I ntrnat. J. Math.
Vol. 2 (1979) 1-13
Math. Si.
CONVEXITY, BOUNDEDNESS, AND ALMOST PERIODICITY
FOR DIFFERENTIAL EQUATIONS IN HILBERT SPACE
JEROME A. GOLDSTEIN
Department of Hathematics
Tulane University
New Orleans, Louisiana 70118
U.S.A.
(Received October 27, 1978)
ABSTRACT.
There are three kinds of results.
First we extend and sharpen
a convexity inequality of Agmon and Nirenberg for certain
inequalities in Hilbert space.
differential
Next we characterize the bounded solutions
of a differential equation in Hilbert space involving and arbitrary unbounded
normal operator.
Finally, we give a general sufficient condition for a
bounded solution of a differential equation in Hilbert space to be almost
periodic.
KEV WORDS AND PHRASES. Differential equns in Hilb space, Convey
rnequality, Sf-adjoin operators, Bounded solution, Almost perdic
sol.
AMS (MOS) SUBJECT CLASSIFICATION (1970) CODES.
Secondary 34C25, 47I5.
Primary 4G0, 47A50
J. A. GOLDSTEIN
2
i.
INTRODUCTION.
Let
a complex Hilbert space
//du(t)/dt
S
1
<
be two commuting self-adjoint operators on
2
H. Let
[a,b]
u
u(t)]/
(S 1 + iS 2)
(t)dt < c
where
S
_<
/
H satisfy the inequality
,(t)]]u(t)]l
a -< t < b
{i .I)
We shall show that this implies the convexity
1/2.
inequality
lluCt)ll
-< K
which holds for some constant
lluca)ll
c
Kc
b-t
b-a
and all
Nirenberg [I] first proved this assuming
extended it to weak solutions of (1.1).
and to the range of values
K
constant
lluCb)ll
0 < c < 1/2
t
c
t-a
b-a
[a,b]. S. Agmon and L.
-3/2 recently S. Zaidman [7]
2
Our results apply to weak solutions
moreover, we obtain a smaller
than did these previous authors.
This result is presented in
Section 2.
Section 3 is devoted to obtaining the structure of the set of all
bounded solutions of
du(t)/dt
(S 1 +
(_oo < t < ).
iS2)u(t
The results generalize and improve a recent result of Zaidman [83.
In Section 4 we study almost periodic solutions of the inhomogeneous
equation
du(t)/dt
here
A
Au(t)
+
f(t)
is a closed linear operator on
(-
H and f
< t <
)
is an H-valued function.
Under a finite dimensionality assumption we show that bounded solutions are
almost periodic.
This generalizes the results obtained by Zaidman in [6].
DIFFERENTIAL EQUATIONS IN HILBERT SPACE
2.
A CONVEXITY THEOREM.
H
Hilbert space
Let
map the real interval [a,b3 into a complex
u
<-,.>.
with inner product
closed, densely defined linear operator,
Ildu(t)/dt- Bu(t)ll
if
f(t)
-=
du/dt
Bu
Ilf(t)ll
satisfies
weak solution of (2.1) if u
tiable functions
D(B*)
-<
D(B)
B
Let
c
H
H be a
is a strong solution of
u
(t)[lu(t)ll
is continuously differentiable on
u
3
_<
(2.1)
[a,b], takes
values in
D(B)
Iluft)ll
a <- t -< b.
u
(t)
and
is a
is continuous and for continuously differen-
@ with compact support in
3a,b[
and with values in
we have
I
b
I
<u(t),’(t)> dt
a
b
{<u(t),B*(t)>
+
<f(t),(t)>}dt
a
IIf(t)ii -< ,(t)Ilu(t)[I
a _< t -< b.
That a strong solution of (2.1) is a weak solution follows from an
integration by parts.
THEOREM 2.1.
is synetric.
Let
H be a weak solution of (2.1) where B
[a,b]
u
If
b
(t)dt -< c < 1/2
(2.2)
a
then the convexity inequality
Iluct)ll
-< K
C
Ilu(a)ll Ilu(b)]]
holds, where
a
b----
Kc
(2.3)
4
J.A. GOLDSTEIN
In particular, when c
1/2
Kc
we get
2-) 1/2
(4 +
Nirenberg [I] proved this result for strong solutions, taking
and obtaining the constant
2
Kc
(> (4
+
2)i/2).
Agmon and
1/2 /
c
This result also
Zaidman [7] extended the Agmon-
appears in Friedman’s book [3, p.219].
The new features of Theorem 2.1 are (i)
1 < c < 1
the result is extended to cover the case
(ii) the constant
Nirenberg result to weak solutions
2-
K
c
is sharpened for each value of
c
(including
By enlarging the Hilbert space H
&djoint operator (cf. Sz.-Nagy
itS
1
adjoint operators (i.e., e
S
1
+ iS
2
and
2
e
commute for all real
_
for each
e > 0
2
2
llul Cb)ll flu2 C)[I
+
/
/
M
sup
(ll=()ll
+ 2.
