On operator interpolation problems

ON OPERATOR INTERPOLATION PROBLEMS

Young Soo Jo ^∗ , Joo Ho Kang and Ki Sook Kim

Abstract. In this paper we obtained the following : Let H be a Hilbert space and L be a subspace lattice on H. Let X and Y be operators acting on H. If the range of X is dense in H, then the following are equivalent:

(1) there exists an operator A in AlgL such that AX = Y , (2) sup

E

^⊥

Y f

E

^⊥

Xf
: f ∈ H, E ∈ L



= K < ∞.

Moreover, if condition (2) holds, we may choose the operator A such that
A
= K.

1. Introduction

On the process of solving operator equation AX = Y for two given X and Y in the algebra B(H), the class of all bounded operators acting on a Hilbert space H, many mathematicians investigated the problem on their field of studies. What is the condition for an operator A satisfying AX = Y to be a member of A which is a specified subalgebra of B(H)?

Douglas [2] used the range inclusion property of operators to show necessary and sufficient conditions for the existence of an operator A satisfying AX = Y . In [7], authors investigated the problem on nest algebra. Hopenwasser [3] studied on CSL and Moore [9] showed the condition is a necessary and sufficient condition for the existence of an interpolating operator in Alg L when L is a CSL and X has dense range.

In [5], authors has found the other necessary and sufficient condition for interpolating operator when L is a subspace lattice and X has dense range. In this paper, we are going to show that the condition given in [9] holds for a subspace lattice L.

Received March 3, 2003.

2000 Mathematics Subject Classification: 47L35.

Key words and phrases: interpolation problem, subspace lattice, Alg L, CSL.

∗

This work was supported by Korea Research Foundation Grant(KRF-99-015-

DP0016).

Let H be a Hilbert space. A subspace lattice L is a strongly closed lat- tice of orthogonal projections in B(H) containing the trivial projections 0 and I. The symbol Alg L denotes the algebra of bounded operators on H that leave invariant every projection in L; AlgL is a weakly closed subalgebra of B(H). A lattice L is a commutative subspace lattice, or CSL, if the projections in L all commute; in this case, AlgL is called a CSL algebra.

We want to introduce the simplest case of the operator interpolation problem that relaxes all restrictions on A, requiring it simply to be a bounded operator. In this case, the existence of A is nicely characterized by the well-known factorization theorem of Douglas.

Theorem A. ([2], [4]) Let X and Y be bounded operators acting on a Hilbert space H. Then the following statements are equivalent:

(1) range Y ^∗ ⊆ range X ^∗ ,

(2) Y ^∗ Y ≤ λ

²

X ^∗ X for some λ ≥ 0,

(3) there exists a bounded operator A on H so that AX = Y . Moreover, if (1), (2), and (3) are valid, then there exists a unique oper- ator A so that

(a) A

²

= inf {µ : Y ^∗ Y ≤ µX ^∗ X}, (b) ker Y ^∗ = ker A ^∗ and

(c) range A ^∗ ⊆ range X ⁻ .

2. Operator interpolation problems in B(H)

Let H be a Hilbert space and M a subset of H. Then M means the closure of M and M ^⊥ is the orthogonal complement of M.

We use the convention

⁰₀

= 0, when necessary.

We will introduce theorems that are obtained easily.

Theorem 2.1. Let X and Y be bounded operators acting on H.

Then the following are equivalent:

(1) there exists λ( ≥ 0) such that Y ^∗ Y ≤ λ

²

X ^∗ X,

(2) there exists λ( ≥ 0) such that Y f ≤ λXf for every f ∈ H, (3) sup

Y f

Xf : f ∈ H



<∞.

We can get the following theorem by Theorems A and 2.1.

Theorem 2.2. Let X and Y be bounded operators acting on H.

Then the following are equivalent:

(1) range Y ^∗ ⊆ range X ^∗ ,

(2) there exists λ( ≥ 0) such that Y ^∗ Y ≤ λ

²

X ^∗ X,

(3) there exists an operator A in B(H) such that AX = Y ,

(4) there exists λ( ≥ 0) such that Y f ≤ λXf for every f ∈ H, (5) sup

Y f

Xf : f ∈ H



< ∞.

