THE NUMERICAL RANGE AND NORMAILIY
OF TOEPLITZ OPERATORS
BOO RIM CHOE AND YOUNG JOO LEE
Abstract. We investigate some relations between the numerical range and normality of Toeplitz operators acting on the Bergman space or pluriharmonic Bergman
space of the ball.
1. Introduction
Let T be a bounded linear operator on a Hilbert space H with a inner product h , i.
The numerical range W (T ) of T is the subset of the complex plane C defined by
W (T ) = {< T x, x >: x ∈ H, hx, xi = 1}.
It is well known that W (T ) is a convex set whose closure contains the spectrum σ(T )
of T . If T is normal, then the closure of W (T ) is the convex hull of σ(T ). Furthermore,
it is also known that each extreme point of W (T ) is an eigenvalue of T . See [2] and
[3] for these facts and more information.
The description of the numerical range of an arbitrary Toeplitz operator on the
Hardy space of the unit disk was given in [1] and [4]. On the Bergman space of the unit
disk, the numerical range of a Toeplitz operator with harmonic symbol was described
in J. K. Thukral([8]) in terms of its spectrum or certain topological conditions. In
this paper, we consider the same problem on two spaces of the ball in n-dimensional
complex space. We first consider Toeplitz operators with M-harmonic symbols acting
on the Bergman space on the ball in Section 2. We investigate some relations between
the numerical range and normality of a Toeplitz operator. Next, we consider the same
problem on the pluriharmonic Bergman space of the ball in Section 3. Our results
show that the similar results as case of the unit disk hold on the ball. As you can
see in [8], J. K. Thukral made use of the well known fact that the spectrum of the
Toeplitz operator Tu with real harmonic symbol u is given by (inf u, sup u). But, we
don’t know whether the same is true on the ball with M-harmonic symbols. So, we
need another methods on the ball.
2. Toeplitz operators on the Bergman space
Let B be the unit ball of the complex n-space Cn . The Bergman space A2 is the
subspace of the Lebesgue space L2 = L2 (B, V ) consisting of all holomorphic functions
on B where the notation V denotes the normalized Lebesgue volume measure on B.
1991 Mathematics Subject Classification. 47B35.
Key words and phrases. Numerical range, Toeplitz operators, Bergman spaces.
This research is partially supported by KOSEF(98-0701-03-01-5).
1
2
BOO RIM CHOE AND YOUNG JOO LEE
It is known that A2 is a closed subspace of L2 and hence is a Hilbert space with the
inner product given by
Z
hf, gi =
f ḡ dV,
f, g ∈ L2 .
B
We let P be the Hilbert space orthogonal projection from L2 onto A2 . For a bounded
measurable function u on B, the Toeplitz operator Tu : A2 → A2 with symbol u is
the linear operator defined by
Tu f = P (uf )
2
for functions f ∈ A . Clearly, Tu is a bounded operator on b2 . In fact, ||Tu || ≤ ||u||∞ .
In this chapter, we consider Toeplitz operators with M-harmonic symbols and
investigate some relations between the numerical range and normality of a Toeplitz
operator. So, we begin with the notion of M-harmonicity. For u ∈ C 2 (B) and z ∈ B,
e
the invariant Laplacian is defined by (∆u)(z)
= ∆(u ◦ ϕz )(0)where ∆ denotes the
ordinary Laplacian and ϕz is the standard automorphism of B such that ϕz (0) = z.
We say that a function u ∈ C 2 (B) is M-harmonic on B if it is annihilated on B by
e See Chapters 2 and 4 of [7] for details. It is known that M-harmonic functions
∆.
have the following mean value property(see Theorem 4.2.4 of [7]):
Z
(1)
(u ◦ ϕa )dV = u(a),
a∈B
B
for every bounded M-harmonic u on B. We begin with following.
Lemma 1. Let u be a bounded M-harmonic function on B. Then u(B) ⊂ W (Tu ).
