(~, ..... ~n) ~ D ~ and the operator R u be defined by the equation (R,~)(x) ~-- ~r/(r (x)) for x~Sr. Let f~E, g~ E* and F(~) ---- (~(1%)], g). Then F is a holomorphic function in the polydisk D n, and, consequently, by the maximum modulus principle we have max {I F (~) ] : p D n} = I F (~) I, w h e r e v - - (v , , . . . , ~n), J ~z I = �9 . �9 = I vn [ = i . T h u s ,

I (q, (n) I, g) I < I (~o (n,.) I, g) I and, since R~ ~ ~ , ' , the lemma is proved.

Lemmas 1 and 2 and the remark on the pw-convergence reduce the proof of Theorem 1 to the inequality ]~ (r)IE < I~ IE for operators T from the set ~,.

An isometric operator of the space E onto itself will be said to be unitary; two operators Tx and T2 are unitarily equivalent if there exists a unitary operator U such that UTIU-* = T2. The unitary equivalence of TI and T= obviously implies that

I �9 (TO I~ = I ~ (T~) I. fo~ r ~ ~

E ~ ~E *"~ Let { r}r=, be a partition of the segment [0, i] into segments, and ~ rH]=, be the partition of the segment E r into nr segments of equal lengths, enumerated from left to right by the index j, and let {kr~},~<r~m, ~<~<~r be a set of complex numbers with modulus one. Let us define an operator Q by the equation

(Q/)(x) (~r~/(x--(n~--l)lF,~l), x ~ F ~ , ] = n~. (4)

We will denote the set of all such operators by ~. If Id, = Ir~, in the definition of Q for all r, j~, and j~ such that l<r~m, l<],<nr, and l~]~nr, then it will be said that Q~$s, and if Irj = 1 for all r and j such that i~ r< m and i~]~nr, then we will say that Q ~ ~g~. The operators of ~ are cyclic shifts on E r with respect to the partition

466

LEMMA 3. If R ~ $~, then there exists an operator T in the set $3 that is unitarily equivalent to R.

Proof. Let the operator R be defined by Eq. (3). We choose a natural number N such that for each j, 0<j<2 N, the segment [j2 -N, (] ~-|)2 -N] is contained in one of the segments ~r and in one of the segments A r from the definition of the operator R. Then {I)([j2-N,~ ~- i) 2-~])~ [k2 -N, (k q-I)2 -N] for a certain k, 0< k< 2 ~. Consequently, ~ realizes a bijection of the family

~N- 1

= {112-~, ( / + l) 2-~1}~=o

onto itself, and therefore this family can be reindexed so that

~; = {?,lh<,<~,,<~<,,, ~,~=, n, = 2 ~

and

l?,~+,, ] ~nr, ~(V,j) = ~y,,, j ' = hr.

Let ~r/----~. if ~r i ~ ~. Let us define a segment Frj by the equation

F,I= [2-n (2~<rnt. + j - - ]), 2-n(2~< n~ + j)].

Let ~ be a mapping of the segment [0, i] onto itself such that �9 ! T,~ is an order-preserving linear mapping of the segment Yr~ onto Fr~, and let (U (f) {z) -- ](~ (x)) for x ~ [0, i]. Then U is a unitary operator, U RU'= Q, where Q is the mapping defined by Eq. (4).

Let us set

I ~ . l (z + I F . j I). x ~ F . j . 1 < , . . ( r / ) (~) = t , .1 (z - (n. - t ) I P.; I). z ~ F .~. j = n.. ( 5 )

where ~r are complex numbers such that It~r = H~=t ~rj, I ~ r ~ m. We will prove that the opera-

tors T and Q are unitarily equivalent. Indeed, if crl = 1 and cr~ =(%r~Crj-,)/[tr, for |(~r~m andthen2VQV_ ~ ~ ~:r~.and a unitary operator V is defined by the equation (V/)(x) : Crif (x) for x ~ Frj ,

Remark. It follows from the proof of the lemma that the operator Q is unitarily equiva-

lent to an operator from ~4 if H: r ~,)~ ] for |<r<m. J I

6 LEMMA 4. If T ~fes and ~-_~A, then

Proof. Let T be defined by Eq. (5) and let ~r = e ~=i~ Obviously, we can assume that the numbers 8r are binary fractions. Let E, = UrF.n for i ~ r < m. Let us subdivide the ~=1 r J,

c l e f - segment Er i n t o 2n r equal segments Prj, i ~ r ~ m, I ~ j ~ nr = 2nT, enumerated by the index j from left to right. Let us set

(T/)(~) = p , l ( ~ + I t , i t ) , ~ ' , ~ , i < ~,, t ~ . / ( ~ - (~. - t ) I r . ~ l ) , x ~ r . j , j = ~,.