Theorem 2.3. Let X

₁

, X

₂

, . . . , X _n and Y

₁

, Y

₂

, . . . , Y _n be bounded operators acting on H. Then the following are equivalent:

(1) there exists an operator A in B(H) such that AX i = Y _i for i = 1, 2, . . . , n,

(2) sup

  _n

i=1 Y _i f _i 

  _n

i=1 X _i f _i  : f _i ∈ H



< ∞.

Proof. If sup

  _n

i=1 Y _i f _i 

  _n

i=1 X _i f _i  : f _i ∈ H



< ∞, then we may assume that sup

  _n

i=1 Y _i f _i 

  _n

i=1 X _i f _i  : f _i ∈ H



= 1. Then



 n i=1

Y _i f _i  ≤ 

 n i=1

X _i f _i 

for all f _i in H. Let M = {  _n

i=1 X _i f _i : f _i ∈ H }. Then M is a linear manifold. We define a function A : M −→ H by A(  _n

i=1 X _i f _i ) =

 _n

i=1 Y _i f _i for f _i in H. Then A is well-defined.

For, if  _n

i=1 f _i f _i =  _n

i=1 X _i g _i , then  _n

i=1 X _i f _i − (  _n

i=1 X _i g _i ) = 0.

So,   _n

i=1 X _i f _i − (  _n

i=1 X _i g _i )  =   _n

i=1 X _i (f _i − g i )  = 0. Since

  _n

i=1 Y _i (f _i − g i )  ≤   _n

i=1 X _i (f _i − g i )  = 0,  _n

i=1 X _i f _i =  _n

i=1 X _i g _i . And A is continuous. Extend A to M by continuity and define Ah = 0 for h ∈ M ^⊥ . Then, A is a bounded operator on H clearly. Moreover, AX _i = Y _i for i = 1, 2, . . . , n.

The converse is proved easily.

We can extend the above fact for infinitely many operators easily.

Theorem 2.4. Let X _i and Y _i be bounded operators acting on H for each i in N. Then the following are equivalent:

(1) there exists an operator A in B(H) such that AX i = Y _i for i = 1, 2, . . . ,

(2) sup

  _n

i=1 Y _i f _i 

  _n

i=1 X _i f _i  : f _i ∈ H and n ∈ N



< ∞.

Theorems 2.3 and 2.4 show that if we get necessary and sufficient conditions which are equivalent to the second case of the above Theo- rems, then it gives conditions for the existence of an operator A such that AX _i = Y _i for any bounded operators X _i and Y _i . So we want to find conditions that are equivalent to the second case.

Theorem 2.5. Let X

₁

, . . . , X _n and Y

₁

, . . . , Y _n be bounded operators acting on H. Assume that Re < X i f, X _j g >≥ 0 for i < j and all f, g in H. If there exists λ(≥ 0) such that Y _i ^∗ Y _i ≤ λ

²

X _i ^∗ X _i for each i = 1, 2, . . . , n, then sup

  _n

i=1 Y _i f _i 

  _n

i=1 X _i f _i  : f _i ∈H



< ∞.

Proof. If Y _i ^∗ Y _i ≤ λ

²

X _i ^∗ X _i for each i = 1, 2, . . . , n, then Y i f

X i f ≤ λ for each i = 1, 2, . . . , n. Since Re <X _i f, X _j g>≥ 0 for i < j, X k f _k  ≤

  _n

i=1 X _i f _i  for each k = 1, 2, . . . , n. For,



 n i=1

X _i f _i 

²

= < X _k f _k , X _k f _k > + < 

i=k

X _i f _i , X _k f _k >

+ < X _k f _k , 

i=k

X _i f _i > + < 

i=k

X _i f _i , 

i=k

X _i f _i >

= X k f _k 

²

+  

i=k

X _i f _i 

²

+ 

i<j

2Re < X _i f _i , X _j f _j >≥ X k f _k 

²

.

Hence nλ ≥

 n i=1

Y i f _i 

X i f _i  ≥

 n i=1

Y i f _i 

  _n

i=1 X _i f _i  =

 _n

i=1 Y i f _i 

  _n

i=1 X _i f _i  ≥   _n

i=1 Y _i f _i 

  _n

i=1 X _i f _i  _. So, sup

  _n

i=1 Y _i f _i 

  _n

i=1 X _i f _i  : f _i ∈ H



< ∞.

From Theorem 2.2, we have the following corollary.

