SOME PROPERTIES OF MATRICES OF BOUNDED

SOME PROPERTIES OF MATRICES OF BOUNDED OPERATORS FROM SPACE i\

TO l\

B. S. Kashi n I z v e s t i y a A k a d e m i i Nauk A r m y a n s k o i V o l . 1 5 , N o . 5, p p . 3 7 9 - 3 9 4 , 1 9 8 0 UDC

SSR. Matematika,

517.51

n

Introduction.

Assume that l , 1 O < o o , n

n

= l,

Is a space of vectors x

\ Vp

with norm W" = I

.

23

\x% \"

I

f

i
q
,2)yn - »,

1'.

If f or ±yn-n(i) Xi

46

я

we h a v e W U < l ,

Indeed, consider function f(x) with / (x) = x \

n for -

t

< x < — , l < i * n. n

i/k' 2 n , г < я , /•, m y

>/>-2we bound the number |G (m, л, r, />)) of sets ^^£'J, such that

|Qf){1, p]\ > r.

We have min

2

|C /л (л, r, p)|=

Using the bound

C * < / ^ — ^»

[p. я )

C*

Cj-4.

we find from the last equation that min

K/(m, л, r, 1

г

< К ; р' (т - р)-

p*- -ex I n- (f) ' " in ^ ] c» 5 ; » C,Q. 24

2

P

;

Lemma 2 is thus proved. Proof of Theorem

1.

It is well known (see, e.g., [3]) that for n = 1,2,..

8

on sphere {x: H*****' there exists a combination of vectors A*= {*} with !Д |< C" such Л

that for every vector x, ||x|| n = 1, there exists a vector е£л„

with

7

\x—el„ -L.

*2

*

4

2 f E"

It is easy to see that for any vector

( 9 )

Therefore,

2 6E« : U (ОЛЬ, n

>

e,

[d 6 ^

(|

« e , )' J

(j

>

y

(10)

1/Л 1/0-1,2

consequently, ? \"« ^

1 -

J / Л

1

1

/

2

1

(

Ш

2

we have for any vector

Sinci

/

m

/

•

\ 2 \ 1/2

and, bounding the right side in (11) using Lemma 2, we obtain (see (11)):

(SO A ^ - В,Д.ф

/(j,KO(i» )-C;(i) f

Theorem 1 is thus proved. Theorem

?.

There exists an absolute constant В > 0 such that for any m x n

matrix A = (ои/| with ИЛа2)< 1 there exists a combination 2 ^ £ " for which Я

H (2)1(2 21

njtJJ 1 < / < м , К / < 2 л , combination

where ,, = а,у for 1 < / < л and л , there exists a y

2'^£^with И i')ita, i) < c

• 2

In" Now, applying Corollary В to matrix 4(2') —

K / < e W i 8 * , for which, in view of

(13), IA (2')|| ,1, < C л ' In'- — , we obtain combination Theorem 2 is thus proved. 1

(2

2.

2

1

t

4

2c=2', 2 c £ ; с И (2)11(2. : > < - ^ In" — я 2

The following problem, formulated long ago by A. N. Xolmogorov, is well

known in the theory of orthogonal series: assume that l?*i*))*H> thonormalized system of functions. series

(13)

— я

-=

\K„\n\,

1) is an or-

Does there exist a permutation of the natural

for which the system {?*„

(x))n^\

is a convergence system (i.e.,

every series

converges almost everywhere)?

The "finite-dimensional" version of this problem

has the following form: does there exist an absolute constant С such that for any

49

orthonormalized combination

Ф =

jtp„

(x))^L,, x£(0, 1) there exists a permutation

k»\ ** *£ S# such that

( I

1 ;

In [ 5 ] ,

/VU)

I

)

oe

r>v H J! ^HH < c ? m

;i.v^«.

(14)

Garsia expressed the hypothesis that a bound stronger than ( 1 4 )

obtains;

specifically, ,

" J , ,„,.v - Р „ .

kn

д

)6

л

» |

, ,| 2

w

p ( 6 — 6 + 1 ) . 2

2.

г

Let us set up the sequence of partitions Д = д< (A), s = 0,1,- • •, s,, of the given 5

matrix A into submatrices. itself.

г

The zero partition A

consists of the matrix A =

Q

If partition ' л , s > l is constructed and consists of submatrices

then to construct partition Д , those matrices of partition Л

1

[Я»-)**-!,

, which intersect

the diagonal i = j of matrix A (and, by definition, they are all square) and are of order > l - f - > can be divided into four parts (Z*-') P

rule described in Ц , stops.

0

by Д. s

0

. 3

Let us specify the desired permutation

!^///_i

5

= o= oH)

first construct the combination of permutations - , l < r < s , r

a

g

in accordance with the

l 0 , the sum of the orders of the matrices 0

2

rank (**)< m*.

(26)

Obviously,

Ш where

(27) 2,9)

& = и /г; 1*611. l * j l

:

2, (?)

;

I

1(2,?)

& ==. и /г„:

c

Я (!*• •*) * « »

l'6|l.

I* l s

:

/ ' ( l * . j ) > f*>'

Using the following inequality to bound the normil/Wtae): 53

|Z*,,,/»|fl|r.,, )

(28)

2)

as well as the inequality |l#Jb, 2) < 1 (which follows from the fact that |#Д . j ) < 2

< max lAjjJfl, ) . < г)

( 3 0 )

< n"~"* max

ra , c,* It follows from (29), (30), and (27) that ;;/'

Iv-.

" Ь г )

^

( 3 D

and hence 1 _

Combining inequalities (32) and (24), we obtain the requisite bound f or И(Л„)-Ц,, «> Г

2

Theorem 3 is thus proved. In concluding this section, we will formula,e an assertion that can be prov using Corollary 2 and the partition of matrix !a» j into binary blocks. I;

Assertion.

For any matrix -4 = |ao},%i with

ИЬ.г> than does the bound С (In n) "«tt

л

1/2

which follows

trivially from (19). 3.

In this section we offer a bound for th« Kolmogorov n-diameter d (5™, I™) n

of octahedron B™ in space % % the method of proof has some common features with that used in §1.

The definition of the diameter, as well as a number of results

regarding bounds for diameters of finite-dimensional sets, may be found in [10, 11]. It is well known ([12]; see also [10], p. 237) that the following equality is valid:

«маг,/?)~(2^) It follows from

(33) that for m > 2 n

and

1 2 л , which is of importance for applica­ tions to bou ids of diameters of functional classes. T h e o r e m 4.

For л = 1, 2,

and

m > 2 n

1

J

• j min (1, m '* n-" )
я *

я

1

( (4 л') '"-' V ' j 2

/2

1

2

-J/2

2

= 4""-" V'