Proof. For each a ∈ B, we let ka be the well known Bergman kernel at a defined by
!n+1
p
1 − |a|2
ka (z) =
,
z∈B
1 − z · ā
where the notation z · ā = z1 a1 + · · · + zn an denotes the usual Hermitian inner product
for points z, a ∈ Cn . Then it is easily checked that ka ∈ A2 and
Z
hka , ka i =
|ka |2 dV = 1
B
for every a ∈ B. So, by the definition of W (Tu ), we see hTu ka , ka i ∈ W (Tu ) for every
a ∈ B. Since |ka |2 is the real Jacobian of ϕa , we have by (1),
hTu ka , ka i = hP (uka ), ka i
= huka , ka i
Z
=
u|ka |2 dV
ZB
=
u ◦ ϕa dV
B
= u(a)
for every a ∈ B. Hence u(B) ⊂ W (Tu ), as desired. The proof is complete.
3
The following proposition shows that Toeplitz operators with M-harmonic symbols
are only positive in an obvious case. Proposition 2 below was given on the unit disk
in Lemma 3 of J. K. Thukral([8]). But, we don’t know whether the same proof works
on the ball with M-harmonic symbols generally. So, we need an another one.
Proposition 2. Let u be bounded M-harmonic on B. Then Tu ≥ 0 if and only if
u ≥ 0.
Recall that Tu ≥ 0 means hTu f, f i ≥ 0 for every f ∈ A2 .
Proof. If Tu ≥ 0, we have W (Tu ) ⊂ [0, ∞) by the definition of W (Tu ). Since u(B) ⊂
W (Tu ) by Lemma 1, we see u ≥ 0. Now, assume u ≥ 0. Then
Z
hTu f, f i = hP (uf ), f i = huf, f i =
u|f |2 dV ≥ 0
B
2
for every f ∈ A . Hence Tu ≥ 0, as desired. The proof is complete.
We note that the adjoint operator of Tu is given by Tu∗ .
Lemma 3. Let u be bounded M-harmonic on B. If W (Tu ) lies in the upper half-plane
and contains 0, then Tu must be self-adjoint.
Proof. Since W (Tu ) lies in the upper half-plane, we have ImhTu f, f i ≥ 0 for every
f ∈ A2 . We note that
Z
ImhTu f, f i = ImhP (uf ), f i = Imhuf, f i = (Im u)|f |2 dV = hTIm u f, f i
B
2
for every f ∈ A . It follows that TIm u ≥ 0 and then by Proposiiton 2 Im u ≥ 0. On
the other hand, since W (Tu ) contains 0 by the assumption, we have hTu g, gi = 0 for
some g ∈ A2 with hg, gi = 1. So, in particular
Z
Z
2
0 = ImhTu g, gi = Im u|g| dV = (Im u)|g|2 dV.
B
B
Since Im u ≥ 0, we must have (Im u)|g|2 = 0 and then Im u = 0 because g 6= 0.
Therefore u is real and hence Tu is self-adjoint, as desired. The proof is complete. Now, we give a sufficient condition for the normality of a Toeplitz operator with
M-harmonic symbol in terms of their numerical range.
Theorem 4. Let u be bounded M-harmonic on B. If W (Tu ) is not open in C, then
Tu is normal on A2 .
Proof. Since W (Tu ) is not open, there exists ξ ∈ W (Tu ) such that ξ ∈ ∂W (Tu ). Here,
∂F is the boundary of a set F ⊂ C. This yields
0 ∈ W (Tu−ξ )
and
0 ∈ ∂W (Tu−ξ ).
Since W (Tu−ξ ) is convex, by the rotation if necessary, one can choose a constant
α of unit modulus such that αW (Tu−ξ ) lies in the upper half-plane. Note that
W (Tαu−αξ ) = αW (Tu−ξ ) and 0 ∈ W (Tαu−αξ ). By Lemma 3, Tαu−αξ is normal and
then Tu is normal. This completes the proof.
Since an open convex set is the interior of its closure, we have a simple corollary.