Corollary 2.6. Let X

₁

, . . . , X _n and Y

₁

, . . . , Y _n be bounded oper- ators acting on H. Assume that Re < X i f, X _j g >≥ 0 for i < j and all f, g in H. If range Y _i ^∗ ⊆ range X _i ^∗ for each i = 1, 2, . . . , n, then sup

  _n

i=1 Y _i f _i 

  _n

i=1 X _i f _i  : f _i ∈ H



< ∞.

Theorem 2.7. Let X

₁

, . . . , X _n and Y

₁

,. . . ,Y _n be bounded operators acting on H. If sup

  _n

i=1 Y _i f _i 

  _n

i=1 X _i f _i  : f _i ∈ H



< ∞, then there exists λ(≥ 0) such that Y _i ^∗ Y _i ≤ λ

²

X _i ^∗ X _i for each i = 1, 2, . . . , n.

Proof. If sup

  _n

i=1 Y _i f _i 

  _n

i=1 X _i f _i  : f _i ∈ H



< ∞, then

sup

Y i f

X i f : f ∈ H



< ∞

for each i = 1, 2, . . . , n. So, by the Theorem 2.2, there exists λ _i ( ≥ 0) such that Y _i ^∗ Y _i ≤λ

²

_i X _i ^∗ X _i for each i = 1, 2, . . . , n. Hence we can take λ = max{λ

1

, λ

₂

,. . . ,λ _n } such that Y _i ^∗ Y _i ≤ λ

²

X _i ^∗ X _i for each i = 1, 2, . . . , n.

From the above theorems, we have the following.

Theorem 2.8. Let X

₁

, . . . , X _n and Y

₁

,. . . ,Y _n be bounded operators acting on H. If Re < X i f, X _j g >≥ 0 for i < j and all f, g in H, then the following are equivalent:

(1) range Y _i ^∗ ⊆ range X _i ^∗ for each i = 1, 2, . . . , n,

(2) there exists λ( ≥ 0) such that Y _i ^∗ Y _i ≤ λ

²

X _i ^∗ X _i for each i = 1, 2, . . . , n,

(3) there exists an operator A in B(H) such that AX i = Y _i for i = 1, 2, . . . , n,

(4) there exists λ( ≥ 0) such that Y i f ≤ λX i f for i = 1, 2, . . . , n, (5) sup

  _n

i=1 Y _i f _i 

  _n

i=1 X _i f _i  : f _i ∈ H



< ∞.

If we look through the proof of the above theorems, then we know that it is not extended for the countable bounded operators by the same type. But we have a sufficient condition as the following theorem and corollary.

Theorem 2.9. Let X _i and Y _i be bounded operators acting on H for i = 1, 2, . . . . If Re < X _i f, X _j g >≥ 0 for i < j and f, g in H and there exists a sequence {λ n } of non-negative numbers such that  _∞

i=1 λ _i < ∞ and Y _i ^∗ Y _i ≤ λ

²

_i X _i ^∗ X _i for i ∈ N, then

sup

  _n

i=1 Y _i f _i 

  _n

i=1 X _i f _i  : f _i ∈ H and n ∈ N



< ∞.

Proof. If Re < X _i f, X _j g >≥ 0 for i < j and f, g in H, then for each n ∈ N, X k f _k  ≤ 

 n i=1

X _i f _i  for every k ≤ n with the same proof as Theorem 2.5. Hence for each n ∈ N,

∞ >  ^∞

i=1

λ _i ≥

 n i=1

Y i f _i 

X i f _i 

≥

 n i=1

Y i f _i 

  _n

i=1 X _i f _i  ≥

 _n

i=1 Y i f _i 

  _n

i=1 X _i f _i 

≥   _n

i=1 Y _i f _i 

  _n

i=1 X _i f _i  _. So, sup

  _n

i=1 Y _i f _i 

  _n

i=1 X _i f _i  : f _i ∈ H and n ∈ N



< ∞.

Corollary 2.10. Let X _i and Y _i be bounded operators acting on H for i = 1, 2, . . . . Assume that Re < X i f, X _j g >≥ 0 for i < j and f, g in H. If there exists a sequence {λ n } of non-negative numbers such that  _∞

i=1 λ _i < ∞ and Y _i ^∗ Y _i ≤ λ

²

_i X _i ^∗ X _i for i ∈ N, then there exists a bounded operator A in B(H) such that AX i = Y _i for any i ∈ N.

