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Владикавказский математический журнал Октябрь-декабрь, 2005, Том 7, Выпуск 4
YffK 517.98
SOME OPEN QUESTIONS ON POSITIVE OPERATORS IN BANACH LATTICES
E. Yu. Emel'yanov
Dedicated to S. S. Kutateladze on occasion of his 60th birthday
Recently, some new results on asymptotic behaviour of positive operators in Banach lattices were obtained. Here we discuss some open problems related to these results.
1. Mean ergodicity and weak almost periodicity of positive operators
1.1. Given an operator T in a Banach space X, denote the Cesaro averages of T by
1 r-1 AT := 1E Tk-
k=0
An operator T is called Cesaro bounded if supn ||AT|| < to, 'power bounded if sup{||Tn|| : n £ N} < to, and doubly 'power bounded if sup{||Tn|| : n £ Z} < to. There exist operators which are Cesaro bounded but not power bounded. An operator T is called mean ergodic if the norm limit lim Anx exists for all x £ X. Each mean ergodic operator T is Cesaro bounded and
satisfies
-1
lim \\n-1Tnx\\ =0 (x G X). (1)
n^rx
An operator T is called weakly almost 'periodic if its orbit {Tnx : n ^ 0} is conditionally weakly compact for all x. It follows from Eberlein's mean ergodic theorem that any weakly almost periodic operator is mean ergodic. For an elementary introduction to the ergodic theorems on operators in Banach spaces, we refer to Krengel [13] and to Lyubich [14].
Below we will use also basic results and terminology from the Banach lattice theory. We refer to [2] and [15] for them.
1.2. The first interesting question arises from the following simple assertion: given a bounded operator T in a Banach space, if Tm is mean ergodic for some natural m, then T is mean ergodic. This fact follows immediately from the equality
m— 1
Aim = mE TkATnm (n,m £ N), m k=0
and from Eberlein's mean ergodic theorem.
© 2005 Emel'yanov E. Yu.
In general, the converse of the assertion above is not true, even for positive contractive operators on a Banach lattice C(K), for appropriate compact Hausdorff space K. Corresponding example was constructed in Sine's paper [19]. However, for a positive power bounded operator S on a Banach lattice with order continuous norm, from the mean ergodicity of S follows the mean ergodicity of Sm for all m g N, as was shown by Derriennic and Krengel in [4]. The construction of example in [19] is quite technical, and cannot be used directly in any C(K). This motivates the following question:
Open question 1. Let K be an infinite compact Hausdorff space. Is there a positive power bounded mean ergodic operator T in C (K) such that Tm is not mean ergodic for some m g N?
1.3. Another interesting question is related to Sine's mean ergodic theorem [18]:
Let T be a Cesaro bounded operator which satisfies (1). Then T is mean ergodic if and only if the space Fix(T) of fixed vectors of T separates the space Fix(T*) of fixed vectors of T *.
What can be said in the case of positive operators in Banach lattices? More precisely:
Open question 2. Let T be a positive Cesaro bounded operator in a Banach lattice such that (1) is satisfied and positive fixed vectors of T separates Fix(T*). Is T mean ergodic?
1.4. It was shown by Komornik [11] and by Kornfeld and Lin [12] that any mean ergodic Markov operator on L1-space is weakly almost periodic. In general this result is not true for contractive positive operators in Banach lattices. To show this we can take the operator T on C(K) constructed in [19]. This operator is mean ergodic but not weakly almost periodic, since T2 is not mean ergodic.
Open question 3. Let T be a mean ergodic positive power bounded operator on a Banach lattice with order continuous norm. Is T weakly almost periodic?
Even for the Banach lattice co, the answer seems to be unknown. In the case if the answer for Co is negative, the following question arises:
Open question 4. Let T be a mean ergodic positive power bounded operator on a KB -space. Is T weakly almost periodic? Does this property of positive power bounded operators characteriz KB-spaces among Banach lattices?
1.5. Any closed ball in a reflexive Banach space is weakly compact. From this fact and Eberlein's mean ergodic theorem it follows that every power bounded operator in a reflexive Banach space is mean ergodic. Is the converse true? This still open question was suggested by Sucheston [20]. The problem was solved in positive for Banach lattices in [6, 8, 16, 21]; and, for Banach spaces with a basis, in [9]. Moreover, in [21] it was shown that every a-Dedekind complete Banach lattice in which every positive power bounded operator is mean ergodic is reflexive. The following question is open.
Open question 5. Let E be a Banach lattice such that every positive power bounded operator on E is mean ergodic. Is E reflexive?
1.6. We finish this section with the following problem. It was proved by Neubrander [5], that any power bounded positive operator T on an AL-space X such that 0 belongs to the weak-closure of the orbit |Tnx}^=i for each x g X satisfies a formally more strong condition:
w- lim Tnx = 0 (x G X).
It is well known fact that in general this is not true for power bounded operators in Banach spaces. However it is an interesting open question to extend this property on positive operators in Banach lattices:
Open question 6. Let T be a positive power bounded operator in a Banach lattice which satisfies 0 £ w-cl{Tnx}^=1 for each x £ X. Does w- lim Tnx = 0 hold for all x £ X?
