.
* V2\\Axf If
.
Ax f 0 we set
We deduce, after dividing by u 2 ,
l|Ar||2 - allxllp^ir 1 + HP^flHUlAir 1 = 2||A||||ar|| . a S 0 , (2.5) implies that
and a unit vector P x = 0 in ( 2 . 5 ) . reduces to ( 2 . 2 ) .
x
C(H; A) £ 2 .
i s an extremal for
But
Moreover,
A i f and only i f
C(H; A) = 2 a = 0 and
P x = 0 i s equivalent to (2.1) and a = 0
Thus (i) and (ii) are proved.
Using (2.5) again,
C(H; A) - 2 if and only i f there i s a sequence
D i s s i p a t i v e o p e r a t o r s and i n e q u a l i t i e s {x } of unit vectors in V[A )
A x^ t 0 and
lim (||P x || 2 -a
(2.6)
where
u = U x n || n\\
\\Ax \\2IA2XJi
and
* 2 .)
a
n
" = 0
= a[\s , x ) . *n n'
(This makes
Unfortunately this condition, which is -both
necessary and sufficient for for
such that
433
{x }
A , is rather cumbersome.
to be an approximate extremal sequence
Thus we turn to the simpler condition of
(Hi) . The hypothesis of (Hi) implies, by (2.5),
Uxf where assume
a
+
=a(X,x)sO. \\A x
II
(llP^H^aJpxJI"1 = 2\A\\ By taking a subsequence if necessary we may
is bounded away from zero.
»ll
By hypothesis,
lim P,x
n^
A n
= 0
and a
= 2 Be((A2+U)x , \Ux +AX )) = 2 Re(-X2x , -A2x ) + o(l) \ n n/
since
P,x •* 0 An
= 2X 2 as
n •+ °° by (2.3).
This completes the proof of part
(Hi).
For the proof of (iv) consider the necessary and sufficient condition (2.6) for
C(H; A) = 2 .
Since
A
and
A'1
are both bounded, by taking a
subsequence if necessary we may assume that
lim
positive.
~ \\A
We now verify (2.3).
hypothesis,
We have
V
A x
II 2x II*
= X
\\ "*A
where
and
»
X
is
434
Herbert
A.
Gindler
and Jerome
A.
Goldstein
= lim \P,x + (y -\)AX + U2-X2 he \\
0 = lim P x = lim P , x Xn J1-W0
since the term in square brackets converges to zero. of (iv) note first that
a
•+ 0 . Next, since
y
yi a
^n'
X
To complete the proof
-+ A
and Pyx
n
•*• 0 ,
An
r) = a ^ ' =2
Reul x , x J -*• 0 as w
I t follows that REMARK 2.3. that if operators
A is
Theorem 2.2 (i«-> can be generalized as follows.
m-dissipative and if
A . = A(I-cA)
+ 61 are bounded, have bounded inverses, are
m-dissipative, and converge to A£&x •* Ax
A in the following senses as for
exp(t/3e(5)x •* exp(W)x Thus if C(H; 4 ») = 2
e, 6 •*• 0
:
a; € 0(4) ,
•* (XI-4)" 1 x
[U-A^'^x
Note
e, 6 are positive numbers, then the
for for
x ? H and x i H,
for sufficiently small
X> 0 ,
t > 0 . E
and
6 , we can apply
(iv) to A , and then use a Cantor diagonalization argument to conclude that (2.3) is a necessary condition for A . REMARK
2.4. For A = L - I with
L
a contraction, the extremal
conditions (2.1) and (2.2) become (2.7)
L2x + (\-2)Lx + (X2-X+l)x = 0 ,
Re = 1 . Similar expressions hold in the approximate extremal case. REMARK 2.5.
By Proposition 2.1 and i t s proof, for
m-dissipative and normal on
H,
C(H; A) = 1
A i f and only if there i s a unit vector
x
A nonzero,
and there is an extremal for
and a positive number
X such
Dissipative operators and
that
A*Ax = Ax ; that is, A
nonzero eigenvalue. A*Ax = Ax
When
becomes, using
inequalities
435
has an extremal if and only if
A = L - I L* = L
where
L
A*A
has a
is unitary, the equation
,
L2x + (\-2)Lx + x = (L-aD(L-BJ)x = 0 . Thus
A
has an extremal if and only if
L
has an eigenvalue other than
one. REMARK A > 0
and
2.6.
Consider the extremal equation (2.1) to be solved for
x € V[A )
when
H
is complex.
Factor this equation as
U-aI)U-&Dx = 0 . If
(X-BJ)x = 0 , then
Ax = &x , whence
\\Axf = |e| 2 W| 2 = IM2x||||x|| . This cannot give A
with
C(H; A) > 1 .
