- Email: [email protected]
More on positive commutators
J. Math. Anal. Appl. 373 (2011) 580–584
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
More on positive commutators Roman Drnovšek a , Marko Kandic´ b,∗ a b
Department of Mathematics, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia Department of Mathematics, Institute of Mathematics, Physics and Mechanics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia
a r t i c l e
i n f o
a b s t r a c t Let A and B be positive operators on a Banach lattice E such that the commutator C = A B − B A is also positive. The paper continues the investigation of the spectral properties of C initiated in J. Braˇciˇc et al. (in press) [3]. If the sum A + B is a Riesz operator and the commutator C is a power compact operator, then C is a quasi-nilpotent operator having a triangularizing chain of closed ideals of E. If we assume that the operator A is compact and the commutator AC − C A is positive, the operator C is quasi-nilpotent as well. We also show that the commutator C is not invertible provided the resolvent set of C is connected. © 2010 Elsevier Inc. All rights reserved.
Article history: Received 23 June 2010 Available online 7 August 2010 Submitted by Richard M. Aron Keywords: Banach lattices Positive operators Commutators Spectrum Compact operators
1. Introduction We first recall some relevant definitions and results. Let X be a Banach space. A subspace of X is non-trivial if it is different from {0} and X . The spectrum and the spectral radius of a bounded operator T on X are denoted by σ ( T ) and r ( T ), respectively. A bounded operator T is said to be power compact if T n is a compact operator for some n ∈ N, and it is called a Riesz operator or an essentially quasi-nilpotent operator if {0} is the essential spectrum of T . A chain C is a family of closed subspaces of X that is totally ordered by inclusion. We say that C is a complete chain if it is closed under arbitrary intersections and closed linear spans. If M is in a complete chain C , then the predecessor M− of M in C is defined as the closed linear span of all proper subspaces of M belonging to C . (In fact, the linear span is the union of these subspaces.) Note that the predecessor is an ideal if the chain consists of ideals. We will need the following form of Ringrose’s theorem (see [3, Theorem 1.5] or [10, Theorem 7.2.7]). Theorem 1.1. Let K be a compact operator on a Banach space X , and let C be a complete chain of closed subspaces invariant under K . For each M ∈ C , define K M to be the quotient operator on M/M− induced by K . Then
σ ( K ) \ { 0} =
σ ( K M ) \ {0}.
M∈C
Let E be a real or complex Banach lattice. An operator T on E is called positive if the positive cone E + is invariant under T . It is well known that every positive operator T is bounded and that r ( T ) belongs to σ ( T ). A bounded operator T on E is said to be ideal-reducible if there exists a non-trivial closed ideal of E invariant under T . Otherwise, it is ideal-irreducible. If the chain C of closed ideals of E is maximal in the lattice of all closed ideals of E and if every its member is invariant under an operator T on E, then C is called a triangularizing chain for T , and T is said
*
Corresponding author. ´ E-mail addresses: [email protected] (R. Drnovšek), [email protected] (M. Kandic).
0022-247X/$ – see front matter doi:10.1016/j.jmaa.2010.07.056
©
2010 Elsevier Inc. All rights reserved.
R. Drnovšek, M. Kandi´c / J. Math. Anal. Appl. 373 (2011) 580–584
581
to be an ideal-triangularizable. Note that such a chain is also maximal in the lattice of all closed subspaces of E (see e.g. [5, Proposition 1.2]). In [3] positive commutators of positive operators on Banach lattices are studied. Let us recall the main result [3, Theorem 2.2]. Theorem 1.2. Let A and B be positive compact operators on a Banach lattice E such that the commutator C = A B − B A is also positive. Then C is an ideal-triangularizable quasi-nilpotent operator. Moreover, it belongs to the radical of the Banach algebra generated by A and B. If, in addition, at least one of the operators A and B is ideal-irreducible, then C = 0. In this paper we continue this investigation by proving several generalizations and giving partial answers to some questions posed in [3]. We conclude this section by proving an easy consequence of the famous de Pagter’s theorem (see [1, Theorem 9.19] or [9]). Theorem 1.3. Every quasi-nilpotent power compact positive operator T on a Banach lattice E is ideal-triangularizable. Proof. We may assume that T = 0. By the ideal-triangularization lemma (see [6]), it suffices to show that T is idealreducible. Since T is power compact, there exists the smallest positive integer n such that T n is a compact operator. If T n = 0, then T commutes with a non-zero quasi-nilpotent compact positive operator T n , and so an application of [1, Theorem 9.19] shows that T is ideal-reducible. If T n = 0, then the null ideal {x ∈ E: T n−1 |x| = 0} of T n−1 is a non-trivial closed ideal invariant under T . 2 2. Auxiliary results Lemma 2.1. Let E be an ordered vector space with a generating positive cone E + . Let A and B be positive operators on E satisfying A B B A. Then, for each n ∈ N,
( A + B )n
n n k =0 n
k
A n−k B k ,
(1)
B n A n ( B A ) ( A B )n A n B n
(2)
( A B − B A )n ( A B )n − ( B A )n An B n − B n An .
