1. Introduction
The motivation for this note is provided by the results obtained in [1-4]. Let T be a bounded linear operator on a complex Hilbert space H. The numerical range of T, denoted by W(T), is the subset of the complex plane and
![](https://www.scirp.org/html/2-5300237\05d1a1b3-c6ed-4a9b-8766-ab2f199fe27a.jpg)
The numerical radius of T is defined as,
![](https://www.scirp.org/html/2-5300237\89dc8b62-7e40-4919-87ee-f37dd4c40ea6.jpg)
The following lemma is known and is an easy consequence of the definitions involved.
Lemma 1.1.
, where T* is the adjoint operator of T and
is the complex conjugate of
.
Berger and Stampfli in [2] have proved that if
and
, for some n, then
. Also, they gave an example of an operator T and an element
such that
implies that
and
. In Theorem 2.1, we present a different proof of their result in [2] and show that
is indeed the best constant.
Theorem 2.1 also generalizes the result in [4] and provides a partial converse to Theorem 1 in [1, p. 372].
Our next main result in Theorem 2.3 gives an alternative and shorter proof of Theorem 1 in [1].
Applying Lemma 2 and Proposition 2 of [1], a new result on the numerical range of nilpotent operators on H is obtained in Theorem 2.4. This gives a restricted version of Theorem 1 in [3].
Finally, two examples are discussed. Example 3.1 deals with the operator
, where 1 is not the eigenvalue of
if
. Example 3.3 justifies why
fails to increase until and unless
.
2. Main Results
Theorem 2.1. The following statements are true for a bounded linear operator T on a Hilbert space H with
.
1)
such that
,
.
2) If
for some integer n, then
and
.
3) The set
forms a nontrivial subspace of T so that its orthogonal complement is invariant.
Proof. 1) For each real number
and a postive integer, n, let
. Then the inner product relation
implies that
![](https://www.scirp.org/html/2-5300237\41513ca8-7f74-4d8b-b147-2db732410a14.jpg)
That is,
![](https://www.scirp.org/html/2-5300237\bfc4a1bf-fa40-4d76-b1a7-10a15ce2b7cf.jpg)
Hence,
![](https://www.scirp.org/html/2-5300237\78a1c11e-c71e-423b-adeb-6af7d5aaa942.jpg)
Since
![](https://www.scirp.org/html/2-5300237\261d302b-3654-4ed3-9f8a-2e2b95f5ec2e.jpg)
it follows that
![](https://www.scirp.org/html/2-5300237\073ad77e-4698-4e22-a6a4-7b45c9af68c1.jpg)
Dividing the above inequality by
, we have
![](https://www.scirp.org/html/2-5300237\fd486c09-c497-4893-a21d-e34ea55b8d20.jpg)
Let
be the following block-diagonal matix of order n and
![](https://www.scirp.org/html/2-5300237\960b35b4-06c6-48fb-b9d8-1b80fffeefb1.jpg)
If γn denotes the determinant of
such that
then the value of γn is positive because all principal minors of
are nonnegative. Suppose that ![](https://www.scirp.org/html/2-5300237\95ffb0a2-eb24-443d-913b-355075f05d68.jpg)
(2.1)
We consider the following cases:
Case 1. If
for the least
then
and
converges to zero.
Case 2. Let
for all
. Then
and by induction
![](https://www.scirp.org/html/2-5300237\43e2b128-1f2b-4d1b-a1dd-e572fde6c2da.jpg)
Further, the inequality
![](https://www.scirp.org/html/2-5300237\62d67b45-282d-4e15-bd1f-b3ec860bb0e3.jpg)
implies that
converges to q as n goes to infinity for some q ≥ 0. Therefore from Equation (2.1),
as
. Thus
. Obviously, q = 1 only if
.
2) By the assumption,
for some positive integer n. Now fom Equation (2.1), we obtain:
![](https://www.scirp.org/html/2-5300237\f7dafee0-919c-404a-b52e-42d8510f23be.jpg)
and
so that
. The equality,
now follows from (a) and thus
. Also,
which gives
since
.
3) To prove this case, we assume that if the vector
is orthogonal to the spanning set
then
. Let
, for
. Then
![](https://www.scirp.org/html/2-5300237\5cc23856-bdb1-4567-9091-b44e54c0256a.jpg)
Hence,
for
and the spanning set
is a non-trivial invariant subspace on T.
In [2, p. 1052], an example of an operator T on
and an element x in H with
, is given where
. Theorem 2.1 above establishes that
is the best constant in this case.
Remark 2.2. An operator A on H is hyponormal if
. Let
then
if A is a hyponormal operator. Hence,
,
and the set of vectors
forms a reducing subspace of A.
A natural connection between Feijer’s inequality and the numerical radius of a nilpotent operator was estaplished by Haagerup and Harpe in [1]. They proved, using positive definite kernals, that for a bounded linear operator T on a Hilbert space H such that
and
then
. The external operator is shown to be a truncated shift with a suitable choice of the vector in H. The inequality is related to a result from Feijer about trigonometric polynomials of the form
with
. Such a polynomial is positive if
for all
. Here, we present a simplified proof of Theorem 1 in [1].
Theorem 2.3. For an operator N on H with
and
, we have
.