C(H; A) = 2
It follows that if
we must have
y = Ax - fte t 0
x and
is an extremal for Ay = ay .
A
similar remark holds for approximate extremal sequences.
3. Series inequalities Let a = 0
for
IK denote the (real or complex) scalar field.
Let
a = -°° or
and set
1 S p < »
with the usual modification for
p = °° . These are, of
course, the standard Lebesgue sequence spaces. THEOREM
3.1
(Copson [2] - note the error in the conclusion of this
theorem on page 109). £e£
numbers such that
I
j=.co
°° £
{x.} ._
2 |x.| ?
ie convergent and 3
be a sequence of real or complex
is convergent.
Then, for
Ax. = x. - x. J J+l 7
Herbert
436
A. Gindler and Jerome A. Goldstein
I I*.I
A 2 x. 3\
(3.1)
£/
Equality holds in (3.1) if and
x. = 0 /or a£Z
inequality (3.1) is best possible. THEOREM
3.2
(Copson
aomplex numbers such that
.
£
Let
|x.|
{*•}"_,, be a sequence of real or 3 3~v
is convergent.
A 2 x.
Then
3=0
3
3=0
convergent and 2
(3.2) 3=0
=0 where
Ax . = x . - x . as before. 3 3 1 J
x. = 0 3
/or all
Proofs. section.
L
Since
.7=0
Equality occurs in (3.2) if and only if
Finally the constant
h
in (3.2) is best possible.
These results follow readily from the results of the previous
To prove Theorem 3.1 let
shift defined by Then
3 .
A2x. 3
Lx = y
is unitary and
Ax = {Ax.}._ 3 3
H = IS-™)
Since
L
has no eigenvalues, A
Lx = y
where
H = IA0)
and define the
y = {y .} .__
and
3 3~ *•'
all
j 2 0 . Then
L
is a contraction on
H , whence for
C{H; A) < 2
by Proposition 2.1, proving (3.2).
C(H; A) = 2
and that
A
To show that
A
y . = x. 3
for
3
A = L - I ,
It remains to show that
has no extremals.
For the moment assume that are no extremals for
j .
Theorem 3.1 is now proved.
For the proof of Theorem 3.2, let by
be the bilateral
y = {y .} ._ and y . = x. for all 0 3 ~ * 3 i?+-»L + I . By Proposition 2.1, C(H; L-I) = 1 .
, (3.1) follows.
L
L
where
has no extremals by Remark 2.5-
unilateral shift
. Let
since
C{H; A) = 2
C{H; A) = 2 . Then, by Remark 2.5, there L
has no eigenvalues.
and that an approximate extremal sequence
exists, we use Theorem 2.2 (Hi).
The extremal equation (2.7) (and the
associated approximate extremal equation) is a second order difference equation whose general solution can easily be found explicitly.
Doing so
Dissipa+ive operators
and
inequalities
leads us to look for an approximate extremal sequence
437
[x f
of the form
x
n =
x
= p£ sin(a.pn+B.) , j > 0 .
Elementary, but rather tedious calculations, which we omit, show that if we k
take
•
p = 1 - £ , a. = 3 j(l-e) •* 0
we write the resulting
x
as
1
,
x
approximate extremal sequence for
6- = -IT/3 3
, and
0 < e < 1 , and if
, then the sequence A . The calculation is the one hinted
at by Copson [2], and this is the one part of Copson's paper that we have been unable to simplify.
The proof of Theorem j.2 is now complete.
•
4. Inequalities for m-dissipative operators For
A
an
m-dissipative operator on a Banach space
X
let
C(A, x) = for k
x € V[A2)
with
A x t 0 , so that
C(A, x)
is the smallest constant
which makes the inequality
IUx|| 2 < *||A2ar|| IMI valid.
(Consequently
C(X; A) = sup C(A, x) .) x
establish some results about space. [3]
In this section we shall
C{A, x) , especially when
X
is an
iP
These results complement and improve some of our earlier results
and some of those in [4]. Examples include the case when
X = Z"(o)
and
A
is the difference operator as in the proof of Theorem
3.2. For our first result we use Hoi brook's measure "Euclidean" a Banach space
X
is [4]. Set
(U.I)
=
sup
a(X)
It is easy to see that and only if
1 5 a(X) < 2
and that
a(X) = 1 . One interprets
a(X)
X
a(X)
of how
is a Hilbert space if
as a measure of how close
X
438
Herbert
A.
i s t o a H i l b e r t space.
Gindler
and
Jerome
A.
