Lipschitz Continuity of the Solution Mapping of Symmetric Cone Complementarity Problems

This paper investigates the Lipschitz continuity of the solution mapping of symmetric cone (cid:2) linear or nonlinear (cid:3) complementarity problems (cid:2) SCLCP or SCCP, resp. (cid:3) over Euclidean Jordan algebras. We show that if the transformation has uniform Cartesian P -property, then the solution mapping of the SCCP is Lipschitz continuous. Moreover, we establish that the monotonicity of mapping and the Lipschitz continuity of solutions of the SCLCP imply ultra P -property, which is a concept recently developed for linear transformations on Euclidean Jordan algebra. For a Lyapunov transformation, we prove that the strong monotonicity property, the ultra P -property, the Cartesian P -property, and the Lipschitz continuity of the solutions are all equivalent to each other. Abstract These classes of symmetric cone complementarity problems provide a uniﬁed framework for the linear or nonlinear complementarity problems (cid:2) LCP or NCP, resp. (cid:3) over the nonnegative orthant cone in R n , that is, V (cid:5) R n and K (cid:5) R n (cid:4) (cid:2) see (cid:6) 1–4 (cid:7)(cid:3) , the second-order cone (cid:2) linear or nonlinear (cid:3) complementarity problems (cid:2) SOCLCP or SOCCP, resp. (cid:3) , that is, V (cid:5) R n and K (cid:5) K n (cid:2) see (cid:6) 5–8 (cid:7)(cid:3) , and the semideﬁnite (cid:2) linear or nonlinear (cid:3) complementarity problems (cid:2) SDLCP or SDCP, resp. (cid:3) , that is, V (cid:5) S n and K (cid:5) S n (cid:4) (cid:2) see (cid:6) 9–12 (cid:7)(cid:3) . It is also known that the complementarity problem is special case of variational inequality problem which has a wide range of applications, see (cid:6) 3, 9 (cid:7) .