(3)
and
Proof. The inequalities (1) and (2) can be proved easily by induction, and so we omit their proofs. Denoting C = A B − B A we have
( A B )n = ( B A + C )n ( B A )n + C n , which yields the left-hand inequality of (3), while the right-hand inequality follows from (2).
2
Lemma 2.2. Let A and B be power compact positive operators on a Banach lattice E with A B B A. Then the operator A + B is also power compact. If A and B are nilpotent, then A + B is also nilpotent and the nilpotency index of A + B is at most the sum of nilpotency indices of A and B. Proof. By the assumption, there exist m, n ∈ N such that A m and B n are compact operators. Then the operator
S=
m +n
m+n
k =0
k
A m+n−k B k
is compact, since for k n the operator B k is compact, while for k < n the operator A m+n−k is compact. Since ( A + B )m+n S by (1), the operator ( A + B )3(m+n) is compact by the Aliprantis–Burkinshaw theorem [2, Theorem 5.14]. This shows that the operator A + B is power compact. If A m = 0 and B n = 0, then S = 0, so that ( A + B )m+n = 0. Also, this implies that the nilpotency index of A + B is at most the sum of nilpotency indices of A and B. 2 Lemma 2.3. Let A and B be positive operators on a Banach lattice E with A B B A. If either of them is power compact, then A B − B A is also a power compact operator. If either of them is nilpotent, then A B − B A is also nilpotent with nilpotency index at most the minimum of nilpotency indices of operators A and B.
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Proof. Suppose that either A n or B n is a compact operator for some n ∈ N. Then the positive operator ( A B − B A )n is dominated by the compact operator A n B n − B n A n by (3). It follows from the Aliprantis–Burkinshaw theorem [2, Theorem 5.14] that the operator ( A B − B A )3n is compact, so that the operator A B − B A is power compact. In the special case when either A n = 0 or B n = 0 we obtain from (3) that ( A B − B A )n = 0, which implies that the nilpotency index of A B − B A is at most the minimum of nilpotency indices of operators A and B. 2 3. On quasi-nilpotency of positive commutators We start with the following general result. Proposition 3.1. Let A and B be positive operators on a Banach lattice E such that A B B A. Then
r ( A B − B A ) r ( A )r ( B ).
(4)
In particular, if one of the operators A and B is quasi-nilpotent, then so is the commutator A B − B A. Proof. By the inequality (3) we have
( A B − B A )n An B n − B n An An B n , so that
( A B − B A )n An B n for all n ∈ N. Taking the n-th roots and letting n go to infinity we obtain the inequality (4).
