Continuous versions of an inequality due to Hoffman and Wielandt

Continuous Versions of an Inequality due to Hoffman and Wielandt Yiiji Sakai Department

of Applied

Mathematics

Faculty of Engineering Shinshu University Nagarw-shi, Dedicated

380 Japan to Professor

Helmut W. Wielandt

on his seventy-fifth

birthday.

ABSTRACT Two continuous versions of a celebrated inequality due to Hoffman are obtained

as a consequence

of an inequality

due to Horn

and Wielandt

and Weyl’s

majorant

theorem.

1.

INTRODUCTION

Hoffman

and Wielandt

[2] obtained the following celebrated inequality 2>‘j2 denoting the Frobenius norm of a matrix

with 111<11s =(X~=,C~=ljkijl K = (kij) of order n.

lf A and B are normal matrices with eigenvalues &, respectively, then there exists a suitable numbering

H-W. LY,, and PI ,...,

THEOREM

a1 ,...,

of the eigenvalues

such that

t

I’Y~-P~I~GIIA-‘II~~

i=l

Zf, in addition,

!RfP,>

..-

A is Hermitian,

then a “best”

arrangement

is a, > . . . >, a,,

>!RC/?“.

In this note, we shall show that two continuous versions (Theorems 1 and 2 in Section 2) of Theorem H-W are obtainable using a well-known inequality LZNEAR ALGEBRA

AND ITS APPLZCATIONS 71:283-287

Q Elsevier Science Publishing Co., Inc., 1985 52 Vanderbilt Ave., New York, NY 10017

(1985)

283

0024-379S/XS/S;3,;30

YUJI SAKAI

284 due to Horn [3] (see also [l, Theorem [4] and [l, Theorem 8.51).

2.

4.21) and Weyl’s majorant

theorem

(see

RESULTS

Let @ be a separable Hilbert space equipped with the inner product (. , .) and K be a completely continuous (abbreviated to c.c.) operator on 4. Then the Schmidt norm of K is, by definition, ]]K]]s = [tr(K*K)]‘/‘, and the Schmidt class is the set of all C.C. operators T such that ]]T]]s < co, where tr denotes the trace of an operator. Further, for each CC. operator K, let s,(K) denote the i th eigenvalue of (K *K)r12 (called the i th singular value of K) arranged in nonincreasing order of magnitude taking account of their multiplicities. The symbol VI e I3 denotes the real part of an operator B, defined by (B + B*)/2,

and A > 0 means that A is a positive

operator.

THEOREM 1. Let A and B be completely continuous opemtors on n sepamble Hilbert space $5 belonging to the Schmidt class with eigenvalues { oi } and { pi } respectively, ta k ing account of their multiplicities, Zf A 2 0, and B is normal with 5%e B >, 0, then there exists a suitable numbering of the eigenvalues such that

cc

C Iai -

PiI

G IIA - Bl122'

i=l

THEOREM

separable

3.

Zf A and B are completely continuous operators 2. Hilbert space @ belonging to the Schmidt class, then

on a

PROOF

We need Weyl’s majorant theorem (see [4] and [l, Theorem following lemma due to Horn [3] (see also [l, Theorem 4.21).

8.51) and the

CONTINUOUS

LEMMA. For any completely arable Hilbert space kj,

continuous

operators

A and B on a sep-

k

k

C si( AB) < C si(A)si(B)t

k=1,2,....

i=l

i=l

Proof of Theorem orthonormal

285

VERSIONS OF THE H-W INEQUALITY

system

Suppose

1.

AB = BA. Then there exists a completely

{ +, } in @ consisting

of the common

eigenvectors

of A

and B. Therefore,

5(

IV-Bll;=

i=l

=

C

Iai-PI

i)

I27

i=l

where II is a suitable permutation over the set of all natural numbers N. Assume Aa0 and ‘%eBaO. Then AB#BA if and only if AYIeB

is

non-self-adjoint. Suppose now that AB f BA. Then it follows, by the Lemma and the Weyl majorant theorem, that O<%etr(A*B)=tr(AiR~eB) < F

s,(ASieB),<

F s,(A)s,(!HcB)~m.

i=l Therefore,

i=l

there exists an n E N such that

!Retr(A*B)<

i

si(A)si(!ReB).

i=l

It is easy to find two permutations

aCI,(i)=si(A)and

I? i and II 2 over { 1,. . . , n} such that

%e&,Ci,=si(%eB),

i=l,...,n.

We can easily extend these permutations II i and II 2 to permutations II i and II 2 over N respectively. Besides, it follows from A > 0 and % e B 2 0 that (Y~!R e pj >, 0 for all i, j E N. Therefore,

2%

YUJI SAKAI

Thus we obtain

5

i=

E

-PII,(i)12=

alI,(i)

"i,(i)+

i=l

1

ii!IPIl,(i)12-22 aII,(~)?‘1cPl12(i)

i=l

i=l


Proof of Theorem 2.

Let

A and

follows from the Weyl’s theorem

B be C.C. operators

and the Lemma

%etr(A*B)
on 4.

Then

it

that

si(A*B)

f i=l

and

E si( A*B) < z i=l

si(A*)si(

B) = f

si(A)‘i(B).

i=l

i=l

Thus we have

E i=l

[&A)

-si(B)12=

f

Sag+

i=l

5

Si(B)2-2

i=l

f

si(A)s;(B)

i=l


= (IA - Bll;.

n

The author would like to thank Professor Tsuyoshi Ando, whose comments led to improvements in the proof of Theorem 1. Also he would like to thank Professor Hisahalrc Umegaki for constant encouragement. Note added in proofi After having submitted the manuscript, we found that a more general form of Theorem 2 had been already obtained in the paper: A. S. Markus, The eigen- and singular values of the sum and product of linear operators, Uspehi Mat. Nauk 19:93-123 (1964) (Russian Math. Surveys 19:91-120 (1964)) (especially Corollary 5.3). But our proof of Theorem 2 is quite different from his, and much simpler.

CONTINUOUS

VERSIONS

OF THE

287

H-W INEQUALITY

REFERENCES 1

I. C.

Gohberg

selfudjoint 2

A. J. Hoffman matrix,

3

Duke

A. Horn, Proc.

4

and M. G. Krein,

Proc.

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J. 20:37-39

Acad.

Inequalities Nut.

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(1953).

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Sci. U.S.A.

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and H. W. Wielandt, Math.

On the singular

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H. Weyl, tion,

(translation),

Operators

36:374-375

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Sci. U.S.A.

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35:408-411

(1949).

of a linear transforma-