Introduction
we use V for short in subsequent content be a Euclidean Jordan algebra and K be the symmetric cone in V. Given a continuous transformation F : V → V and q ∈ V, the symmetric cone complementarity problem denoted by SCCP F, K, q is to find a vector x ∈ V such that x ∈ K, F x q ∈ K, x, F x q 0. 1.1 When F reduces to a linear transformation L, the above problem is called the symmetric cone linear complementarity problem and is denoted by SCLCP L, K, q , that is, the symmetric cone linear complementarity problem is to find a vector x ∈ V such that x ∈ K, L x q ∈ K, x, L x q 0. 1.2 These classes of symmetric cone complementarity problems provide a unified framework for the linear or nonlinear complementarity problems LCP or NCP, resp.over the nonnegative orthant cone in R n , that is, V R n and K R n see 1-4 , the second-order cone linear or nonlinear complementarity problems SOCLCP or SOCCP, resp., that is, V R n and K K n see 5-8 , and the semidefinite linear or nonlinear complementarity problems SDLCP or SDCP, resp., that is, V S n and K S n see 9-12 .It is also known that the complementarity problem is special case of variational inequality problem which has a wide range of applications, see 3, 9 .
One of the important issues in complementarity problems is to characterize the Lipschitz continuity of its solutions or called the Lipschitz continuity of solution mapping with respect to q.For q ∈ V, let φ F q be the set of all solutions to SCCP F, K, q .Then, we intend to know under what conditions the multivalued solution mapping φ F : q → φ F q of SCCP F, K, q is Lipschitz continuous.In other words, under what conditions, there will exist κ > 0 such that φ F q 1 ⊆ φ F q 2 κ q 1 − q 2 B 1.3 for all q 1 , q 2 ∈ V satisfying φ F q 1 / ∅ and φ F q 2 / ∅, where B is the closed unit ball in V.That is, if x 1 ∈ φ F q 1 there exists x 2 ∈ φ F q 2 such that Note that the Lipschitz constant κ depends only on the continuous transformation F. Below is a brief history regarding this issue.For LCP M, q , it is well known that the Lipschitz continuity of the solution mapping with respect to q ∈ V can be described in any one of the following ways: i the matrix M is P -matrix see 13, 14 ; ii LCP M, q has a unique solution for all q ∈ R n i.e., GUS-property of M ; iii for any q ∈ R n , the solution set φ M q / ∅ and the set-valued mapping q → φ M q are Lipschitzian.
In particular, Mangasarian and Shiau 14 showed that if M is a P -matrix, then solutions of linear inequalities, programs, and LCP are Lipschitz continuous.Murthy et al. 15 showed that M is a P -matrix if and only if the LCP M, q has a solution for all q ∈ R n and the solution mapping is Lipschitzian.Gowda and Sznajder 16 generalized the above result to affine variational inequality problems, while Yen 17 studied Lipschitz continuity of the solution mapping of variational inequalities with a parametric polyhedral constraint.As for NCP, Levy 18 obtained that the solution mapping is locally single-valued and Lipschitz continuous under suitable conditions.How about when K is nonpolyhedral?Balaji et al. 19 proved that L being monotone and the Lipschitz continuity of the solution mapping of SDLCP imply the GUS-property, while Chen and Qi in 9 employed Cartesian P -property to guarantee the GUS-property and the locally Lipschitzian property of the solution mapping of SDLCP.These make a complete extension of i -iii to their counterparts in SDLCP.A natural question arises here: can the above results be extended to a general symmetric cone case which is a unified framework?
In fact, there has been some papers dealing with the SCLCP over Euclidean Jordan algebras.For example, Balaji 20 established the result that if L has the Lipschitzian Qproperty, then L has the positive principal minor property.Gowda et al. 21 showed that if L has P -property, then SCLCP L, K, q has a nonempty compact set for all q ∈ V.In addition, Tao and Gowda 22 used degree-theoretic arguments to show that under a certain R 0 -type condition, every P 0 symmetric cone nonlinear complementarity problem SCCP F, K, q has a solution.However, it still remains open under what conditions the solution map φ F : q → φ F q of SCCP F, K, q is Lipschitz continuous.In this paper, we explore new results regarding Lipschitz continuity of the solution mapping of the SCLCP L, K, q or SCCP F, K, q over Euclidean Jordan algebras.In Theorem 3.1, we show that if the transformation F has the uniform Cartesian P -property with modulus ρ > 0, then the solution mapping φ F is Lipschitz continuous with respect to q ∈ V.Meanwhile, we give examples to show that the solution mapping of nonstrong monotone SCLCP L, K, q is not Lipschitz continuous with respect to q, and GUS-property does not imply Lipschitz continuity of the solution mapping.
On the other hand, various P -properties and GUS-property have been investigated in the literature 4, 9, 10, 13, 16, 19, 21-24 .Relations among them are well studied as well.In 19, Theorem 2.2 , it is proved that if the linear transformation L in SDLCP has the monotonicity property and φ L is Lipschitzian, then L has the P 2 -property and the GUS-property.The concept of P 2 -property in S n was extended to a general Euclidean Jordan algebra, called ultra P -property 23 .Hence, it is desirable to know whether 19, Theorem 2.2 can be true or not in SCLCP L, K, q if P 2 -property is replaced by ultra P -property.In this paper, we answer this question positively, see Theorem 3.8.Further, for the Lyapunov transformation L a , we present several equivalent conditions for the ultra P -property of L a .
Next are a few words about notations and some basic concepts employed.For a vector x ∈ V, the norm is denoted by x : x, x , where •, • denotes the Euclidean inner product.For the Euclidean Jordan algebra V, let L V denote the set of all continuous linear transformation L : V → V, and Aut K denote the set of all invertible linear transformations Γ : V → V such that Γ K K.For the convex set K, let int K denote the interior of the K. L T means the adjoint operator of L. The identical transformation on V will be denoted by I.For the SCCP F, K, q , the solution set of SCCP F, K, q is denoted by φ F q .For the SCLCP L, K, q , the solution set of SCLCP L, K, q is denoted by SOL L, K, q or φ L q .

Preliminaries
In this section, we briefly recall some basic concepts and background materials in Euclidean Jordan algebras, which will be used in the subsequent analysis.More details can be found in 21-23, 25 .An Euclidean Jordan algebra is a triple V, •, •, • V for short , where V is a finitedimensional inner product over R and x, y → x • y : V × V → V is a bilinear mapping satisfying the following three conditions: We call x • y the Jordan product of x and y.In addition, if there is an element e ∈ V such that x • e x for all x ∈ V, the element e is called the identity element in V.In a given Euclidean Jordan algebra V, the set of squares K : {x 2 : x ∈ V} is a symmetric cone 25, Theorem III.2.1 .In other words, K is a self-dual closed convex cone, and, for any two elements x, y ∈ int K , there exists an invertible linear transformation Γ : V → V such that Γ x y and Γ K K.For any x ∈ V, we write nonzero and cannot be written as a sum of two nonzero idempotents.We say that a finite set {e 1 , e 2 , . . ., e r } of primitive idempotents in V is a Jordan frame if where r is called the rank of V. Now, we recall the spectral and Peirce decompositions of an element x in V.

2.3
Here, the numbers λ i for i 1, . . ., r are the eigenvalues of x and the expression λ 1 e 1 • • • λ r e r is the spectral decomposition (or the spectral expansion) of x.
In an Euclidean Jordan algebra V, corresponding to the convex cone K, let Π K denote the metric projection onto K, namely, for an x ∈ V, x * Π K x if and only if x * ∈ K and x−x * ≤ x−y for all y ∈ K.It is well known that x * is unique.For any x ∈ V, combining the spectral decomposition of x with the metric projection of x onto K, we have the expression of metric projection Π K x as follows see 21 : The Peirce Decomposition Fix a Jordan frame {e 1 , e 2 , . . ., e r } in an Euclidean Jordan algebra V.For i, j ∈ {1, 2, . . ., r}, we define the following eigenspaces: x x • e j for i / j.