-
C
II fC)ll
ds
llfC)ll
ds)
a
<-
-< b}.
This implies
2
2
-I 2
M < 8 + eM + e
N
where
8
lluc)ll2 + llucb)ll 2
b
llc)ll
N=
ds.
a
M
2-<
and
s),
is replaced by the (unbounded)
We use Zaidman’s notation.
lluCb)ll2 ilUlC)I12 2+
here
B
t
The proof follows Zaidman [7, pp. 236-244] with
(cf. [7, p.242, line 33)we get
liuCt)ll2
to be a self-
[5]). Also, for S 1 and S 2 commuting selfisS
the following changes on pp. 242-244.
_<
).
according to the observation made in [I, p.138].
PROOF OF THEOREM 2.1.
lluCt)ll 2
B
we can extend
we may extend the theorem to the case where
normal operator
c < 1/2
(8 +
e- 1 N 2)(I
e)
-1
Consequently
From
DIFFERENTIAL EQUATIONS IN HILBERT SPACE
0 < e < 1.
for
u
follows that
where
B
u’
(t)
e
t
1/2.)
-< (t)
llu(t)l[)
{lie t u(t)li 2
2 <
(8
-1
2) (1
N
(t)Ilu(t)ll
{ll es u<s)ll
sup
ot
f(t)
e)
u) yields
-1
But by (2.1) and (2.2),
< 1.
0 <
e
b
t
e
-<
o
a
-< M
+ e
e
dt,
rather than
o
it
a<t<b},
lie a u(a)l[ 2 + I1 eOb u(b)ll 2
lie t f(t)ll
B
Since
Letting
Bo
and all
_<
p.242]).
sup
we have that (2.4) (applied to
N
,
M
M
’
is a weak solution of
(cf. [7, Lena
B- oI
IIft)l[
f [where
Bu
u(t)
No
for all real
e
(This becomes [7, p.242, eqn. (*)] when
is a weak solution of
5
dt
O(t) dt
a _< s _< b}
C.
Squaring this gives
2
N
2
<M
c
2
Plugging into (2.5) yields
Mo
2 <
(Bo.
+
-1
c
2
Mo
2) (i
E:)
-1
or
M
2
-<
E(1-)
C
2
(2.6)
J.A. GOLDSTEIN
6
0 < e < 1
provided
]2- 1[
<
implies
u
{1-4c
2)1/2
2
) > c
e(1
(b- a)
-I
log(llu(a)ll/lluCb)ll)
(llu()ll/llu(b)ll)
for all
t
E
u(a)
As in [7, pp. 243, 244],
t
t
0 < c < 1/2
i.e.,
u{b}
or
0
0
u(b) # 0.
makes
II = uCa)ll II
and
and
u(a) # 0
so to prove the theorem we may suppose
0
o
Choosing
and
u(b)ll.
Thus 2.6)becomes,
[a,b]
/llu()ll)
2b
2a
2t
b-a
+
llu(b)ll /
\llu(b)ll
llu(b)ll
lluCb)ll
1
b-a
2L
L
where
e((1-e)
c
2)-1.
llu(t)ll
a < t < b.
holds for
llu<a)ll
Ilu(b)ll 2
Consequently
-< (2L) I/2
llu(a)ll
g()
Regard
b-t
b-a
(2L)
t-a
Ilufb)l[
b-a
1/2
2e
e(1-e)
function of
e.
1"-2
12e
I
<
(1
It is minimized when
c
holds in this case.
)
’2
as a
in which case
This is a legitimate choice of
4c2) 1/2
C
1/2
since
The proof of the theorem is
now complete.
3.
BOUNDED SOLUTIONS.
on
H.
Let
We study functions
(strong) solutions of
S1 $2 be
CI(]I, H)
u
E
commuting self-adjoint operators
(If
]-,[)
which are bounded
DIFFERENTIAL EQUATIONS IN HILBERT SPACE
du(t)/dt
+ iS
(S 1
2)
u(t)
t
7
lR.
itS
LENNA S.l.
for all
Let
t e
PROOF.
and some
Let
tS
u(t)
tS
(Recall that
e
h e
u(0).
h
e
1
itS
1
oJ’
be a bounded solution
u
H
Then
u(t)
e
2
0}.
Slf
Then
itS
(e
{f e
Ker(S1)
(3.1).
itS
2
e
h)
2
tS
(e
1
h).
2
are defined by the operational calculus associated
tS
itS
2
1
with the spectral theorem.) Since e
is unitary, Ilu(t)ll
e
lie
tS
follows.