3. Interpolation problems in Alg L

In [9], the authors showed a necessary and sufficient condition that there exists an operator A in Alg L such that AX = Y when L is a commutative subspace lattice. We state the theorem.

Theorem B [9]. Let L be a commutative subspace lattice in B(H) and let X be an operator with dense range, and let Y be a bounded operator acting on H. The following conditions are equivalent:

(1) there exists an operator A in Alg L such that AX = Y , (2) ρ(X, Y ; L) = ^def sup

E ^⊥ Y f

E ^⊥ Xf : E ∈ L and f ∈ H



< ∞ .

Furthermore, if condition (2) holds, then there is an operator A ∈AlgL such that AX = Y and whose norm is equal to ρ(X, Y ; L).

In the following theorem, we obtain the above result for non-commuta-

tive subspace lattice case.

Theorem 3.1. Let L be a subspace lattice on H and let X and Y be in B(H). If the range of X is dense in H, then the following are equivalent:

(1) there exists an operator A in Alg L such that AX = Y , (2) sup

E ^⊥ Y f

E ^⊥ Xf : f ∈ H, E ∈ L



= K < ∞.

Moreover, if condition (2) holds, we may choose the operator A such that A = K.

Proof. Assume that sup

E ^⊥ Y f

E ^⊥ Xf : f ∈H, E ∈L



= K < ∞. Then for each E in L, there exists an operator A E in B(H) such that A E (E ^⊥ X) = E ^⊥ Y and A E  ≤ K by Theorem 2.2. In particular, if E = 0, then we have an operator A

₀

in B(H) such that A

₀

X = Y and A

₀

 ≤ K. And for each E ∈ L, A E (E ^⊥ X) = E ^⊥ Y = E ^⊥ (A

₀

X). Since the range of X is dense in H, A E E ^⊥ = E ^⊥ A

₀

for each E in L. Hence for each E in L,

E ^⊥ A

₀

E ^⊥ = A _E E ^⊥ E ^⊥ = A _E E ^⊥ = E ^⊥ A

_{0 .}

So A

₀

is in Alg L. Since A

0

X = Y and E ^⊥ A

₀

E ^⊥ = E ^⊥ A

₀

,

E ^⊥ Y f = E ^⊥ A

₀

Xf = E ^⊥ A

₀

E ^⊥ Xf ≤ A

0

E ^⊥ Xf .

So, K ≤ A

0

. Therefore, A

0

 = K.

The proof of the converse is obvious.

Theorem 3.2. Let X

₁

, . . . , X _n and Y

₁

, . . . , Y _n be bounded operators acting on H. If the range of X k is dense in H for some k in {1, 2, . . . , n}, then the following are equivalent:

(1) there exists an operator A in Alg L such that AX i = Y _i for i = 1, 2, . . . , n,

(2) sup

E ^⊥ (  _n

i=1 Y _i f _i ) 

E ^⊥ (  _n

i=1 X _i f _i )  : f _i ∈ H, E ∈ L



= K < ∞.

Proof. If sup

E ^⊥ (  _n

i=1 Y _i f _i ) 

E ^⊥ (  _n

i=1 X _i f _i )  : f _i ∈ H, E ∈ L



= K, then

E ^⊥ (  _n

i=1 Y _i f _i ) 

E ^⊥ (  _n

i=1 X _i f _i )  ≤ K

for each E ∈ L and all f i ∈ H. So for each E ∈ L, there exists an oper- ator A _E in B(H) such that A E (E ^⊥ X _i ) = E ^⊥ Y _i for each i = 1, 2, . . . , n by Theorem 2.3. If E = 0, then A

₀

X _i = Y _i for i = 1, 2, . . . , n. Hence

A _E (E ^⊥ X _i ) = E ^⊥ Y _i = E ^⊥ A

₀

X _i for each i = 1, 2, . . . , n.

Since range X _k = H for some k in {1, 2, . . . , n}, A E E ^⊥ = E ^⊥ A

₀

for each E ∈ L. Also,

E ^⊥ A

₀

E ^⊥ = A _E E ^⊥ E ^⊥ = A _E E ^⊥ = E ^⊥ A

_{0 .}

So A

₀

is in Alg L.

We omit the proof of the converse since it can be proved easily.
We can generalize Theorem 3.2 to the countable case easily.