It is sufficient to prove that I �9 (T) IE ~ I �9 (~) I~ for every polynomial ~, since, on applying, if necessary, the process of doubling the number of segments again, we can obtain the equal-

ity ~r = I for all r, i~ r~ m. Then by virtue of the remark to Lemm~ 3 the operator T is unitarily equivalent to an operator from the set ~.

Let N be a natural number. We divide each segment Frl, i <r~m, t < j ~ nr into 2N

g(~) | ~ k ~2N of equal lengths and enumerate them from left to right by the index segments ~r~, k. Let us define an operator TN by the equation

467

'M (z + I F,~ I), x ~ F , j , ~ < n,,

~/(x - - (n, - - ! + t/(2N))I F, j J), ~=~(~' (rN/)(x) = i = n,, k ~ ~

~ , / ( x (n , - - t I / ( 2 N ) ) ] F r j [), - ~ ( ~ ) -- ~t~Urj,

] = n~, k is evea .

The sequence {TN}N>* converges strongly to the operator T. Indeed, if /~C [0, I] and mf is the modulus of continuity of the function f, then

1 (TN/)(z) - - (r/)(x) I < ~/(I/N), x ~ [0, 11.

Let us prove that the operator TN is unitarily equivalent to the operator T. For this we divide each segment Frj into N segments =r~A(O of equal lengths, l<k.~<N, enumerated from

left to right by the index k. Let ~ be a mapping of the segment [0, I] onto itself such A(~) A(~) ~ (~k-1) . (~) that (I) I ~r, is an order-preserving linear mapping of ~rj onto ~rj for ] < mr, and �9 ] mrj+%

(~) (~) is a similar mapping of Arj+n r onto 8~ . Then the operator U, defined by the equation (U~(x) = I (~ (x)), is unitary and U-IT~U = ~.

By virtue of the remark about the pw-convergence we have

1 r ( r ) I~ < lira ! ~ (T~) !~ = I ~ (T) I~" N

Let ~ be a one-to-one (mod 0) mapping of the segment [0, I] onto itself such that for each measurable subset e of [0, i] the sets �9 (e) and ~-~ (e) are measurable and [ �9 (e) I = I e I. Such a # is called an automorphism of a measure space. A sequence of automorphism {~}n>t is said to converge weakly to an automorphism ~ if for each measurable set e the measure of the symmetric difference l~ (e) A~(e) l converges to zero (see [ii]). If ~ is an automorphism of a measure space, then the operator U#, defined by the equation (Ur = f(~(x)) , is unitary. Let us observe that if the sequence {~},,>~ converges weakly to ~, then the sequence {Uo,},,>t converges to the operator U~ in the strong operator topology. Indeed, if f = Xe, then

lim II uoJ Ur I1~ - l i m II x%(,)ar I1~ - o

by virtue of the statement c) about separable symmetric spaces.

Proof of Theorem i. By virtue of Lemmas 1-4 it is sufficient to prove that [ ~ (T)]E-~< [~ [E for an arbitrary operator T from the set ~. Obviously, in this case T = U~ for a certain automorphism ~. Now let ~ be an aperiodic automorphism, i.e.,~rnx~=x for almost all x, x~[0~ I], for any natural number n. By the Rokhlin--Halmos theorem there exists a sequence of automorphisms {Sn},>~, such that the sequence {S~ ~ ~Sn}~x converges weakly to the automorphism ~ [Ii]. Then the sequence {Us~ws },>~ converges to the operator U~ in the strong

operator topology; moreover the operator Us~ws" is unitarily equivalent to the operator U~.

Consequently,

l ~ (r)I~ <~!m I~ (us~s~)I~ - I~ (uv)I~.

Let us consider an automorphism ~0, 0~O~l, of the space [0, i], defined by the equation

Then

where = = ~=~0. consequently,

~o (x) = x + 0 (rood 1).

If 0 is an irrational number, then the automorphism ~0 is aperiodic and,

I r (r ) I~ < I r ( U ~ ) 1~ = I r IE.