But
lie
tS
in which case
e
1
1
hil
is bounded for
if and only if
t e
h
itS
u(t)
and so
h
e
2
h e
hll
Ker(S)
as advertised.
h
A special case occurs when
Ker(Sl
where
M.
S
2
1 < j < n.
for
M.
restricted to
M
1
@
0
Mn
is a real constant
X.
times the identity on
Then any bounded solution of (S.l) is of the form
n
u(t)
[
itX.
(3.2)
h.
e
j=l
h. is the orthogonal projection of u(O) onto M.
1 -< j -< n.
J
J
covers the result obtained by Zaidman in [8]. More precisely, let
where
{E(O)
0
It}
This
be a resolution of the identity and let
S
1
x(O) dE(O)
S
2
y(O) dE(O)
be associated commuting self-adjoint operators, where
x
and
y
are continuous
J.A. GOLDSTEIN
8
JR.
real functions on
S
then
is
2
1 -< j < n
h.
4.
{O
is the finite set
times the identity on
y(Oj)
1 _< j _< n.
Mj
(E(Oj +)
O
1
n
E(Oj-)) (H)
Za:xdman’s
This is
result [8].
ALMOST PERIODIC SOLUTIONS.
Let A
THEOREM 4.1.
f
x
and so any bounded solution of (3.1) is of the form (3.2) with
Mj
E
kj
If the zero set of
H
/
H be almost periodic.
]R -+
du(t)/dt
H be
Let
Au(t)
a bounded linear operator and let
u
+
]R -+
H be
f(t)
a bounded (i.e.
(t
E
).
Suppose there is a finite dimensional subspace H 1 of
such that H
{Af(s) s } u {Au(0)} and
1
e
Then
tA
u
(H1)
H 1 for all
c
t
(4.1)
(4.2)
I.
is almost periodic.
When
H
is" finite dimensional, this is the classical Bohr-Neugebauer-
Bochner theorem (cf. Amerio-Prouse [2, p.SS]).
operator we can take
H 1 to be the range of A
When
A
is a finite rank
and Theorem 4.1 becomes
the theorem of Zaidman [63 in this case.
of H
and let
I
uj(t)
Pju(t)
P.
j
1,2.
(s e I), then for all real
L
whence
u
I
and
u
H2
H O H 1 be the orthogonal complement
be the orthogonal projection onto H.
j
1,2. Let
PROOF OF THEOREM 4.1.
2
2 >
Let
Note that if
L
is as upper bound for
t
flu(t)]] 2
are bounded.
[]u l(t)l] 2
Also,
+
]]u2(t)]] 2
Ilu(s)ll
DIFFERENTIAL EQUATIONS IN HILBERT SPACE
dUl/dt + du2/dt
du/dt
P1
Applying
Au
+ Au
1
2
9
P1 f + P2 f
+
to both sides gives
dUl/dt PIAUl + PIAU 2 + P1 f
The function
u
u(t)
(4.3)
admits the variation of parameters representation
etAu(0)
+
e {t-s)A f(s) ds
u(0)
+
f(s) ds
e
(t-s)
[
+
Anf(s)
n’.
n:l
ds
H 1 by (4.2). Applying
The last (summation) term belongs to
P2
to this
expression gives
P2 etAu.{0)
+
du2(t)/dt P2 etAAu(0)
+
u2(t)
P2
f(s) ds
differentiating yields
by (4.2).
du2/dt
Since
P2
is almost periodic and
f
is almost periodic.
Since
u
2
f(t)
P2
f(t)
P2
is bounded it follows that
is bounded,
u
2
itself is almost
periodic (see [2, p.553).
Next, by (4.3),
du
where
and
g(t)
P1Au2(t)
P1A /’/1 /’/1
/
l(t)/dt
+
Plf(t)
PlAUl(t)
+
(4.4)
g(t)
is almost periodic.
Since
u
1
is bounded
is linear, (4.4) is a linear system in the finite
J. A. GOLDSTEIN
i0
H
dimensional Hilbert space
Bohr-Neugebauer-Bochner theorem [ 2] that
u
u
I
+ u
It follows from the classical
(see [4.2)).
1
u
is almost periodic.
I
is almost periodic, and the proof is complete.
2
A
Theorem 4.1 can be easily extended to the case when
Consequently
is unbounded,
as fo i lows.
linear operators
(A)
Let A
THEOREM 4.2.
{T[t)
t e ]}
bounded solution of [4.1) where
H
c
on
f
H generate
+
[Co)
a
H [cf. [4]). Let
is almost periodic.
group of bounded
be a
u
Suppose there is a
finite dimensional subspace H 1 of H such that
/41
and T[t) (H I)
m
{(T(t)
I) f(s)
H 1 for all
c
t
.
s e JR, t e } u {Au(0)}
Then u
is almost periodic.
