5_497_bulut.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 4, No. 3, 2011, 244-250

ISSN 1307-5543 – www.ejpam.com

A New Generalization of the Operator-Valued Poisson Kernel

Serap BULUT

Kocaeli University, Civil Aviation College, Arslanbey Campus, 41285 İzmit, Kocaeli, TURKEY

Abstract. The purpose of this paper is to give a new generalization of the operator-valued Poisson

kernel and discuss its some applications.

2000 Mathematics Subject Classifications: 45P05, 47A60; 46E40, 47B38

Key Words and Phrases: Poisson Kernel, Operator-valued Poisson Kernel

1. Introduction

Let H be a Hilbert space which will be always complex and let L (H ) be the algebra of

all bounded linear operators from H to H . We write I for the identity operator on H . For

T ∈ L (H ), we denote by σ(T ) the spectrum of T .

For two operators S, T ∈ L (H ), we write S ≥ T to indicate that S − T is positive, i.e.,

〈(S− T ) x , x〉 ≥ 0 for all x ∈H .

Let A ∈ L (H ). For a complex valued function f analytic on a domain E of the complex

plane containing the spectrum σ(A) of A we denote f (A) as Riesz-Dunford integral [2, p.

568], that is,

f (A) :=
1

2πi

∫

C

f (z)(zI − A)−1dz, (1)

where C is positively oriented simple closed rectifiable contour containing σ(A).

Throughout the paper D will denote the open unit disc D = {z : |z| < 1} in the complex

plane C.

2. The (Scalar) Poisson Kernel and The Operator-valued Poisson Kernel

For rei t ∈ D, the (scalar) Poisson kernel Pr,t is defined by

Pr,t(e
iθ ) =

1− r2

�

1− rei t e−iθ
��

1− re−i t eiθ
� (2)

Email addresses: serap.bulut�ko
aeli.edu.tr
http://www.ejpam.com 244 c© 2011 EJPAM All rights reserved.



S. BULUT / Eur. J. Pure Appl. Math, 4 (2011), 244-250 245

=
1

1− rei t e−iθ
+

1

1− re−i t eiθ
− 1

=
∑

n≥0

rneint e−inθ +
∑

n≥0

rne−int einθ − 1.

It is the well-known property of the (scalar) Poisson kernel that the integral formula

1

2π

2π
∫

0

Pr,t(e
iθ )dθ = 1

holds.

In [1], the author gave the definition of the operator-valued Poisson kernel Kr,t(T ) ∈ L (H )
for T ∈ L (H ) such that σ(T ) ⊂ D and for rei t ∈ D, in the following way:

Kr,t(T ) = (I − rei t T ∗)−1 + (I − re−i t T )−1 − I . (3)

For an operator T ∈ L (H ) and a polynomial p(z) =
n
∑

k=0

akzk ∈ C [z]|D, p(T ) ∈ L (H ) is

defined by

p(T ) =

n
∑

k=0

akT k.

Remark 1. T 0 is defined to be the identity operator, whatever the operator T .

Another way to define p(rT ) for 0≤ r < 1 is to use the operator-valued Poisson kernel.

Lemma 1 ([1]). Let T ∈ L (H ) such that σ(T )⊂ D. For all r ∈ [0,1), we have:

p(rT ) =
1

2π

2π
∫

0

p(ei t)Kr,t(T )d t , p ∈ C [z]|D . (4)

Remark 2. Note that in the case p identically equal to 1 we have

1

2π

2π
∫

0

Kr,t(T )d t = I .

Remark 3. Since the definition of the (scalar) Poisson kernel Pr,t(e
iθ ) in (2) is also valid for

|r|< 1, the definition of the operator-valued Poisson kernel Kr,t(T ) in (3) is valid for |r| < 1 too.

Thus we have the following definition and theorem.

Definition 1. Let T ∈ L (H ) such that σ(T )⊂ D. The operator-valued Poisson kernel is defined

by

Kr,t(T ) = (I − rei t T ∗)−1 + (I − re−i t T )−1 − I . (5)

Here r is a real parameter satisfying |r|< 1.



