CC BY
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Vladikavkaz Mathematical Journal 2009, Vol. 11, No 2, pp. 27-30
UDC 517.98
WEAKLY COMPACT-FRIENDLY OPERATORS1
M. Çaglar, T. Misirlioglu
To §afak Alpay, on the occasion of his sixtieth birthday
We introduce weak compact-friendliness as an extension of compact-friendliness, and and prove that if a non-zero weakly compact-friendly operator B : E ^ E on a Banach lattice is quasi-nilpotent at some non-zero positive vector, then B has a non-trivial closed invariant ideal. Relevant facts related to compact-friendliness are also discussed.
Mathematics Subject Classification (2000): primary 47A15.
Key words: invariant subspace, positive operator, weakly compact-friendly, locally quasi-nilpotent.
Introduction
Throughout the paper all operators will be assumed to be one-to-one and to have dense range, since otherwise the kernel in the former case and the closure of the range in the latter case of the corresponding operator is the required non-trivial closed invariant subspace for it.
The letters X and Y denote infinite-dimensional Banach spaces while the letters E and F infinite-dimensional Banach lattices. As usual, L(X, Y) stands for the algebra of all bounded linear operators between X and Y, and L(X) := L(X, X). An operator P £ L(X, Y) is said to be a quasi-affinity if P is one-to-one and has dense range. The operators S £ L(X) and T £ L(Y) are quasi-similar, denoted by S ~ T, if there exist quasi-affinities P £ L(X, Y) and Q £ L(Y, X) such that TP = PS and QT = SQ. The notion of quasi-similarity was first introduced by Sz.-Nagy and Foias in connection with their work on the harmonic analysis of operators on Hilbert space [6]. Note that quasi-similarity is an equivalence relation on the class of all operators. For more about this concept, see [4, 5].
An operator T on E is said to be dominated by a positive operator B on E, denoted by T — B, provided |Tx| ^ B(|x|) for each x £ E. An operator which is dominated by a multiple of the identity operator is called a central operator. A positive operator B on E is said to be compact-friendly [1] if there exist three non-zero operators R, K, and C on E with R, K positive and K compact such that R and B commute, and C is dominated by both R and K. It is worth mentioning that the notion of compact-friendliness is of substance only on infinite-dimensional Banach lattices, since every positive operator on a finite-dimensional Banach lattice is compact. Also, if B is compact, letting R = K = C = B in the definition, it is seen that compact operators are compact-friendly, but the converse is not true as the identity
© 2009 gaglar M., Misirlioglu T.
1The authors would like to thank the referee for valuable comments.
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Qaglar M, Misirlioglu T.
operator on an infinite-dimensional space shows. Lastly, it is straightforward to observe that any power of a compact-friendly operator is also compact-friendly. A fairly complete treatment of compact-friendly operators is given in [1, 2].
For a positive operator B on a Banach lattice E, the super right-commutant [B) and the super left-commutant (B] are defined, respectively, by
[B) := {A G L(E)+ | AB - BA ^ 0} and (B] := {A G L(E)+ | AB - BA < 0}.
For all unexplained notation and terminology, we refer to [1, 2].
In analogy with the notions of quasi-similarity and compact-friendliness, we define the concepts of weak positive quasi-similarity and weak compact-friendliness.
Definition 1.1. Two positive operators B G L(E) and T G L(F) are weakly positively quasi-similar, denoted by B ~ T, if there exist positive quasi-affinities P G L(F, E) and Q G L(E, F) such that BP ^ PT and TQ ^ QB.
Definition 1.2. A positive operator B G L(E) is called weakly compact-friendly if there exist three non-zero operators R, K, and C on E with R, K positive and K compact such that R G [B), and C is dominated by both R and K.
The following simple but important fact is the basic structural property of weak positive quasi-similarity, and for the sake of completeness, we include it in this section.
Theorem 1.3. w is an equivalence relation on the class of all positive operators.
< Taking P = Q = I, it is readily seen that w is reflexive. As for the symmetry, the relation B ~ T brings the positive quasi-affinities P and Q such that BP ^ PT and TQ ^ QB; hence, taking Po := Q and Qo := P, one sees that TPo ^ PoB and BQo ^ QoT, i. e., T w B. To see that ~ is transitive, assume that B ~ T and T ~ S, whence BPi ^ P\T and TQi ^ QiB, and TP2 ^ P2S and SQ2 ^ Q2T for some positive quasi-affinities Pi,Qi (i = 1, 2). Then one gets BP1P2 < P1TP2 < P1P2S and SQ2Q1 < Q2TQi < Q2Q1B. Thus, P3 := P1P2 and Q3 := Q2Q1 are the required positive quasi-affinities to conclude that BP3 ^ P3S and SQ3 < Q3B, i. e., B w S. >
Main Results
We begin with the following fact, which will be used in the sequel and which is also of interest in itself.
Theorem 2.1. Let B and T be two positive operators on a Banach lattice E. If B is weakly compact-friendly and is weakly positively quasi-similar to T, then T is also weakly compact-friendly.