2
Remark. Proposition 3.1 shows that the commutator in [3, Example 3.3] is necessarily quasi-nilpotent. Remark. The inequality (4) follows easily from the submultiplicativity of the spectral radius that was shown in [13] in more general setting (see also [14]). Indeed, since 0 A B − B A A B, we have r ( A B − B A ) r ( A B ) r ( A )r ( B ). The next theorem gives sufficient conditions under which a positive commutator of positive operators must be 0. Theorem 3.2. Let A and B be positive operators on a Banach lattice E with A ideal-irreducible. Suppose that either A B B A or A B B A. If there exists a non-zero positive operator P such that A P = P A = P , then A B = B A. This condition is satisfied if the operator A is mean ergodic with a non-zero mean ergodic projection P , i.e., the sequence { n1 ( I + A + A 2 + A 3 + · · · + A n−1 )}n∈N converges strongly to a non-zero operator P , or, in particular, if the sequence { A n }n∈N converges strongly to a non-zero operator. Proof. We observe that
P ( A B − B A ) P = P A B P − P B A P = P B P − P B P = 0. We only prove the case when the commutator A B − B A is positive, since the other case can be shown similarly. Since A P = P A = P , the null ideal {x ∈ E: P |x| = 0} of P and the range ideal { y ∈ E: ∃x ∈ E + such that | y | P x} of P are both invariant under A. Since P = 0, the first one must be trivial and the second one must be dense in E. So, it follows from P ( A B − B A ) P = 0 first that ( A B − B A ) P = 0, and then that A B − B A = 0. This completes the proof in this case. For the proof of the addition we note that the strong limit P of the sequence { n1 ( I + A + A 2 + A 3 + · · · + A n−1 )}n∈N satisfies the equalities A P = P A = P . Also, it is well known that this limit exists and it is equal to the strong limit of the sequence { A n }n∈N whenever the latter exists. 2 Corollary 3.3. Let A and B be positive operators on a Banach lattice E with A ideal-irreducible. Suppose that either A B B A or A B B A. If r ( A ) > 0 and r ( A ) is a pole of the resolvent of A, then A B = B A. In particular, if A is power compact, then A B = B A. Proof. Assume that r ( A ) > 0 is a pole of the resolvent of A. Multiplying A by an appropriate scalar if necessary, we can assume that r ( A ) = 1. By [11, Theorem V.5.2 and Corollary], the number 1 is an eigenvalue of A of algebraic multiplicity one, and the corresponding eigenspace is spanned by a quasi-interior point u ∈ E + . Furthermore, there exists a non-zero f (x) positive functional f on E such that A ∗ f = f . Define the operator P on E by P x = f (u ) u. Note that f (u ) > 0, as the vector u is a quasi-interior point and the positive functional f is non-zero. Then P is a positive projection such that A P = P A = P . So, we can apply Theorem 3.2.
R. Drnovšek, M. Kandi´c / J. Math. Anal. Appl. 373 (2011) 580–584
583
If A is power compact, then r ( A ) > 0 by [1, Corollary 9.21], as A is ideal-irreducible. Therefore, the last assertion is a special case. 2 The following theorem and its corollary extend (the essential part of) Theorem 1.2. Theorem 3.4. Let A and B be positive operators on a Banach lattice E such that the sum S = A + B is a Riesz operator. If the commutator C = A B − B A is a power compact positive operator, then it is an ideal-triangularizable quasi-nilpotent operator. Proof. Note that C = S B − B S. Let C be a chain (of closed ideals) that is maximal in the lattice of all closed ideals invariant under S. By maximality, this chain is complete. Since S A 0 and S B 0, every member of C is also invariant under A and B. Fix any M ∈ C with M− = M. It follows from [4, Theorems 3.21 and 3.23] that the quotient operator S M is a positive Riesz operator on M/M− . By maximality of the chain C , it is also ideal-irreducible. If S M is quasi-nilpotent, then so is C M = S M B M − B M S M by the inequality (4). If r ( S M ) > 0, then r ( S M ) is a pole of the resolvent of S M , so that C M = S M B M − B M S M = 0 by Corollary 3.3. Therefore, the operator C M is quasi-nilpotent in both cases. Since C is n power compact, there exists n ∈ N such that the operator C n is compact. Therefore, C M = (C M )n is a quasi-nilpotent compact operator. By Theorem 1.1, we conclude that C n is a quasi-nilpotent operator, and so C is quasi-nilpotent as well. By Theorem 1.3, it has a triangularizing chain of closed ideals of E. 2 Corollary 3.5. Let A and B be positive power compact operators on a Banach lattice E such that A B B A. Then A B − B A is an ideal-triangularizable quasi-nilpotent operator. Proof. By Lemmas 2.2 and 2.3 the operators A + B and A B − B A are power compact. Since power compact operators are Riesz operators, we finish the proof applying Theorem 3.4. 