Theorem 3.3. Let X _i and Y _i be bounded operators acting on H. If the range of X _k is dense in H for some k ∈ N, then the following are equivalent:

(1) there exists an operator A in Alg L such that AX i = Y _i for i = 1, 2, . . . ,

(2) sup

E ^⊥ (  _n

i=1 Y _i f _i ) 

E ^⊥ (  _n

i=1 X _i f _i )  : f _i ∈ H, E ∈ L, n ∈ N



= K < ∞.

Proof of the following theorem can be obtained by the same way as the proof of Theorem 2.5.

Theorem 3.4. Let X

₁

, . . . , X _n and Y

₁

, . . . , Y _n be bounded operators acting on a Hilbert space H. Assume that Re <E ^⊥ X _i f, E ^⊥ X _j g >≥ 0 for each E ∈ L, i < j and all f, g in H. If there exists λ(≥ 0) such that (E ^⊥ Y _i ) ^∗ (E ^⊥ Y _i ) ≤ λ

²

(E ^⊥ X _i ) ^∗ (E ^⊥ X _i ) for each E ∈ L and i = 1, 2, . . . , n, then sup

E ^⊥ (  _n

i=1 Y _i f _i ) 

E ^⊥ (  _n

i=1 X _i f _i )  : f _i ∈ H, E ∈ L



< ∞.

Theorem 3.5. Let X

₁

, . . . , X _n and Y

₁

, . . . , Y _n be bounded operators acting on a Hilbert space H. If

sup

E ^⊥ (  _n

i=1 Y _i f _i ) 

E ^⊥ (  _n

i=1 X _i f _i )  : f _i ∈ H, E ∈ L



< ∞, then there exists λ( ≥ 0) such that

(E ^⊥ Y _i ) ^∗ (E ^⊥ Y _i ) ≤λ

²

(E ^⊥ X _i ) ^∗ (E ^⊥ X _i )

for each E ∈ L and i = 1, 2, . . . , n.

Proof. If sup

E ^⊥ (  _n

i=1 Y _i f _i ) 

E ^⊥ (  _n

i=1 X _i f _i )  : f _i ∈ H, E ∈ L



= λ < ∞ for some nonnegative real number λ, then

E ^⊥ (

 n i=1

Y _i f _i )  ≤ λE ^⊥ (

 n i=1

X _i f _i ) 

for each E ∈ L and all f i in H. If f i = 0 for i = k, then E ^⊥ Y _k f _k  ≤ λE ^⊥ X _k f _k  for k = 1, 2, . . . , n. So

(E ^⊥ Y _i ) ^∗ (E ^⊥ Y _i ) ≤ λ

²

(E ^⊥ X _i ) ^∗ (E ^⊥ X _i )

for each E ∈ L and i = 1, 2, . . . , n.

We have the following theorem from the above theorems.

Theorem 3.6. Let X

₁

, . . . , X _n and Y

₁

, . . . , Y _n be bounded operators acting on a Hilbert space H. If Re < E ^⊥ X _i f, E ^⊥ X _j g >≥ 0 for each E ∈ L, i < j and all f, g in H and if the range of X k is dense in H for some k, then the following are equivalent:

(1) range (E ^⊥ Y _i ) ^∗ ⊆range (E ^⊥ X _i ) ^∗ for each E ∈L and i = 1, 2, . . . , n, (2) there exists λ( ≥ 0) such that (E ^⊥ Y _i ) ^∗ (E ^⊥ Y _i ) ≤λ

²

(E ^⊥ X _i ) ^∗ (E ^⊥ X _i ) for each E ∈ L and i = 1, 2, . . . , n,

(3) there exists λ( ≥ 0) such that E ^⊥ Y _i f ≤ λE ^⊥ X _i f for each E ∈ L and i = 1, 2, . . . , n,

(4) sup

E ^⊥ (  _n

i=1 Y _i f _i ) 

E ^⊥ (  _n

i=1 X _i f _i )  : f _i ∈ H, E ∈ L



< ∞,

(5) there exists an operator A in B(H) such that AE ^⊥ X _i = E ^⊥ Y _i for each E ∈ L and i = 1, 2, . . . , n,

(6) there exists an operator A in Alg L such that AX i = Y _i for each i = 1, 2, . . . , n.

From Theorem 2.9, we can get the following theorem.