Goldstein
Using Clarkson's i n e q u a l i t i e s [ / ] , Holbrook [4]
showed t h a t a(X) < 2 U-2/p|
(U.2)
if
X is a subspace of an If THEOREM 4 . 1 .
For
space.
X> 0
let
B, = (\I+A)(\I-A)~l
be the
Cayley
A
transform of an m-dissipative M = sup{||B x || : X > 0} .
operator
Then for all
A on X . x € V[A2)
\\Ax\\2 5a(X)[l-nf)\\A2x\\\\x\\
(U.3) Proof.
x € V[A2)
For
Let
,
.
and X > 0 we have the identity
2\Ax = (A2x+\2x) + Bx(-42x-X2x) , from which i t follows that 2\\\Ax\\ < \\A2x+\2x\\ + ||B x l||U 2 x-X 2 x|| .
{k.h)
2
AY
The Cauchy inequality
a.b. 33
UX2||/lx||2 S S2a(X)(l + ||B x || 2 )(||4 2 x|| 2 + xN|x|| 2 ) by (U.I).
If
-42x * 0 , setting
(U.5)
IMx||2 < a (X
and (U.3) follows.
•
REMARK
4.2.
When
X
Kato's theorem [6] because gives an estimate on
X2 = ||/52x||Hxlf1
yields
is a Hilbert space the above result reduces to a(X) = M = 1
in this case.
Also, (U.5) (which
C(A, x) ) generalizes Theorem 2.2 in [3] because, in
the notation of [3], 1 + llB^H2 < 2M[x; \J
and (U.2) holds.
Moreover,
(U.5) generalizes Theorem 9 of [4] since one can readily check that, in the notation of [4], HB^H < 2>(X) .
Dissipative
THEOREM 4 . 3 . space, and let x € V[A2)
q
Let
A be
m-dissipative
and let
439
on X , a subspace of an If
be the conjugate exponent, p
a2x # 0
with
o p e r a t o r s and i n e q u a l i t i e s
+ q
= 1 .
X = X^ = (||42a:|i/||a:||)^ .
Let
If
2 5 p 2 .
Let
and let
x\\2/\\x\\\
: p between 2 and p\ .
Then
C[Ap, x J ^ I where
e = 1 Replacing
if s
\ > 1 by
while
M~ , or
s = ((3-X)/(l+X))2 A^
if
0 < X < 1 .
gives the estimates (U.8) and (U.9).
The theorem is proved by the proof technique of Theorem h.5;
we omit the
details.
References [J]
James A. Clarkson, "Uniformly convex spaces", Trans. Amer. Math. Soc. 40 (1936), 396-itlU.
[2]
E.T. Copson, "Two series inequalities", Proc. Roy. Soc. Edinburgh Sect. A 83 (1979), 109-llU.
442
Herbert
A.
Gindler
and Jerome
A.
Goldstein
[3]
Herbert A. Gindler and Jerome A. Goldstein, "Dissipative operator versions of some classical inequalities", J. Analyse Math. 28 (1975), 213-238.
[4]
John A.R. Holbrook, "A Kallman-Rota inequality for nearly Euclidean spaces", Adv. in Math. 14 (197M, 335-3U5.
[5]
Robert A. Kallman and Gian-Carlo Rota, "On the inequality ll/'ll 2 S Hf||• ||/"|| " , Inequalities , I I , 187-192 (Proc. Second Sympos., US Air Force Acad., Colorado, 1967. Academic Press,. New York, 1970).
[6]
Tosio Kato, "On an inequality of Hardy, Littlewood, and Polya", Adv. in Math. 7 (1971), 217-218.
[7]
Man Kam Kwong and A. Zettl , "Landau's inequality", submitted.
[8]
Man Kam Kwong and A. Z e t t l , "Ramifications of Landau's inequality", Proc. Roy. Soc. Edinburgh Seat. A 86 (1980), 175-212.
[9] H.H. /k)6ns [ J u . l . Ljubic], "0 HepaseHCTBax r-iewfly cTeneHRMH /iHHePiHoro one-paeopa" [On inequalities between the powers of a linear operator], Izv. Akad. Nauk SSSF Ser. Mat. 24 (i960), 825-86U; English t r a n s l : Transl. Amer. Math. Soa. (2) 40 (196I*), 39-8U. [70]
G. Lumer and R.S. P h i l l i p s , "Dissipative operators in a Banach space", Pacific J. Math. 11 (1961), 679-698.
Department of Mathematics,, San Diego State University, San Diego, California
92182,
USA; Department of Mathematics, Tulane University, New Orleans, Louisiana USA.
70118,