2 The following result can be viewed as a Banach lattice analogue of the famous Kleinecke–Shirokov theorem. It also gives a partial answer to the open question (1) in [3] whether it is enough to assume in Theorem 1.2 that only one of the operators A and B is compact. Theorem 3.6. Let A and B be bounded operators on a Banach lattice E with A compact and positive. Suppose that the commutator C = A B − B A is also positive. If either AC C A or AC C A, then C is an ideal-triangularizable quasi-nilpotent operator. Proof. Since the operator A is compact, the positive operator T = A + C is also compact. Let C be a chain of closed ideals that is maximal in the lattice of all closed ideals invariant under T . By maximality, this chain is complete. Since T C 0 and T A 0, every member of C is also invariant under A and C . For each M ∈ C with M− = M, the quotient operator T M on M/M− induced by T is a positive compact operator. By maximality of the chain C , the operator T M is also ideal-irreducible. Since the commutator T M C M − C M T M = A M C M − C M A M is either positive or negative, we can apply Theorem 1.2 for the operators T M and C M to obtain that T M C M − C M T M = 0, and so A M C M − C M A M = 0. Then the operator C M is quasi-nilpotent by the Kleinecke–Shirokov theorem (see e.g. [7]). By Theorem 1.1, we conclude that C is a quasi-nilpotent operator. It is also ideal-triangularizable by Theorem 1.3. 2 4. On invertibility of positive commutators The following theorem is inspired by Wielandt’s proof of the result that 0 is the only scalar commutator in a normed algebra (see e.g. [7]). Theorem 4.1. Let A, B and T be positive operators on a Banach lattice E such that T A B − B A and either T A AT or T A AT . If T is invertible, then the inverse is not a positive operator. Proof. We prove only the case when T A AT , as the opposite case can be proved similarly. Let us show by induction that we have, for each n ∈ N,
A n B − B A n nT A n−1 .
(5)
Since this is trivially true for n = 1, we assume this inequality holds for some n. Then
A n +1 B − B A n +1 = A n ( A B − B A ) + A n B − B A n A
An T + nT An−1 A (n + 1) T An .
584
R. Drnovšek, M. Kandi´c / J. Math. Anal. Appl. 373 (2011) 580–584
Suppose now that T is an invertible operator having a positive inverse. It follows from (5) that A is not nilpotent. Indeed, if n would be the nilpotency index of A, then A n = 0 and T A n−1 = 0 which is not possible by (5). The inequality (5) also implies that
T −1 A n B − B A n n A n −1 , and so
n A n−1 T −1 A n B − B A n 2 T −1 A n−1 A B .
(6)
Since A is not nilpotent, we conclude that
n 2 T −1 A B for all n ∈ N. This contradiction completes the proof of the theorem.
2
The open question (3) in [3] asks whether a positive commutator of positive operators can be invertible. We now show that the commutator is not invertible provided its resolvent set is connected. Theorem 4.2. Let A and B be bounded operators on a Banach lattice E such that A B B A 0. Then the number 0 cannot belong to the unbounded connected component of the resolvent set of the commutator C = A B − B A. In particular, if the resolvent set of C is connected, then the operator C is not invertible. Proof. Assume the contrary. By Zhang’s theorem (see [12] or [8]), there exist a > 0 and n ∈ N such that C n aI , where I denotes the identity operator on E. Therefore, we have
( A B )n = ( B A + C )n ( B A )n + C n ( B A )n + aI , and so
n
r( A B )
n = r ( A B )n r ( B A )n + aI = r ( B A )n + a = r ( B A ) + a.
Since r ( A B ) = r ( B A ), this contradiction completes the proof of the theorem.
2
Acknowledgments The authors were supported by the Slovenian Research Agency. They also thank Aljoša Peperko for a useful comment.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
Y.A. Abramovich, C.D. Aliprantis, An Invitation to Operator Theory, American Mathematical Society, Providence, 2002. C.D. Aliprantis, O. Burkinshaw, Positive Operators, reprint of the 1985 original, Springer, Dordrecht, 2006. J. Braˇciˇc, R. Drnovšek, Y.B. Farforovskaya, E.L. Rabkin, J. Zemánek, On positive commutators, Positivity, doi:10.1007/s11117-009-0028-1, in press. H.R. Dowson, Spectral Theory of Linear Operators, London Math. Soc. Monogr. Ser., vol. 12, Academic Press, London, New York, 1978. R. Drnovšek, Triangularizing semigroups of positive operators on an atomic normed Riesz space, Proc. Edinb. Math. Soc. (2) 43 (1) (2000) 43–55. ´ Ideal-triangularizability of semigroups of positive operators, Integral Equations Operator Theory 64 (4) (2009) 539–552. R. Drnovšek, M. Kandic, P.R. Halmos, A Hilbert Space Problem Book, second ed., Grad. Texts in Math., vol. 19, Springer-Verlag, New York, Berlin, 1982. P. Meyer-Nieberg, Banach Lattices, Universitext, Springer-Verlag, Berlin, 1991. B. de Pagter, Irreducible compact operators, Math. Z. 192 (1) (1986) 149–153. H. Radjavi, P. Rosenthal, Simultaneous Triangularization, Springer-Verlag, New York, 2000. H.H. Schaefer, Banach Lattices and Positive Operators, Grundlehren Math. Wiss., vol. 215, Springer-Verlag, New York, Heidelberg, 1974. X. Zhang, On spectral properties of positive operators, Indag. Math. (N.S.) 4 (1) (1993) 111–127. M. Zima, On the local spectral radius in partially ordered Banach spaces, Czechoslovak Math. J. 49 (4) (1999) 835–841. M. Zima, Positive Operators in Banach Spaces and Their Applications, Wydawnictwo Uniwersytetu Rzeszowskiego, Rzeszów, 2005.