Proof. We will follow the notations of Theorem 1 in [1]. Let S be the operator on
and
,
be the basis in
. We define the operator S as follows:
and
for ![](https://www.scirp.org/html/2-5300237\74b68237-2211-40f4-80f9-b6bb6f2e3238.jpg)
The matrix for S gives a dialation for T. Let A be the matrix for S and
![](https://www.scirp.org/html/2-5300237\453a4516-2517-4298-a451-421fe02b2184.jpg)
If
is a unitary operator on
with diagonal
then
. By Lemma 1, we have:
![](https://www.scirp.org/html/2-5300237\087469d7-fd09-4548-890c-da4a2ee2292e.jpg)
This helps to define the characteristic function of a contraction.
For the operator N on H, let
then
is a positive operator and
depends on N. Let the range of
be denoted by
. Then the tensor product,
, is a Hilbert space. We define the map
so that F is an isometry.
For λ, let
where
I is the identity operator, and
is an operator on
.
Therefore
and
.
Now, we claim that
, for we hope that
By Lemma 1.1
![](https://www.scirp.org/html/2-5300237\a153c579-8b93-4363-80ec-d496c574d05b.jpg)
That is,
.
Since
, we have:
![](https://www.scirp.org/html/2-5300237\f7741965-7ff6-4094-8386-0416d75facb8.jpg)
and
![](https://www.scirp.org/html/2-5300237\77256940-c09a-4d13-9ee2-be3a08a4147a.jpg)
where
is the spectral radius of
. By the definition of the spectral radius, we have the characteristic polynomial f such that
by [5, p. 179, Example 9], the roots of
are given by
,
and
and
.
Karaev in [3] has proved, using Theorem 1 in [1] and the Sz.-Nagy-Foias model in [6] that the numerical range
of an arbitrary nilpotent operator N on a complex Hilbert space H is an open or closed disc centered at zero with radius less than or equal to
, ![](https://www.scirp.org/html/2-5300237\296ca24b-8af4-4403-a3b8-b98dcd5d0376.jpg)
Using Theorem 2 and the assumption that
,
, we have
as a closed or an open disc centered at zero with radius equal to
. In fact, we have the following theorem.
Theorem 2.4. For a nilpotent operator N on H with
,
and
, the numerical range
is a disc centered at zero with radius
.
Proof. For any
we must claim that
, for
and
is a vector in
.
From [1, p. 374, Proposition 2], we have
. Also, for some
,
. Now by [1] [P.375, Lemma 2], we obtain:
![](https://www.scirp.org/html/2-5300237\8980c10c-89db-4d9a-802c-e0d7a4cf410d.jpg)
and
![](https://www.scirp.org/html/2-5300237\73948073-be81-4df8-8825-3506187bdf22.jpg)
Let
. Then:
![](https://www.scirp.org/html/2-5300237\b8c8d244-ed60-485a-ac72-edd29fa8b589.jpg)
and the theorem follows from above since
is arbitrarily chosen.
3. An Application
An operator A is a unilateral weghted shift if there is an orthonormal basis
and a sequence of scalers
such that
for all
. It is easy to see that
where S is the unilateral shift and D is the diagonal operator with
, for all n.
Thus,
and
for all n. So
is the basis of eigenvectors for
. Also, note that A is bounded if
is bounded.
If A is a unilateral shift then
and
for
. Consequently, for a hyponormal operator A,
and
for
. A wighted shift is hyponormal if and only if its weight sequence is increasing.
Example 3.1. Let
be an operator on
such that
and
for
and
. Here, we show that
is not an eigenvalue of
if
. We prove our claim by contradiction Let
be an eigenvalue of
. Then, there exists
with
and
, n = 2, 3, ···. It is not hard to see that:
![](https://www.scirp.org/html/2-5300237\ee08af08-bf5c-4c07-8cfa-7a4b429693ef.jpg)
For
, we have
and thus
, which shows that
, contrary to our assumption. Thus,
is not an eigenvalue of
if
.
Remark 3.2. Following [2], if
then
![](https://www.scirp.org/html/2-5300237\25fb5633-131c-4652-8025-c116f79ab219.jpg)
Therefore, the numerical radius,
is equal to 1.
The example below shows that there exists an operator
such that
for
.
Example 3.3. Let
be a unilateral shift. If
is the orthogonal projection of
onto the spanning set of vectors
then
and
has the usual matrix representation. Let
![](https://www.scirp.org/html/2-5300237\d7cc00ff-cbdd-4333-a127-fef108389445.jpg)
Then the characteristic polynomial of
is given by a Chebyshev polynomial
of the first kind. Let
where
. Then:
![](https://www.scirp.org/html/2-5300237\b79e42d5-1b7a-4e60-9451-6d4448da72ab.jpg)
(easily proven by trigonometric identities) and
for
is a linear combination of powers of xk. Also, det
. If
then the roots are given by the Chebyshev polynomial of the first kind. The roots can be found by finding the eigenvalues of matrix B. By [2, p. 179, Example 9], the eigenvalues of B are given by
, for
.
Suppose that
![](https://www.scirp.org/html/2-5300237\5e5cf6f7-f222-4cc6-8c63-13cf2a81f155.jpg)
then
. Hence,
if
.