Appl. Math.-JCU l i b (1996), 247-249

E X I S T E N C E O F A N O N - S T R I C T M O R P H I S M

O F Q U O T I E N T B A N A C H S P A C E S *

ZHANG HAITAO

A b s t r a c t . In this paper we prove the existence of a morphism of quotient Banach spaces that is not strict.

1. I n t r o d u c t i o n

The concepts of quotient Banach spaces (q.B.s.) and quotient Frechet spaces were introduced by L.Waelbroeck ([6]). In the study of morphisms of q.B.s, and that of quotient Frechet spaces (see [1], [3]-[4], etc.), the class of strict morphisms ([6]) is very important. Since a strict morphism carries many properties of the inducing operator, it is much easier to deal with. A natural question of theoretic importance is: "Is every morphism of q.B.s, strict?"

The aim of this paper is to answer the above question. We will give an example of morphism of q.B.s, which is not strict. We have learned that a similar example was given independently by L. Waelbroeck.

In Section 2 we recall some definitions and list some theorems needed. In Section 3 we will give the main result and the example.

2. D e f i n i t i o n s and T h e o r e m s N e e d e d

2.1 D e f i n i t i o n [6]. A linear subspace Y of a Banach space X is called a Banach subspace of X if there is a norm q on Y such that q makes Y complete and q is stronger than the norm induced from X. If X is a Banach space and Y is a Banach subspace of X, then the algebraic quotient X / Y is called a quotient Banach space (q.S.s.).

2.2 D e f i n i t i o n [3],[6]. Let X/Xo and Y/Yo be two q.B.s. A linear mapping T from X/Xo to Y/Yo is called morphism (or an operator [3]) of q .B.s . , if the lifted graph of T

Go(T) = {(x,y) e X x Y, T(x + Xo) = Y + Yo}

Received April 14, 1994. 1991 MR Subject Classification: 46B99. Keywo~ds: Quotient Banach space, strict morphism. *Project supported by the National Natural Science Foundation of C.hina.

247

248 Z H A N G H A I T A O

is a Banach subspace of X x Y. A morphism is called strict if there is a bounded linear operator To from X to Y such that To(Xo) C Y0, and T is induced by To, i.e. To(x) ÷ Yo = T(x + Xo) for each x E X.

2.3 D e f i n i t i o n [2]. A Banach space X is said to have the lifting property if, for any Banach space II, Z and any bounded linear operators S : Y --* Z, T : X --* Z with S surjective, there is an operator T1 : X --* Y such tha t T = S o T1.

2.4 T h e o r e m [2]. A Banach space X has the lifting property iff X is isomorphic to an 11 space.

2.5 T h e o r e m [6]. Any morphism from a q.B.s. X / X o is strict if X is isomorphic to an 11 space.

3. The Example

3.1 D e f i n i t i o n . A q.B.s. X / X o is said to have the strict lifting property if any morphism from X / X o to a q.B.s. Y/Yo is a strict morphism.

Lemma 3.2. A Banach space X has the lifting property iff the (quotient) Banach space X/{0} has the strict lifting property.

Proof. By Theorem 2.5, if X has the lifting property, then X is isomorphic to an 11 space, and any q.B.s. X / X o has the strict lifting property.

Conversely, suppose X/{0} has the strict lifting property. Let Y, Z be two Banach spaces and S : Y -~ Z, T : X --* Z be bounded linear operators with S surjective. Since S is surjective, it induces an isomorphism S : Y / N ( S ) --~ Z defined by S(y + N(S)) = S(y), y e Y, where N(S) is the null space of S. Let To: X/{0} --* Y / N ( S ) be defined by To(x) = :~- l (T(x)) , x E X. By the assumption there is a bounded linear operator T1 from X to Y such tha t Tl(x) + N(S) = To(x), x E X. Therefore S(TI(x)) = S (S - I (T(x ) ) ) = T(x), x e X . The proof is completed. []

3.3 T h e o r e m . There ex/st a q.B.s. X / X o and a morphism T of X / X o such that T is not strict.

Proof. Let X1 be a Banach space not isomorphic to an 11 space. By Theorem 2.4 there exist a (quotient) Banach space Y/Yo (like Y / N ( S ) of the proof of Lemma 3.2) and an operator T1 : X1 --* Y/Yo that can not be lifted (i.e. T1 is not strict).

Let X = X1 x Y, X0 -- {0} x Y0 and T : X / X o --* X / X o be

(0 T = T1 '

then T is a morphism of q.B.s. X/Xo , but T is not strict. []

EXISTENCE OF A NON-STRICT MORPHISM 249

R e f e r e n c e s

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Department of Applied Mathematics, Zhejiang University, Hangzhou 310027.