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
More on positive commutators Roman Drnovšek a , Marko Kandic´ b,∗ a b
Department of Mathematics, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia Department of Mathematics, Institute of Mathematics, Physics and Mechanics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia
a r t i c l e
i n f o
a b s t r a c t Let A and B be positive operators on a Banach lattice E such that the commutator C = A B − B A is also positive. The paper continues the investigation of the spectral properties of C initiated in J. Braˇciˇc et al. (in press) [3]. If the sum A + B is a Riesz operator and the commutator C is a power compact operator, then C is a quasi-nilpotent operator having a triangularizing chain of closed ideals of E. If we assume that the operator A is compact and the commutator AC − C A is positive, the operator C is quasi-nilpotent as well. We also show that the commutator C is not invertible provided the resolvent set of C is connected. © 2010 Elsevier Inc. All rights reserved.
Article history: Received 23 June 2010 Available online 7 August 2010 Submitted by Richard M. Aron Keywords: Banach lattices Positive operators Commutators Spectrum Compact operators
1. Introduction We first recall some relevant definitions and results. Let X be a Banach space. A subspace of X is non-trivial if it is different from {0} and X . The spectrum and the spectral radius of a bounded operator T on X are denoted by σ ( T ) and r ( T ), respectively. A bounded operator T is said to be power compact if T n is a compact operator for some n ∈ N, and it is called a Riesz operator or an essentially quasi-nilpotent operator if {0} is the essential spectrum of T . A chain C is a family of closed subspaces of X that is totally ordered by inclusion. We say that C is a complete chain if it is closed under arbitrary intersections and closed linear spans. If M is in a complete chain C , then the predecessor M− of M in C is defined as the closed linear span of all proper subspaces of M belonging to C . (In fact, the linear span is the union of these subspaces.) Note that the predecessor is an ideal if the chain consists of ideals. We will need the following form of Ringrose’s theorem (see [3, Theorem 1.5] or [10, Theorem 7.2.7]). Theorem 1.1. Let K be a compact operator on a Banach space X , and let C be a complete chain of closed subspaces invariant under K . For each M ∈ C , define K M to be the quotient operator on M/M− induced by K . Then
σ ( K ) \ { 0} =
σ ( K M ) \ {0}.
M∈C
Let E be a real or complex Banach lattice. An operator T on E is called positive if the positive cone E + is invariant under T . It is well known that every positive operator T is bounded and that r ( T ) belongs to σ ( T ). A bounded operator T on E is said to be ideal-reducible if there exists a non-trivial closed ideal of E invariant under T . Otherwise, it is ideal-irreducible. If the chain C of closed ideals of E is maximal in the lattice of all closed ideals of E and if every its member is invariant under an operator T on E, then C is called a triangularizing chain for T , and T is said
*
Corresponding author. ´ E-mail addresses: [email protected] (R. Drnovšek), [email protected] (M. Kandic).