Remark. It follows from the proof of the theorem thatl~(U$)l~ = I �9 IE for each aperiodic automorphism ~. Using the approximation of such a ~ by periodic automorphisms [ii], we can easily show that

468

Iv la = sup i V (U~,/=) I~-

3. Functional Calculus for Absolute Contraction Operators. If T is a contraction operator in a Hilbert space~, then~----~0~ ~, where ~,and ;~ftare invariant subspaces of the operator T, T I ~0 is a unitary operator, and T [ ;~, is a completely nonunitary contraction operato_r (i.e., does not have unitary components). If the spectral measure of the operator T [~0 is absolutely continuous with respect to the Lebesgue measure, then with each function ~, ~ ~//0o, we can associate an operator ~ {T) [2] so that

and

~) (r + r = r (T) + W (T), 2) (r162 (T) = r (T)r (r), 3) q9 (T) --- ~>~0 ~ (n) T =, when ~',,~01 ~ (n) I <~- + c~,

I.m (r) I~ < II ~11~- = sup {I v (g) 1: I g I < t}.

In t h i s s e c t i o n we w i l t o b t a i n t h e a n a l o g s of t h i s c a l c u l u s f o r a b s o l u t e c o n t r a c t i o n o p e r - a t o r s .

THEOREM 2. Let T be an absolute contraction operator in the space L p (~ ~, p,). Let the unitary component of the operator T, considered as an operator in the space L ~ {~, ~, p,), have an absolutely continuous spectral measure with respect to the Lebesgue measure. Then with each function V~ My, there can be associated an operator ~ (T) in the space L p (~, if, p,) so that the conditions 1)-3) and the following condition are fulfilled:

4) I (P (T)ILv ~ [ r [p.

P r o o f . I f {V,~}-)~ i s a sequence of f u n c t i o n s from H ~ such t h a t r ~ H% supll W l I ~ < q - o o I t

and r ~ (~) f o r a l l ~, ~ E D, t h e n t h e sequence o f o p e r a t o r s .{~ (T)}.>,, c o n s i d e r e d as op- e r a t o r s i n t h e space L 2, conve rges to o p e r a t o r ~ ( T ) i n t he weak o p e r a t o r t o p o l o g y [ 2 ] . The s ta t&ment o f t h e theorem i s obv ious f o r p = 1, +~. L e t p ~ ( l , + c o ) . I f ~ E M v, t h e n t h e r e e x i s t s a se_quence {/~}.>1, f , ~ c~A, such t h a t I f . I v ~ I~1~ and f . ( ~ ) - * / ( ~ ) f o r a l l Q E D [12] . Then II/.11~ ~ I/~ I~ ~ Iv Ij, and the sequence o f o p e r a t o r s {f,~(T)},,>~ conve rges i n t h e space L ~ to t h e o p e r a t o r V ( T ) i n t h e weak o p e r a t o r t o p o l o g y . Hence we g e t

j ' (/,~ C T) g) hdl.t --> j ' (q) (2/') g) ]u/p, (6)

for all g and h in L* ~ L ~. Since ][n (T)lcp ~ l[a ]p ~ Iv (see the inequality (i)), it follows from (6) that the sequence of operators {fn(T)} converges weakly in the space LP, since

the set Lx N L~ is dense in LP as well as in LP'. Let ~ (T) be the limit of this sequence. It is obvious that the definition of ~ (T) does not depend on choice of the sequence of polynomials {~,},>~ and that if the sequence of functions {~},>i~ ~n ~ M~, converges pointwise in D to a function ~Mv, and sup |~ Ip< q-o~ then ~,(T)-+~(T) in the weak operator topoi-

ogy. The desired properties i), 3), and 4)are obvious. Let us establish the multiplicativ- ity of the calculus.

Let ~ ~ M~, , ~ M~ and the sequences of polynomials {~,},>, and {,,}.~ converge pointwise in D to the functions ~ and ~ respectively, sup[~n Ip<~176 and sup[~Iv<oo. If g~L v, and h ~ L v" ( i /p -4- l /p ' = t), t h e n

((~. ,~) (T) g, ~) = ( v . (T) , ~ (r) g, h) = ( , . (T) g, ( ~ . (T))* ~). We have

-and

lira ((Vra%0(T) g, h) = (((p.,,) (T)g i h), n

nm (,,, (T) g, (~,,, (T))* .h) = ( , (T) g, (,r,,, (7))* h) = (,r,,, ( i "~ (2') e, h). n

Hence ((~m~)(T)g, h) = {~m (T)~ (T)g, h). The sequence of functions {~m ~}m>1 converges pointwise in D to the function ~ and sup [q)m~ Ip<oo, which gives

lira ((~m*)(T) g, h) = ((q~p) (T) g, h),

lira (q~m (T)* (7) g, h) = (~ (T) * (T) g, h). f t l

46.9

Since g and h are arbitrary, it follows that (q~)(T)= ~ (T~ (T).