Theorem 3.7. Let X _i and Y _i be bounded operators on a Hilbert space H for i = 1, 2, . . . . If Re <E ^⊥ X _i f, E ^⊥ X _j g >≥ 0 for each E ∈ L, i < j and all f, g in H and if there exists a sequence {λ n } of non-negative numbers such that  _∞

i=1 λ _i < ∞ and (E ^⊥ Y _i ) ^∗ (E ^⊥ Y _i ) ≤ λ

²

_i (E ^⊥ X _i ) ^∗ (E ^⊥ X _i ) for each E ∈ L and i ∈ N, then

sup

E ^⊥ (  _n

i=1 Y _i f _i ) 

E ^⊥ (  _n

i=1 X _i f _i )  : f _i ∈ H, E ∈ L, n ∈ N



< ∞.

Proof. Let Re <E ^⊥ X _i f, E ^⊥ X _j g >≥ 0 for each E ∈ L, i < j and all f, g in H and let n ∈ N. Then E ^⊥ X _k f _k  ≤ E ^⊥ (

 n i=1

X _i f _i )  for every k ≤ n. Indeed,

E ^⊥ (

 n i=1

X _i f _i ) 

²

= < E ^⊥ X _k f _k , E ^⊥ X _k f _k > + < E ^⊥ ( 

i=k

X _i f _i ), E ^⊥ X _k f _k >

+ < E ^⊥ X _k f _k , E ^⊥ ( 

i=k

X _i f _i ) > + < E ^⊥ ( 

i=k

X _i f _i ), E ^⊥ ( 

i=k

X _i f _i ) >

= E ^⊥ X _k f _k 

²

+ E ^⊥ ( 

i=k

X _i f _i ) 

²

+ 

i<j

2Re < E ^⊥ X _i f _i , E ^⊥ X _j f _j >

≥ E ^⊥ X _k f _k 

²

for i, j = 1, 2, . . . , n _. Hence for each n ∈ N,

∞ >

 ∞ i=1

λ _i ≥

 n i=1

E ^⊥ Y _i f _i 

E ^⊥ X _i f _i 

≥

 n i=1

E ^⊥ Y _i f _i 

E ^⊥ (  _n

i=1 X _i f _i )  ≥

 _n

i=1 E ^⊥ Y _i f _i 

E ^⊥ (  _n

i=1 X _i f _i ) 

≥ E ^⊥ (  _n

i=1 Y _i f _i ) 

E ^⊥ (  _n

i=1 X _i f _i )  _. So, sup

E ^⊥ (  _n

i=1 Y _i f _i ) 

E ^⊥ (  _n

i=1 X _i f _i )  : f _i ∈ H, E ∈ L, n ∈ N



< ∞.

References

[1] M. Anoussis, E. Katsoulis, R. L. Moore and T. T. Trent, Interpolation problems for ideals in nest algebras, Math. Proc. Camb. Phil. Soc. 111 (1992), 151–160.

[2] R. G. Douglas, On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc. 17 (1966), 413–415.

[3] A. Hopenwasser, The equation T x = y in a reflexive operator algebra, Indiana Univ. Math. J. 29 (1980), 121–126.

[4] , Hilbert-Schmidt Interpolation in CSL-Algebras, Illinois J. Math. 33

(1989), no. 4, 657–672.

[5] Y. S. Jo and J. H. Kang, Interpolation problems in Alg L, to appear.

[6] R. Kadison, Irreducible Operator Algebras, Pro. Nat. Acad. Sci. U.S.A. (1957), 273–276.

[7] E. Katsoulis, R. L. Moore and T. T. Trent, Interpolation in nest algebras and applications to operator Corona Theorems, J. Operator Theory 29 (1993), 115–

123.

[8] E. C. Lance, Some properties of nest algebras, Proc. London Math. Soc. 19 (1969), no. 3, 45–68.

[9] R. Moore and T. T. Trent, Linear equations in subspaces of operators, Proc.

Amer. Math. Soc. 128 (2000), no. 3, 781–788.

[10] N. Munch, Compact causal data interpolation, J. Math. Anal. Appl. 140 (1989), 407–418.

Young Soo Jo

Department of Mathematics Keimyung University Daegu 704-701, Korea E-mail: [email protected]

Joo Ho Kang and Ki Sook Kim Department of Mathematics Daegu University

Daegu 712-714, Korea

E-mail: [email protected]