0022-247X/$ – see front matter doi:10.1016/j.jmaa.2010.07.056
©
2010 Elsevier Inc. All rights reserved.
R. Drnovšek, M. Kandi´c / J. Math. Anal. Appl. 373 (2011) 580–584
581
to be an ideal-triangularizable. Note that such a chain is also maximal in the lattice of all closed subspaces of E (see e.g. [5, Proposition 1.2]). In [3] positive commutators of positive operators on Banach lattices are studied. Let us recall the main result [3, Theorem 2.2]. Theorem 1.2. Let A and B be positive compact operators on a Banach lattice E such that the commutator C = A B − B A is also positive. Then C is an ideal-triangularizable quasi-nilpotent operator. Moreover, it belongs to the radical of the Banach algebra generated by A and B. If, in addition, at least one of the operators A and B is ideal-irreducible, then C = 0. In this paper we continue this investigation by proving several generalizations and giving partial answers to some questions posed in [3]. We conclude this section by proving an easy consequence of the famous de Pagter’s theorem (see [1, Theorem 9.19] or [9]). Theorem 1.3. Every quasi-nilpotent power compact positive operator T on a Banach lattice E is ideal-triangularizable. Proof. We may assume that T = 0. By the ideal-triangularization lemma (see [6]), it suffices to show that T is idealreducible. Since T is power compact, there exists the smallest positive integer n such that T n is a compact operator. If T n = 0, then T commutes with a non-zero quasi-nilpotent compact positive operator T n , and so an application of [1, Theorem 9.19] shows that T is ideal-reducible. If T n = 0, then the null ideal {x ∈ E: T n−1 |x| = 0} of T n−1 is a non-trivial closed ideal invariant under T . 2 2. Auxiliary results Lemma 2.1. Let E be an ordered vector space with a generating positive cone E + . Let A and B be positive operators on E satisfying A B B A. Then, for each n ∈ N,
( A + B )n
n n k =0 n
k
A n−k B k ,
(1)
B n A n ( B A ) ( A B )n A n B n
(2)
( A B − B A )n ( A B )n − ( B A )n An B n − B n An .
(3)
and
Proof. The inequalities (1) and (2) can be proved easily by induction, and so we omit their proofs. Denoting C = A B − B A we have
( A B )n = ( B A + C )n ( B A )n + C n , which yields the left-hand inequality of (3), while the right-hand inequality follows from (2).
2
Lemma 2.2. Let A and B be power compact positive operators on a Banach lattice E with A B B A. Then the operator A + B is also power compact. If A and B are nilpotent, then A + B is also nilpotent and the nilpotency index of A + B is at most the sum of nilpotency indices of A and B. Proof. By the assumption, there exist m, n ∈ N such that A m and B n are compact operators. Then the operator
S=
m +n
m+n
k =0
k
A m+n−k B k
is compact, since for k n the operator B k is compact, while for k < n the operator A m+n−k is compact. Since ( A + B )m+n S by (1), the operator ( A + B )3(m+n) is compact by the Aliprantis–Burkinshaw theorem [2, Theorem 5.14]. This shows that the operator A + B is power compact. If A m = 0 and B n = 0, then S = 0, so that ( A + B )m+n = 0. Also, this implies that the nilpotency index of A + B is at most the sum of nilpotency indices of A and B. 2 Lemma 2.3. Let A and B be positive operators on a Banach lattice E with A B B A. If either of them is power compact, then A B − B A is also a power compact operator. If either of them is nilpotent, then A B − B A is also nilpotent with nilpotency index at most the minimum of nilpotency indices of operators A and B.
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R. Drnovšek, M. Kandi´c / J. Math. Anal. Appl. 373 (2011) 580–584
Proof. Suppose that either A n or B n is a compact operator for some n ∈ N. Then the positive operator ( A B − B A )n is dominated by the compact operator A n B n − B n A n by (3). It follows from the Aliprantis–Burkinshaw theorem [2, Theorem 5.14] that the operator ( A B − B A )3n is compact, so that the operator A B − B A is power compact. In the special case when either A n = 0 or B n = 0 we obtain from (3) that ( A B − B A )n = 0, which implies that the nilpotency index of A B − B A is at most the minimum of nilpotency indices of operators A and B. 2 3. On quasi-nilpotency of positive commutators We start with the following general result. Proposition 3.1. Let A and B be positive operators on a Banach lattice E such that A B B A. Then
r ( A B − B A ) r ( A )r ( B ).
(4)
In particular, if one of the operators A and B is quasi-nilpotent, then so is the commutator A B − B A. Proof. By the inequality (3) we have
( A B − B A )n An B n − B n An An B n , so that
( A B − B A )n An B n for all n ∈ N. Taking the n-th roots and letting n go to infinity we obtain the inequality (4).