S. BULUT / Eur. J. Pure Appl. Math, 4 (2011), 244-250 246

Theorem 1. Let T ∈ L (H ) such that σ(T )⊂ D. Then we have

1

2π

2π
∫

0

Kr,t(T )d t = I , (6)

where r is a real parameter satisfying |r|< 1.

The purpose of this paper is to give generalizations of (5) and (6).

Firstly, in the next Section we recall the generalization of the (scalar) Poisson kernel.

3. The Generalization of the (Scalar) Poisson Kernel

In [3], Haruki and Rassias gave the new generalizations of the Poisson kernel of the form

P(θ , r) =
1− r2

�

1− reiθ
��

1− re−iθ
� ,

where r is a real parameter satisfying |r| < 1.

One of this generalizations which is taken into consideration by us as follows:

Definition 2. Set

Q (θ ; a, b)
def
=

1− ab

(1− aeiθ )(1− be−iθ )
, (7)

where a, b are complex parameters satisfying |a| < 1 and |b| < 1.

Then they proved the following integral formula for Q (θ ; a, b).

Theorem 2.

1

2π

2π
∫

0

Q (θ ; a, b) dθ = 1,

where a, b are complex parameters satisfying |a| < 1 and |b| < 1.

Remark 4. Note that we can express the generalization of the (scalar) Poisson kernel in (2) as

Qa,b,t

�

eiθ
�

=
1− ab

(1− aei t e−iθ )(1− be−i t eiθ )
.

4. A New Generalization of the Operator-valued Poisson Kernel

In this Section, we shall treat generalizations of (5) and (6).



S. BULUT / Eur. J. Pure Appl. Math, 4 (2011), 244-250 247

Definition 3. For T ∈ L (H ) such that σ(T ) ⊂ D, define the generalization of the operator-

valued Poisson kernel Kr,t(T ) in the following way:

Qa,b,t(T )
def
= (I − aei t T ∗)−1+ (I − be−i t T )−1 − I , (8)

where a, b are complex parameters satisfying |a| < 1 and |b| < 1.

Remark 5. Note that Qa,b,t(T ) ∈ L (H ).

Remark 6. By taking a = r and b = r in (8), we find that (8) is a generalization of (5).

Lemma 2. We have the following equalities:

�

Qa,b,t(T )
�∗
= Q b̄,ā,t(T ) = Q ā, b̄,−t(T

∗).

Lemma 3. For T ∈ L (H ) such that σ(T ) ⊂ D, we have:

Qa,b,t(T ) = (I − aei t T ∗)−1(I − abT ∗T )(I − be−i t T )−1 (9)

= (I − be−i t T )−1(I − abT T ∗)(I − aei t T ∗)−1 (10)

=

∞
∑

n=0

aneint T ∗n +

∞
∑

n=0

bne−int T n − I . (11)

Proof. By (8), we get

Qa,b,t(T ) = (I − aei t T ∗)−1 + (I − be−i t T )−1 − I

= (I − aei t T ∗)−1
�

I + (I − aei t T ∗)(I − be−i t T )−1 − (I − aei t T ∗)
�

= (I − aei t T ∗)−1
�

(I − be−i t T ) + (I − aei t T ∗)− (I − aei t T ∗)(I − be−i t T )
�

(I − be−i t T )−1

= (I − aei t T ∗)−1(I − abT ∗T )(I − be−i t T )−1.

Thus we obtain (9).

Similarly, the equality

Qa,b,t(T ) = (I − be−i t T )−1 + (I − aei t T ∗)−1 − I

gives proof of (10).

On the other hand, since



aei t T ∗



< 1 and



be−i t T



 < 1, we have

∞
∑

n=0

aneint T ∗n = (I − aei t T ∗)−1

and
∞
∑

n=0

bne−int T n = (I − be−i t T )−1,

respectively [see 4, Theorem 7.10]. By the last two equalities above and (8), we get (11).



S. BULUT / Eur. J. Pure Appl. Math, 4 (2011), 244-250 248

Lemma 4. Let T ∈ L (H ) such that σ(T )⊂ D. Then

‖T‖ ≤ 1⇐⇒ Qa,ā,t(T )≥ 0.