< Since T w B, there exist quasi-affinities P and Q such that BP ^ PT and TQ ^ QB. As B is compact-friendly, there exist three non-zero operators R, K, and C on E with R, K positive and K compact such that R G [B), C R, and C K. Therefore, it follows from BR < RB that BRP < RBP < RPT, which implies QBRP < QRPT, which in turn implies that T(QRP) ^ QBRP ^ (QRP)T. On the other hand, the dominations C - R and C — K yield QCP — QRP and QCP — QKP, respectively. It is, thus, enough to take R1 := QRP, K1 := QKP, and C1 := QCP as the required three operators for the weak compact-friendliness of T. >
Remark 2.2. Theorem 2.1 shows that, although quasi-similarity doesn't preserve compactness of an operator as shown by T.B. Hoover in [4, p. 683], weak positive quasi-similarity preserves weak compact-friendliness.
Weakly Compact-Friendly Operators
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Theorem 2.3. If a non-zero weakly compact-friendly operator B : E ^ E on a Banach lattice is quasi-nilpotent at some xo > 0, then B has a non-trivial closed invariant ideal. Moreover, for each sequence (Tn)ngN in [B) there exists a non-trivial closed ideal that is invariant under B and under each Tn.
< The proof follows the same lines of thought with the proof of [1, Theorem 10.55], and therefore we will only outline it. Assume, without loss of generality, that ||B|| < 1, and take arbitrary scalars an > 0 that are small enough so that the positive operator T := ^^^ anTn exists and ||B + T|| < 1. It then follows that T £ [B), and by the same reasoning, the operator (B + T)n also belongs to [B) for each n £ N, whence the positive operator A := o(B + T)n is an element of [B), too. For each x > 0, denote by Jx the principal ideal generated by Ax, and observe that Jx is a non-zero ideal which is (B + T)-invariant. It then follows that Jx is invariant under B and T, and under each Tn. Hence, provided that Jx = E for some x > 0, the ideal Jx is the sought one which is invariant under B and under each Tn. Thus, one can assume that Jx = E for each x > 0, which is the same as saying that Ax is a quasi-interior point in E for each x > 0. Since B is weakly compact-friendly, one can fix three non-zero operators R, K, C : E ^ E with R, K positive and K compact such that
R £ [B), C x R, and C X K.
The rest of the proof uses verbatim the arguments in [1, Theorem 10.55]. >
The following corollaries are then immediate.
Corollary 2.4. Let B and T be two positive operators on a Banach lattice E such that B is weakly compact-friendly and T is locally quasi-nilpotent at a non-zero positive element of E .If B ~ T, then T has a non-trivial closed invariant subspace.
< By virtue of Theorem 2.1, the locally quasi-nilpotent operator T is also weakly compact-friendly, and hence it has a non-trivial closed invariant subspace by Theorem 2.3. >
Corollary 2.5. Let T be a locally quasi-nilpotent operator such that there exists a nonzero positive operator in [T) dominated by a compact operator. Then T has a non-trivial closed invariant ideal.
< Being also weakly-compact friendly by Theorem 2.1, the locally quasi-nilpotent operator T has a non-trivial closed invariant ideal by Theorem 2.3. >
Lemma 2.6. If a positive operator B is weakly positively-quasi similar to a compact operator, then [B) contains a compact operator.
< Let B w K with K compact, so that there exist positive quasi-affinities P and Q such that BP < PK and KQ < QB. Thus, BPKQ < PK2Q < PKQB, i. e., the compact operator Ko := PKQ belongs to [B). >
Theorem 2.7. Let T be a locally quasi-nilpotent positive operator which is weakly positively quasi-similar to a compact operator. Then T has a non-trivial closed invariant subspace.
< By Lemma 2.6, [T) contains a compact operator, which is dominated by itself. Hence, Corollary 2.5 yields the existence of a T-invariant subspace. >
References
1. Abramovich Y. A., Aliprantis C. D. An Invitation to Operator Theory // Amer. Math. Soc.—Rhode Island: Providence, 2002.—Vol. 50 (Graduate Studies in Mathematics).
2. Abramovich Y. A., Aliprantis C. D., Burkinshaw O. The invariant subspace problem: some recent advances // Workshop on Measure Theory and Real Analysis.—Grado, 1995; Rend. Inst. Mat. Univ. Trieste.—1998.—Vol. 29.—P. 3-79.
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3. Aliprantis C. D., Burkinshaw O. Positive operators.—The Netherlands: Springer, 2006.—376 p.
4. Hoover T. B. Quasi-similarity of operators // Illinois J. Math.—1972.—Vol. 16.—P. 678-686.
5. Laursen K. B., Neumann M. M. An Introduction to local spectral theory. London Mathematical Society Monographs. New ser. 20.—Oxford: Clarendon Press, 2000.—577 p.
6. Sz.-Nagy B., Foias C. Harmonic analysis of operators on Hilbert space.—New York: American Elsevier Publishing Company, Inc., 1970.—389 p.
Received March 1, 2009. Mert Çaglar
Department of Mathematics and Computer Science, Istanbul Kültür University, Assistant Prof. Bakirkoy 34156, Istanbul, Turkey E-mail: m.caglar@iku.edu.tr
Tunç Misirlioglu Department of Mathematics, IIstanbul University, Assistant Prof. Vezneciler 34134, Istanbul, Turkey E-mail: tmisirli@istanbul.edu.tr