4
BOO RIM CHOE AND YOUNG JOO LEE
Corollary 5. Let u be bounded M-harmonic on B. If Tu is not normal on A2 , then
W (Tu ) is the interior of its closure.
It is known that for any bounded linear operator T on a Hilbert space, if W (T ) is
a part of a line segment, then T must be normal. See Theorem 1.4-1 of [2]. In the
rest of this chapter, we consider the problem of when the converse of this fact is also
true. We first need a couple of lemmas.
Proposition 6. Let u be bounded real M-harmonic on B. If u is nonconstant, then
m, M ∈
/ W (Tu ) where m = inf u and M = sup u.
Proof. If m ∈ W (Tu ), m is an extreme point of W (Tu ) and then an eigenvalue of Tu .
So, there exists a nonzero f ∈ A2 such that Tu f = mf . It follows that P (uf −mf ) = 0
and hence
Z
0 = hP (uf − mf ), f i = h(u − m)f, f i = (u − m)|f |2 dV.
B
2
Since u − m ≥ 0 on B, we must have (u − m)|f | = 0 and then u = m because
f is nonzero. Hence u is constant, which is a contradiction. So, m ∈
/ W (Tu ). The
similar argument can be applied to prove that if M ∈ W (Tu ), then u is constant.
This completes the proof.
Lemma 7. Let u be a bounded real function on B. Then σ(Tu ) ⊂ [m, M ] where
m = inf u and M = sup u.
Proof. If λ ∈
/ [m, M ], then we have either u − λ > 0 or u − λ < 0 on B. We first
assume u − λ > 0 and choose > 0 such that
sup |(u(z) − λ) − 1| < 1.
z∈B
It follows that
||T(u−λ) − I|| = ||T(u−λ)−1 || ≤ ||(u − λ) − 1||∞ < 1.
Hence T(u−λ) is invertible, so is Tu−λ . Since Tu−λ = Tu − λ, we have λ ∈
/ σ(Tu ). If
u − λ < 0 on B, then −u + λ > 0 and hence T−u+λ is invertible by the case we have
done above. It follows that Tu − λ = Tu−λ = −T−u+λ is also invertible and hence
λ∈
/ σ(Tu ). The proof is complete.
Now, we compute the numerical range for Toelitz operators with real M-harmonic
symbol.
Theorem 8. Let u be nonconstant real M-harmonic on B. Then W (Tu ) is an
open line segment in the real line. In fact, W (Tu ) = (m, M ) where m = inf u and
M = sup u.
Proof. Since u is real, Tu is self-adjoint and hence normal. Hence W (Tu ) is the convex
hull of σ(Tu ). It follows from Lemmas 1 and 7 that
[m, M ] = u(B) ⊂ W (Tu ) = convex hull of σ(Tu ) ⊂ [m, M ].
5
Therefore W (Tu ) is a convex set whose closure is [m, M ]. Note that m, M ∈
/ W (Tu )
by Proposition 6 because u is nonconstant. It follows that W (Tu ) = (m, M ). The
proof is complete.
Theorem 2 of J. K. Thukral([8]) shows that Theorem 8 is still remain valid on
the disk for general harmonic symbols. Unfortunately, we were not able to prove or
disprove whether the same is true on the ball with M-harmonic symbols. But, we
can prove the same with pluriharmonic symbols. Recall that a function u ∈ C 2 (B) is
called pluriharmonic if its restriction to an arbitrary complex line that intersects the
ball is harmonic as a function of single complex variable. Note that each pluriharmonic
functions is M-harmonic one.
We first need the following characterization of normal Toeplitz operators with pluriharmonic symbols.
Lemma 9. Let u be bounded pluriharmonic on B. Then Tu is normal on A2 if and
only if u(B) is a part of a line in C.
Proof. See Corollary 9 of [5].
Given a bounded linear operator T on a Hilbert space, we note
W (αT + β) = αW (T ) + β,
α, β ∈ C.