2
Remark. Proposition 3.1 shows that the commutator in [3, Example 3.3] is necessarily quasi-nilpotent. Remark. The inequality (4) follows easily from the submultiplicativity of the spectral radius that was shown in [13] in more general setting (see also [14]). Indeed, since 0 A B − B A A B, we have r ( A B − B A ) r ( A B ) r ( A )r ( B ). The next theorem gives sufficient conditions under which a positive commutator of positive operators must be 0. Theorem 3.2. Let A and B be positive operators on a Banach lattice E with A ideal-irreducible. Suppose that either A B B A or A B B A. If there exists a non-zero positive operator P such that A P = P A = P , then A B = B A. This condition is satisfied if the operator A is mean ergodic with a non-zero mean ergodic projection P , i.e., the sequence { n1 ( I + A + A 2 + A 3 + · · · + A n−1 )}n∈N converges strongly to a non-zero operator P , or, in particular, if the sequence { A n }n∈N converges strongly to a non-zero operator. Proof. We observe that
P ( A B − B A ) P = P A B P − P B A P = P B P − P B P = 0. We only prove the case when the commutator A B − B A is positive, since the other case can be shown similarly. Since A P = P A = P , the null ideal {x ∈ E: P |x| = 0} of P and the range ideal { y ∈ E: ∃x ∈ E + such that | y | P x} of P are both invariant under A. Since P = 0, the first one must be trivial and the second one must be dense in E. So, it follows from P ( A B − B A ) P = 0 first that ( A B − B A ) P = 0, and then that A B − B A = 0. This completes the proof in this case. For the proof of the addition we note that the strong limit P of the sequence { n1 ( I + A + A 2 + A 3 + · · · + A n−1 )}n∈N satisfies the equalities A P = P A = P . Also, it is well known that this limit exists and it is equal to the strong limit of the sequence { A n }n∈N whenever the latter exists. 2 Corollary 3.3. Let A and B be positive operators on a Banach lattice E with A ideal-irreducible. Suppose that either A B B A or A B B A. If r ( A ) > 0 and r ( A ) is a pole of the resolvent of A, then A B = B A. In particular, if A is power compact, then A B = B A. Proof. Assume that r ( A ) > 0 is a pole of the resolvent of A. Multiplying A by an appropriate scalar if necessary, we can assume that r ( A ) = 1. By [11, Theorem V.5.2 and Corollary], the number 1 is an eigenvalue of A of algebraic multiplicity one, and the corresponding eigenspace is spanned by a quasi-interior point u ∈ E + . Furthermore, there exists a non-zero f (x) positive functional f on E such that A ∗ f = f . Define the operator P on E by P x = f (u ) u. Note that f (u ) > 0, as the vector u is a quasi-interior point and the positive functional f is non-zero. Then P is a positive projection such that A P = P A = P . So, we can apply Theorem 3.2.
R. Drnovšek, M. Kandi´c / J. Math. Anal. Appl. 373 (2011) 580–584
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If A is power compact, then r ( A ) > 0 by [1, Corollary 9.21], as A is ideal-irreducible. Therefore, the last assertion is a special case. 2 The following theorem and its corollary extend (the essential part of) Theorem 1.2. Theorem 3.4. Let A and B be positive operators on a Banach lattice E such that the sum S = A + B is a Riesz operator. If the commutator C = A B − B A is a power compact positive operator, then it is an ideal-triangularizable quasi-nilpotent operator. Proof. Note that C = S B − B S. Let C be a chain (of closed ideals) that is maximal in the lattice of all closed ideals invariant under S. By maximality, this chain is complete. Since S A 0 and S B 0, every member of C is also invariant under A and B. Fix any M ∈ C with M− = M. It follows from [4, Theorems 3.21 and 3.23] that the quotient operator S M is a positive Riesz operator on M/M− . By maximality of the chain C , it is also ideal-irreducible. If S M is quasi-nilpotent, then so is C M = S M B M − B M S M by the inequality (4). If r ( S M ) > 0, then r ( S M ) is a pole of the resolvent of S M , so that C M = S M B M − B M S M = 0 by Corollary 3.3. Therefore, the operator C M is quasi-nilpotent in both cases. Since C is n power compact, there exists n ∈ N such that the operator C n is compact. Therefore, C M = (C M )n is a quasi-nilpotent compact operator. By Theorem 1.1, we conclude that C n is a quasi-nilpotent operator, and so C is quasi-nilpotent as well. By Theorem 1.3, it has a triangularizing chain of closed ideals of E. 2 Corollary 3.5. Let A and B be positive power compact operators on a Banach lattice E such that A B B A. Then A B − B A is an ideal-triangularizable quasi-nilpotent operator. Proof. By Lemmas 2.2 and 2.3 the operators A + B and A B − B A are power compact. Since power compact operators are Riesz operators, we finish the proof applying Theorem 3.4. 