10_bulut.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 2, No. 2, 2009, (296-301)

ISSN 1307-5543 – www.ejpam.com

A Note on 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 different proof of the integral formula

1

2π

2π
∫

0

Kr,t(T )d t = I ,

where Kr,t(T ) is the operator-valued Poisson kernel.

AMS subject classifications: Primary 45P05, 47A60; Secondary 46E40, 47B38

Key words: 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 . T is called a

unitary operator if it satisfies T ∗T = T T ∗ = I where T ∗ is the adjoint of T .

Email address: serap.bulut�ko
aeli.edu.tr
http://www.ejpam.com 296 c© 2009 EJPAM All rights reserved.



S. Bulut / Eur. J. Pure Appl. Math, 2 (2009), (296-301) 297

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

complex plane C.

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 teiθ
�

=
1

1− rei te−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 for-

mula

1

2π

2π
∫

0

Pr,t(e
iθ )dθ = 1 (1.1)

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 . (1.2)

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

ak T k.

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

On the other hand, for 0 ≤ r < 1, p(rT ) is defined by means of the operator-

valued Poisson kernel as follows.

Lemma 1.1. [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

.



S. Bulut / Eur. J. Pure Appl. Math, 2 (2009), (296-301) 298

Note that in the case p identically equal to 1, we have

Main Theorem.

1

2π

2π
∫

0

Kr,t(T )d t = I (1.3)

for 0 ≤ r < 1 and T ∈ L (H ) such that σ(T ) ⊂ D.

The purpose of this paper is to give a different proof of (1.3) independently a

polynomial.

In [2] which is a motive of our present paper, a proof of (1.1) is given by using

the functional equation

F(r2n

) = F(r), n = 1, 2, . . .

where

F(r) =
1

2π

2π
∫

0

1− r2

�

1− reiθ
��

1− re−iθ
�dθ , 0≤ r < 1.

In this note, we use a similar method for the operator-valued Poisson kernel

Kr,t(T ).

2. Proof of the Main Theorem

Let rei t ∈ D, 0 ≤ r < 1 and let T ∈ L (H ) such that σ(T )⊂ D. Set

F(rT )
def
=

1

2π

2π
∫

0

Kr,t(T )d t . (2.1)

Then F is a continuous function. Also, it is obvious that F(0) = I for r = 0.

Let us write

F(rT ) =
1

2π

π
∫

0

Kr,t(T )d t +
1

2π

2π
∫

π

Kr,x(T )d x .



S. Bulut / Eur. J. Pure Appl. Math, 2 (2009), (296-301) 299

Making the substitution x = t +π in the second integral, and using (1.2), we obtain

F(rT ) =
1

2π

π
∫

0

�

(I − rei t T ∗)−1 + (I − re−i t T )−1− I
�

d t

+
1

2π

π
∫

0

�

(I + rei t T ∗)−1 + (I + re−i t T )−1− I
�

d t .

Hence we get

F(rT ) =
1

2π

π
∫

0

�

(I − rei t T ∗)−1+ (I + rei t T ∗)−1
�

d t (2.2)

+
1

2π

π
∫

0

�

(I − re−i t T )−1+ (I + re−i t T )−1
�

d t

−
1

2π

π
∫

0

2Id t .

On the other hand, we have the equalities

(I − rei t T ∗)−1 + (I + rei t T ∗)−1 = 2
�

I − r2e2i t T ∗2
�−1

(2.3)

and

(I − re−i t T )−1+ (I + re−i t T )−1 = 2
�

I − r2e−2i t T 2
�−1

. (2.4)

Thus, by (2.3) and (2.4), (2.2) is of the form

F(rT ) =
1

π

π
∫

0

h

�

I − r2e2i t T ∗2
�−1
+
�

I − r2e−2i t T 2
�−1
− I
i

d t .

Making the substitution φ = 2t in the above integral, we find

F(rT ) =
1

2π

2π
∫

0

h

�

I − r2eiφT ∗2
�−1
+
�

I − r2e−iφT 2
�−1
− I
i

dφ.



S. Bulut / Eur. J. Pure Appl. Math, 2 (2009), (296-301) 300

By (1.2), we get

F(rT ) =
1

2π

2π
∫

0

Kr2,φ(T
2)dφ. (2.5)

In view of (2.1) and (2.5), we obtain

F(rT ) = F(r2T 2). (2.6)

By repeated applications of (2.6), we see that

F(rT ) = F((rT )
2n

), n = 1, 2, . . . .

Since ‖rT‖< 1, we have

F(rT ) = lim
n→∞

F((rT )
2n

) = F(0) = I .

Thus the proof is completed.

3. Results

Corollary 3.1. Note that

F(rT ∗) = I .

Lemma 3.2. Let T ∈L (H ) such that σ(T ) ⊂ D. If T is invertible then

Kr−1,t(T
−1) =−Kr,−t(T )

for 0< r < 1.

Corollary 3.3. Let T ∈ L (H ) such that σ(T ) ⊂ D.

(i) If T is invertible then

F(r−1T−1) =−F(rT ∗)

for 0< r < 1.



REFERENCES 301

(ii) If T is a unitary operator then

F(r−1T−1) = −F(rT−1)

for 0< r 6= 1.

When we consider the Corollary 3.3, we have the following

Theorem 3.4. Let T ∈ L (H ) such that σ(T )⊂ D.

(i) If T is invertible then

1

2π

2π
∫

0

Kr−1,t(T
−1)d t = −I

for 0< r < 1.

(ii) If T is a unitary operator then

1

2π

2π
∫

0

Kr,t(T
−1)d t =−I

for r > 1.

References

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

51 (2003), 21–44.

[2] A. E. Taylor, A Note on the Poisson Kernel, Amer. Math. Monthly, 57 (1950), 478–479.