The proof, which differs from the proof of Theorem 4.1 only in
inessential ways, is omitted.
COROLLARY 4.3.
A
I)(A)
c
/
be the span
of
Let
H and let
n"
i’
periodic, provided f
i
l
Xn
’n
be eigenvalues of the linear operator
be corresponding eigenvectors.
Then any bounded solution
/
of 4.1)
H I is almost periodic and u{0)
Let
is almost
e
HI"
This follows immediately from Theorem 4.2.
COROLLARY 4.4.
u(0) e H 1
provided that one assumes that A is a compact normal operator.
PROOF.
Applying
gives
In Corollary 4.3 one can omit the hypothesis that
P.
Let
P1 P2 Ul u2
to (4.1) and noting that
be as in the proof of Theorem 4.1.
A
commutes with
P.
in this case
DIFFERENTIAL EQUATIONS IN HILBERT SPACE
du
l(t)/dt
Au l(t) + f(t)
du2(t)/dt Au2(t)
u
1
(t
l)
(4.5)
is almost periodic by the Bohr-Neugebauer-Bochner theorem.
remains to show that
to
11
H 2.
B
u
{m
be the restriction of
H 2 and complex numbers
for
m <’ m >
aqb
m=l
for all
H2"
Let
of
m"
Let
1’
Qm be the
Vm 0nu 2.
dv /dt
m
Also,
lim
m-+o
Vm(t)
v
m
Qm du2/dt
Qm Au2
u
0
H2
onto the span
such that
Bv
m
is) and takes values in a finite
2
is almost periodic.
uniformly for
m
Then
We claim that
It then follows that
t
periodic [23 and the proof is done.
u
2
d
(u2(t)
Vm(t))
therefore
u (t)
2
v (t)
m
Consequently
Ilu2(t) Vm(t)ll
2
B(u2(t)
.
.
e
tvk
k=m+l
k=m+l
t Re
e
is almost
So it only remains to prove the elaim.
We have
d-
A
m
orthogonal projection (in
is bounded (since
Vm
dimensional space whence
u2(t
B
is a compact normal operator, hence by the spectral theorem there
is an orthonormal basis
by (4.5).
Let
is almost periodic.
2
Thus it only
v (t))
m
<(u 2
k
(B
QmB)(u2(t
Vm)(0), k > Ck
< (u
2
Vm) (0), Ck > 12
Vm(t))
12
J. A. GOLDSTEIN
Since
llu2(t) Vm(t)ll -< llu2(t)ll
follows that for every
Pk
k
m(t)ll 2
n
+
ACKNOWLEDGEMENT:
<(u 2
v
uniformly for
t
for some
L
and all
Vm)(0), k >
t
m,
it
# 0 for some m
Therefore
k=m+l
I1(
as
<
for which
must be purely imaginary
Ilu2(t)
-< L
[<{u 2
Vm)(03, k >l
)u2(0)ll 2
2
o
Q E D
This research is an,outgrowth of discussions I had with
Samuel Zaidman in April, 1978 when I had the pleasure to visit the Universit
de Montreal.
I thank Professor Zaidman for some stimulating dicussions, and
I gratefully acknowledge that my trip to Montreal was supported by his NRC
grant.
Finally, I gratefully acknowledge the support of an NSF grant
DIFFERENTIAL EQUATIONS IN HILBERT SPACE
REFERENCES
[I]
Agmon, S. and L. Nirenberg, Properties of solutions of ordinary
differential equations in Banach spaces, Comm. Pure Appl. Math.
16 (1963), 121-239.
[2]
Amerio, L. and G. Prouse, Almost-Periodic Functions and Functional
Equations, Van Nostrand Rheinhold, New York, 1971.
[3]
Friedman, A., Partial Differential Equations, Holt, Rinehart and
Winston, New York, 1969.
[4]
Hille, E. and R. S. Phillips, Functional Analysis and Semi-Groups,
Amer. Math. Soc. Colloq. Publ. 31___, Providence, R. I., 1957.
[5]
Sz.-Nagy, B., Extensions of linear transformations in Hilbert space
which extend beyond the space, appendix to F. Riesz and B. Sz.-Nagy,
Functional Anal),sis, Ungar, New York, 1960.
[6]
Zaidman, S., Bohr-Neugebauer theorem for operators of finite rank in
Hilbert spaces, Atti Acad. Sci. Torino 109 {1974/1975), 183-185.
[7]
Zaidman, S., A convexity result for weak differential inequalities,
Canad. Math. Bull. 19 1976), 235-244.
[8]
Zaidman, S., Structure of bounded solutions for a class of abstract
differential equations, Ann. Univ. Ferrara 22 (1976), 43-47.
13