2 The following result can be viewed as a Banach lattice analogue of the famous Kleinecke–Shirokov theorem. It also gives a partial answer to the open question (1) in [3] whether it is enough to assume in Theorem 1.2 that only one of the operators A and B is compact. Theorem 3.6. Let A and B be bounded operators on a Banach lattice E with A compact and positive. Suppose that the commutator C = A B − B A is also positive. If either AC C A or AC C A, then C is an ideal-triangularizable quasi-nilpotent operator. Proof. Since the operator A is compact, the positive operator T = A + C is also compact. Let C be a chain of closed ideals that is maximal in the lattice of all closed ideals invariant under T . By maximality, this chain is complete. Since T C 0 and T A 0, every member of C is also invariant under A and C . For each M ∈ C with M− = M, the quotient operator T M on M/M− induced by T is a positive compact operator. By maximality of the chain C , the operator T M is also ideal-irreducible. Since the commutator T M C M − C M T M = A M C M − C M A M is either positive or negative, we can apply Theorem 1.2 for the operators T M and C M to obtain that T M C M − C M T M = 0, and so A M C M − C M A M = 0. Then the operator C M is quasi-nilpotent by the Kleinecke–Shirokov theorem (see e.g. [7]). By Theorem 1.1, we conclude that C is a quasi-nilpotent operator. It is also ideal-triangularizable by Theorem 1.3. 2 4. On invertibility of positive commutators The following theorem is inspired by Wielandt’s proof of the result that 0 is the only scalar commutator in a normed algebra (see e.g. [7]). Theorem 4.1. Let A, B and T be positive operators on a Banach lattice E such that T A B − B A and either T A AT or T A AT . If T is invertible, then the inverse is not a positive operator. Proof. We prove only the case when T A AT , as the opposite case can be proved similarly. Let us show by induction that we have, for each n ∈ N,
A n B − B A n nT A n−1 .
(5)
Since this is trivially true for n = 1, we assume this inequality holds for some n. Then
A n +1 B − B A n +1 = A n ( A B − B A ) + A n B − B A n A
An T + nT An−1 A (n + 1) T An .
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Suppose now that T is an invertible operator having a positive inverse. It follows from (5) that A is not nilpotent. Indeed, if n would be the nilpotency index of A, then A n = 0 and T A n−1 = 0 which is not possible by (5). The inequality (5) also implies that
T −1 A n B − B A n n A n −1 , and so
n A n−1 T −1 A n B − B A n 2 T −1 A n−1 A B .
(6)
Since A is not nilpotent, we conclude that
n 2 T −1 A B for all n ∈ N. This contradiction completes the proof of the theorem.
2
The open question (3) in [3] asks whether a positive commutator of positive operators can be invertible. We now show that the commutator is not invertible provided its resolvent set is connected. Theorem 4.2. Let A and B be bounded operators on a Banach lattice E such that A B B A 0. Then the number 0 cannot belong to the unbounded connected component of the resolvent set of the commutator C = A B − B A. In particular, if the resolvent set of C is connected, then the operator C is not invertible. Proof. Assume the contrary. By Zhang’s theorem (see [12] or [8]), there exist a > 0 and n ∈ N such that C n aI , where I denotes the identity operator on E. Therefore, we have
( A B )n = ( B A + C )n ( B A )n + C n ( B A )n + aI , and so
n
r( A B )
n = r ( A B )n r ( B A )n + aI = r ( B A )n + a = r ( B A ) + a.
Since r ( A B ) = r ( B A ), this contradiction completes the proof of the theorem.
2
Acknowledgments The authors were supported by the Slovenian Research Agency. They also thank Aljoša Peperko for a useful comment.
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