Proof. The proof is same as proof of the Lemma 2.4 in [1].

Now we give a similar result to Lemma 1 by means of (11).

Lemma 5. Let T ∈ L (H ) such that σ(T )⊂ D. For q(z) ∈ C [z]|D, we have

q(bT ) =
1

2π

2π
∫

0

q(ei t)Qa,b,t(T )d t,

where a, b are complex parameters satisfying |a| < 1 and |b| < 1.

Proof. Let q(z) =
N
∑

k=0

akzk. Using (11) and considering the equality
∫ 2π

0
eimt d t = 0 for

m ∈ Z\{0}, we obtain

1

2π

2π
∫

0

q(ei t)Qa,b,t(T )d t =
1

2π

2π
∫

0

 

N
∑

k=0

ak bkT k

!

d t

=

N
∑

k=0

ak bkT k

= q(bT ).

Corollary 1. Note that in the case q identically equal to 1 we have

1

2π

2π
∫

0

Qa,b,t(T )d t = I . (12)

Now we give another proof of (12) independently a polynomial. For this purpose we will

use the Riesz-Dunford integral.

Theorem 3. For T ∈ L (H ) such that σ(T )⊂ D, we have

1

2π

2π
∫

0

Qa,b,t(T )d t = I , (13)

where a, b are complex parameters satisfying |a| < 1 and |b| < 1.



S. BULUT / Eur. J. Pure Appl. Math, 4 (2011), 244-250 249

Proof. By (8), we have

1

2π

2π
∫

0

Qa,b,t(T )d t =
1

2π

2π
∫

0

�

(I − aei t T ∗)−1 + (I − be−i t T )−1− I
�

d t. (14)

We set

I1 =
1

2π

2π
∫

0

(I − aei t T ∗)−1d t, (15)

I2 =
1

2π

2π
∫

0

(I − be−i t T )−1d t (16)

and

I3 =
1

2π

2π
∫

0

Id t. (17)

So, by (15), (16) and (17), (14) is of the form

1

2π

2π
∫

0

Qa,b,t(T )d t = I1 + I2 − I3. (18)

It is clear that

I3 = I . (19)

Next we shall calculate I1 and I2.

Firstly, we have

I1 =
1

2π

2π
∫

0

(I − aei t T ∗)−1d t =
1

2π

2π
∫

0

e−i t(e−i t I − aT ∗)−1d t.

Making substitution z = e−i t in the last integral, we find

I1 = −
1

2πi

∫

|z|=1

(zI − aT ∗)−1dz,

where the integral along the |z| = 1 is in the negative direction. Hence, by the Riesz-Dunford

integral (1), we have

I1 = I . (20)



REFERENCES 250

Similarly, we get

I2 =
1

2π

2π
∫

0

(I − be−i t T )−1d t =
1

2π

2π
∫

0

ei t(ei t I − bT )−1d t.

If we set z = ei t then the last integral is of the form

I2 =
1

2πi

∫

|z|=1

(zI − bT )−1dz,

where the integral along the |z| = 1 is in the positive direction. So, by the Riesz-Dunford

integral (1), we obtain

I2 = I . (21)

Therefore, by (18), (19), (20) and (21) we get (13).

Remark 7. By taking a = r and b = r in (13), we find that (13) is a generalization of (6).

Corollary 2. If we set a = r and b = r in Theorem 3 then we obtain Theorem 1. Hence Theorem

3 gives another proof of Theorem 1.

Remark 8. Note that Qa,b,t (T ) in (8) is an operator-valued form of Qa,b,t

�

eiθ
�

in Remark 4.

References

[1] I. Chalendar, The operator-valued Poisson kernel and its applications, Ir. Math. Soc. Bull.

51, 21–44. 2003.

[2] N. Dunford and J. T. Schwartz, Linear Operators, Part I, General Theory, Interscience,

New York, 1958.

[3] H. Haruki and Th. M. Rassias, New Generalizations of the Poisson Kernel, J. Appl. Math.

Stochastic Anal. 10, 191–196. 1997.

[4] N. Young, An Introduction to Hilbert Space, Cambridge University Press, 1988.