Theorem 10. Let u be bounded nonconstant pluriharmonic on B. If Tu is normal
on A2 , then W (Tu ) is an open line segment.
Proof. Since Tu is normal, u(B) is a part of a line in C by Lemma 9. Hence, there
exist a nonconstant bounded real pluriharmonic function v and constants s, t ∈ C
such that u = sv + t on B. By Theorem 8, we have W (Tv ) = (m, M ) where m = inf v
and M = sup v. Since Tu = sTv + t, we have
W (Tu ) = sW (Tv ) + t = (sm + t, sM + t).
This completes the proof.
3. Toeplitz operators on the Pluriharmonic Bergman space
In this section, we consider the same problem for Toeplitz operators acting on the
pluriharmnoic Bergman space. The pluriharmonic Bergman space b2 is the closed
subspace of L2 consisting of all pluriharmonic functions on B. We let Q be the
Hilbert space orthogonal projection from L2 onto b2 . For the explicit formula of the
projection Q and related facts, see [6]. For a bounded measurable function u on B,
the Toeplitz operator tu : b2 → b2 with symbol u is the linear operator defined by
tu f = Q(uf )
for functions f ∈ b2 . Clearly, tu is a bounded operator on b2 . Using the exactly same
arguments as in Lemmas 1 and 3 of Section 2, we see that for bounded M-harmonic
u on B, u(B) ⊂ W (tu ). Also, if W (Tu ) lies in the upper half-plane and contains 0,
then Tu must be normal. So, we see that the same argument of Theorem 4 still works
on b2 to prove the following.
6
BOO RIM CHOE AND YOUNG JOO LEE
Theorem 11. Let u be bounded M-harmonic on B. If W (tu ) is not open in C, then
tu is normal on b2 .
Also, by the same methods as in proposition 6 and lemma 7, we can prove that for
bounded nonconstant real M-harmonic u, m, M ∈
/ W (tu ) and σ(tu ) ⊂ W (tu ) where
m = inf u and M = sup u. Hence we also have the following as before.
Theorem 12. Let u be nonconstant real M-harmonic on B. Then we have W (tu ) =
(m, M ) where m = inf u and M = sup u.
We also need a corresponding result of Lemma 9.
Lemma 13. Let u be a bounded pluriharmonic on B. Then tu is normal on b2 if and
only if u(B) is a part of a line in C.
Proof. See Theorem 13 of [6].
Finally, we have the following by using the similar argument as in Theorem 10.
Theorem 14. Let u be bounded nonconstant pluriharmonic on B. If Tu is normal
on b2 , then W (tu ) is an open line segment.
References
[1] A. Brown and P. R. Halmos, Algebraic Properties of Toeplitz Operators, J. Reine Angew. Math.
213 (1963/64), 89-102.
[2] K. Gustafson and D. Rao, Numerical Range:The Field of Values of Linear Operators and Matrices, Springer-Verlag, New York, 1997.
[3] P. R. Halmos, A Hilbert Space Problem Book, 2nd ed., Springer-Verlag, New York, 1982.
[4] E. M. Klein, The Numerical Range of a Toeplitz Operator, Proc. Amer. Math. Soc. 35 (1972),
101-103.
[5] Y. J. Lee, Pluriharmonic Symbols of Commuting Toeplitz Type Operators on the Weighted
Bergman Spaces, Canadian Math. Bull. 41 (1998) 129-136. ,
[6] Y. J. Lee and K. Zhu, Some Differential and Integral Equations with Applications to Toeplitz
Operators, preprint.
[7] W. Rudin, Function Theory in the Unit Ball of Cn , Springer-Verlag, Berlin, Heidelberg, New
York, 1980.
[8] J. K. Thukral, The Numerical Range of a Toeplitz Operator with Harmonic Symbol, J. of Operator Theory 34 (1995), 213-216.
Department of Mathematics, Korea University, Seoul 136-701, KOREA
E-mail address: [email protected]
Department of Mathematics, Mokpo National University, Chonnam 534-729, Korea
E-mail address: [email protected]