THEOREM '3. Let T be an absolute contraction operator in the space L p (~, ~, F). Let one of the following two conditions be fulfilled:

i) If [ ~ L v (~7, 2, ~), ] ~ O, then there exists an m E IN- such that I] T'~] HLP < H ][ILp"

2) If ~ ~/2'(~7, ~:, ~), [ ~= O, then there exists an m ~ N such that ]I (T*)m][ILV'< II /IILV'.

Then the conclusion of Theorem 2 is valid.

Proof. It is sufficient to examine the case i). Let us verify that in this case the operator T, considered as an operator in the Hilbert space L 2, is a completely nonunitary contraction operator. Let it be not so. Then there exists a nonzero function ~ ~ L 2 such that ~I T"~HL, = ][~][L, for all n ~ N. Let us consider the space X = LP + L 2, which consists of all functions g ffi gl + g2, where gl ~ L p and g~ ~ L ~. We will consider the following inter- polational norm in this space X:

H g [Ix de_f inf {I] g~ I[L~ + M H g~ IlL' : g = g, + g~, gt ~ L~, g~ ~ L~},

where M is a positive number.

We will show that there exist functions g~L p and h~L 2 such that f = g + h and [I/llx = II g]]LP q- M]~h[{L,. Let g,~ L~',hn~L~,]= gn q-hn and llg-[Iz2 nu MI]h,,IIL'"~H]IIx. Choosing, if necessary, subsequences, we may assume that the sequences {Ug,,IILv }n>~l and {[lh~IIL,}n>l converge. Since the spaces LP and L 2 are reflexive (it is assumed that | < p < q- co) we can select subsequences {gn~}k>~l and {hn~}~>1 , that converge weakly to the functions g and h, respectively. Then

[[ g IILp = tim I]g~ HL,, ][ h []L, = l im ]l h~ lip, ] - - g q- h

and |If~x = [] g[]ry-]-M[]h]IL,. We will prove that the number M can be chosen such that g ~= O.

def. There exist positive numbers c, and c2 such that the function ~ = ]X{=:r162 belongs

to the space LP and I] ~][~v >0. Let ~e___r ]_ g. Since llhll~, <I[ JILL,, the number M can be chosen

so large that ~[]]Ix<~Igl~L~ q- M[[hIIL,< M[I]I[L,. If g = O, then l[ f[Ix ----- MH]IIL,. Consequently, g~O.

By virtue of the condition I) there exists an m~N such that l]TmgHnp <I]gI[LP. Then

II rrn f HX "":- t[ Tm g + ~ h HX < I] Tm g HL p -~ M H T" h IlL, < [[ g ]tLP q- M H h HL' - - H / [IX.

L e t R be t h e o p e r a t o r , a d j o i n t t o t h e o p e r a t o r T, c o n s i d e r e d a s an o p e r a t o r i n t h e H i l b e r t s p a c e L ~, i . e . , ] ' ( T ~ - - ~ ( ~ ) , ~, ~ n ~. Then RnTnf = f f o r a l l e ~ N , s i n c e [] T n]|]L,-[If][L, and t h e o p e r a t o r R i s an a b s o l u t e c o n t r a c t i o n o p e r a t o r a n d , c o n s e q u e n t l y , a c o n t r a c t i o n o p - e r a t o r in the space X also. We have [] ]llx = [I RmTm] Hx "~ [[ Tm ] Hx < II ][Ix. This is a contra- diction.

COROLLARY. If T is an absolute contraction operator in the space LP and one of the sequences {T"}n>0 and {(T*)"}n.>~ converges strongly to the zero operator, then the conclusion of Theorem 2 is valid.

Remark. Let E be a separable symmetric space. We can consider the space ME of analytic functions f in ~ for which there exists a sequence of polynomials {]n},~>~o, converging pointwise in ~ to f, such that sup l~n IE = c < co. We call the infimum of such numbers c over all

choices of the sequence {]~}n>~0 the norm[~ IE of the function ~ in the space M E . Then, if the adjolnt space E* is separable, Theorem 2 and the corollary (and if the space E is reflexive, then Theorem 3 also) can be carried over to the case of absolute contraction operators T in the space E. Moreover, a functional calculus on the class ME is constructed. The proofs of the named statements are carried over verbatim to this case.