2. Positive operator whose spectrum is singleton
2.1. Let T be a positive operator on E = Rn such that a(T) = {1}. Using the elementary linear algebra one can show that T ^ Ie where Ie is the identity operator on E. This simple result leads to the following natural question:
Open question 7. Let T be a positive operator on an infinite-dimensional ordered Banach space E such that a(T) = {1}. Does T satisfy T ^ Ie?
This question is probably one of the several most intrigue questions in operator theory like the invariant subspace problem for operators on separable Hilbert space, because of there is no infinite-dimensional ordered Banach space for which this question is solved in positive or negative.
2.2. For Banach lattices, Open question 7 was solved in positive in the class of lattice homomorphisms by Arendt, Schaefer, and Wolff [17]. For this and related results, we refer to Meyer-Nieberg's book [15]. It is still unknown if this true for other interesting classes of positive operators on Banach lattices. Sometimes, it is better to consider the following versions of this problem:
Open question 8. Let T be a contractive (i. e., ||T|| ^ 1) positive operator on a Banach lattice E such that a(T) = {1}. Does T satisfy T = Ie ?
Open question 9. Let T be a Markov operator on a C(K) such that a(T) = {1}. Does T satisfy T = Ic(k)?
2.3. There are another interesting open question, closely related to ones discussed above. A Banach lattice E is called a Banach lattice algebra if, in addition to linear and lattice operations, the operation of multiplication (associative, but not necessary commutative) is defined on E and satisfies:
a ^ 0,6 ^ 0 ^ a ■ b ^ 0, ||a ■ b|| ^ ||a||||b||.
The second condition means that E is a Banach algebra. We assume also that E as a Banach algebra possesses a positive unit element. Usually, Banach lattice algebras are considered over R, though this definition can be modified for the complex case.
The famous theorem due to Gelfand and Mazur says that if in a Banach algebra every non-zero element is invertible then this algebra is isometrically isomorphic to C. Despite the Gelfand-Mazur's theorem is known for the long time, its analogue for Banach lattice algebras is a long standing open question.
Open question 10. Let E be a Banach lattice algebra in which every nonzero positive element has inverse. Is E isomorphic to R?
This problem was solved in positive only for some special classes of Banach lattice algebras. As far as the author knows, the answer is positive for commutative algebras and for algebras in which every nonzero positive element has positive inverse. The last result is due to Huijsmans [10].
3. Invariant subspace problem for positive operators
3.1. The classical form of the invariant subspace problem is the following: does every bounded operator in a separable Hilbert space possess a non-trivial invariant subspace? This problem is still open, however it was solved in positive for many interesting classes of operators in Hilbert spaces. More general such a question can be asked for bounded operators in separable Banach spaces. It was proved by Enflo that the invariant subspace problem for Banach spaces has a negative solution. It is interesting that examples of operators without nontrivial invariant subspaces exist in a such well studied Banach lattice as l1. But all operators in such examples are non-positive. This leads to the following question:
Open question 11. Does any positive operator in a separable Banach lattice possess a non-trivial invariant subspace?
For discussion of this and many related problems we refer to the book [2].
3.1. It was shown by Abramovich, Aliprantis, and Burkinshaw (1993) that every locally quasi-nilpotent positive operator in lp, for 1 ^ p < to, has a non-trivial invariant subspace. This motivates the following two problems:
Open question 12. Does every (locally) quasi-nilpotent positive operator in a separable Banach lattice possess a non-trivial invariant subspace?
Open question 13. Does every positive operator in lp, for 1 ^ p < to, possess a nontrivial invariant subspace?
4. The renorming problem for positive operators
4.1. Let T be a doubly power bounded operator on a Banach space X. Then there exists an equivalent norm on X under which T is a surjective isometry. We can define such a norm || ■ ||t as follows:
\\x\\t := sup{||Tnx|| : n g Z} (x g X). (2)
However, the situation is different if we consider a positive operator T on a Banach lattice. The norm in (2) may not be a Banach lattice norm, since we do not know if T-1 ^ 0. If this occurs, then it is well known that T is a lattice automorphism and this will make the norm defined in (2) an equivalent lattice norm.
It is known that the positivity of T-1 occurs for any doubly power bounded positive operator in a finite-dimensional Banach lattice. On the other hand due to Abramovich's theorem [1] the positivity of T-1 occurs also if T is a surjective positive isometry. An example of a positive doubly power bounded operator T in L1(R) such that T-1 ^ 0 was constructed in [7]. Recently [3] this result was extended to all AL-spaces. But the technique which was used in [7] and [3] cannot be applied to other types of Banach lattices. This motivates the following question:
Open question 14. Let E be an infinite-dimensional Banach lattice. Is there a positive doubly power bounded operator T in E such that T-1 is not positive?
4.2. It follows from the above-mentioned result of [3] that in a dual infinite dimensional C(K)-space there is a positive doubly power bounded operator with non-positive inverse. However, not all C(K)-spaces are dual.
Open question 15. Let K be an infinite compact Hausdorff space. Is there a positive doubly power bounded operator T in C(K) such that T-1 is not positive?
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Received by the editors August 6, 2005.
Prüf. Emel'yanov Eduard Yu. Turke, Ankaray, Department of Mathematics Middle East Technical University, 06531 E-mail: [email protected]