We can prove the following corollary.

.COROLLARY. Let E be a separable symmetric space with separable adjoint and ~ be a complex-valued Borel measure on T such that [~ [(T)< | and I ~(n) l< | for alln, n~Z. Then for an arbitrary function ~ ~ ME, the sequence {~ (~ (n))},~z is a multiplier of the space E and

[] {r (~ (n))}n~z I]M(E) < I r

470

An analogous statement is valid for multipliers of the space LP(G) for an arbitrary locally compact Abelian group G.

The author thanks N. K�9 Nikol'skii for the formulation of the problem and assistance.

LITERATURE CITED

i. J. Von Neumann, "Eine Spektraltheorie fHr allgemeine Operatoren eines unitgren Raumes," Math. Nachrichten, 4, 258-281 (1951)�9

2. B. S. Nagy and C. Foias, Harmonic Analysis of Operators in Hilbert Spaces, North-Hol- land, Amsterdam (1971).

3. E. Nelson, "The distinguished boundary of the unit operator ball," Proc. Am. Math. Soc., 1--2, No. 6, 994-995 (1961).

4. V. V. Peller, "Estimates of operator polynomials in the space LP by the multiplier norm," Zap. Nauchn. Sem. Leningr. 0td. Mat. Inst. Akad. Nauk SSSR, 6--5, 133-148 (1976).

5. N. K. Nikol'skii, "Five problems on invariant subspaces," Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR, 2--3, 115-127 (1971).

6. V. V. Peller, "An analog of the J. Von Neumann inequality for the space LP," Dokl. Akad. Nauk SSSR, 231, No. 3, 539-542 (1976).

�9 "A �9 7 V.V. Peller, pproxxmation by isometries and a hypothesis of V. I. Matsaev for absolute contraction operators of the space LP," Funkts. Anal. Prilozhen., 12, No. I, 38- 50 (1978).

8. E.M. Semenov, "Embedding theorems for Banach spaces of measurable functions," Dokl. Akad. Nauk SSSR, 156, No. 6, 1292-1295 (1964).

9. B. S. Mityagin, "An interpolation theorem for modular spaces," Mat. Sh., 66, No. 4, 473-482 (1965).

i0. E.M. Semenov, "Interpolation of linear operators in symmetric spaces+" Doctoral Dis- sertation, Voronezh (1968).

ii. P. R. Halmos, Lectures on Ergodie Theory' Chelsea, New York (1962). 12 N. K. Nikol'skii, "On the spaces and algebras of the Toeplitz matrices, acting in ~P,"

Sib. Mat. Zh., ~, No. l, 146-158 (1966).

BOUNDEDNESS OF VECTOR-VALUED EXTERIOR MEASURES

A. N. Sazhenkov

We will denote the ring of sets by ~, the set of natural numbers by �9 , a topological Abelian group by G, the neighborhood filter of the zero in G by WG, and the set {vlnu...q - v~ lu i~V , i = t . . . . . n} b y nV.

Definition i. A subset Q of G is said to be bounded if for each V~W~:; there exists an n ~ such that Q~_+nV.

Definition 2. A set function ~t:~ -+G is said to be bounded if the set {~(E)[E~} is bounded in G.

Definition 3. A set function ~: ~ --~G is called a K-exterior measure, where J[~, if the following statement is valid for eaah V~Wa : ~ (E U F)--~ ~F)~_KV for arbitrary disjoint F, E ~ such that ~ (E)~V.

.Remark. Additive set functions, numerical exterior measures, quasi-Lipschitz set functions, i.e., functions for which 11 ++ (E U F) -- ~ (F) [1 < NII~ (E) li, where N is a number, ~ �9 m -+ and X is a normed group, are K-exterior measures.

Proposition i. Let M ben family of K-exterlor measures. Let the set E ~ be such that the sets

{+(B) IB n E = +, B ~ m , + ~ M }

PP. Novosibirsk State University. Translated from Matematicheskie Zametki, Vol. 25, No.

913-917, June, 1979. Original article submitted April 18~ 1977. ,

0001-4346/79/2556-0471507.50 �9 1979 Plenum Publishing Corporation 471

Documents

Estimates of operator polynomials in symmetric spaces. Functional calculus for absolute contraction operators