2.5
Theorem 2.2 see 25, Theorem IV.2.1 .The space V is the orthogonal direct sum of spaces V ij i ≤ j .Furthermore,

2.6
Hence, given any Jordan frame {e 1 , e 2 , . . ., e r }, we can write any element x ∈ V as where Next, we recall concept of Lyapunov transformation and its relevant conclusions which will be used in our analysis later.In an Euclidean Jordan algebra V, for any x ∈ V, we define the corresponding Lyapunov transformation L x : V → V by L x z x • z for any z ∈ V.As remarked in 21, page 209 , traditionally, the notation L x has been used the Lyapunov transformation 25 .As employed in 21 , we also reserve the notation L x for the Lyapunov transformation and write L x to denote the image of an element x ∈ V under a linear transformation L : V → V. We say that elements x and y operator commute if L x L y L y L x .It is well known that x and y operator commute if and only if x and y have their spectral decompositions with respect to a common Jordan frame 25, Lemma X.2.2 .
Moreover, in this case, elements x and y operator commute.That is, x and y have their spectral decompositions with respect to a common Jordan frame.
In fact, from Property 1 and definition of 1.1 , it can be seen that SCCP F, K, q is equivalent to find a x ∈ V such that In addition, if x is a solution of SCCP F, K, q , then x and F x q operator commute.Now, we review various monotonicity and P -property for a continuous transformation F : V → V.
Definition 2.3.Let V be an Euclidean Jordan algebra.A continuous transformation F : It is said to have d GUS-property if SCCP F, K, q has a unique solution for any q ∈ V; e P -property if Remark 2.4.i When F is linear, strict monotonicity and strong monotonicity coincide.When F is nonlinear, strong monotonicity implies strict monotonicity.
ii Whether F is linear or nonlinear, we have the following implications 22-24 : strong monotonicity ⇒ strict monotonicity ⇒ P -property ⇒ Q-property, strong monotonicity ⇒ GUS-property ⇒ P -property.

2.11
iii When V R n and K R n , GUS-property and P -property coincide.But, once V and K are the other cases, for example, V R n and K K n , where K n denotes the secondorder cone, or V S n and K S n , and so forth.GUS-property is not equivalent to P -property.
Given an Euclidean Jordan algebra V with dim V n > 1, from 25, Proposition III 4.4-4.5 and Theorem V.3.7 , we know that any Euclidean Jordan algebra V and its corresponding symmetric cone K are, in a unique way, a direct sum of simple Euclidean Jordan algebras and the constituent symmetric cone therein, respectively, that is, where every V i is a simple Euclidean Jordan algebra which cannot be direct sum of two Euclidean Jordan algebras with the corresponding symmetric cone K i for i 1, . . ., m, and

2.13
Through the above description and Cartesian P -properties proposed by Chen and Qi 9 in the setting of semidefinite matrices, Kong et al. 26 introduced the concept of uniform Cartesian P -property for the general transformation F in the setting of Euclidean Jordan algebra.This concept is used to study the Lipschitz continuity of the solution mapping in SCCP.Definition 2.5.Consider a linear or nonlinear transformation F : V → V. We say that F has the uniform Cartesian P -property if for any x, y ∈ V and x / y, there exist an index ν ∈ {1, 2, . . ., m} and a scalar ρ > 0 such that

2.14
Remark 2.6.It is easy to observe that when m 1, the uniform Cartesian P -property becomes the strong monotonicity of transformation F. If m n and V R n , it becomes the P -property in the context of NCP.
When the continuous transformation F : V → V is linear i.e., F L , we will introduce another concept, the ultra P -property of L, which is a new concept recently developed for linear transformations on Euclidean Jordan algebra.In fact, the ultra Pproperty is an equivalently straightforward extension of P 2 -property in the setting of the semidefinite matrices 23 .Since P 2 -property involves the ordinary associative product of three square matrices and there may not have an associative triple product in an Euclidean Jordan algebra, for this reason, P 2 -property cannot be extended in a natural way to an Euclidean Jordan algebra 23 .However, the P 2 -property is introduced in Euclidean Jordan algebra using the concepts of principal subtransformation and cone automorphisms of V 23 .
Given a Jordan frame {e 1 , e 2 , . . ., e r } in Euclidean Jordan algebra V, we define

2.15
It is known that V l is a subalgebra of V with rank l, see 25, Proposition IV.1.1 .By means of Peirce decomposition, we have the following representation 21 :

2.16
Let P l denote the orthogonal projection from V onto V l .For a linear transformation L :

2.17
We call L l a principal subtransformation of L. The determinant of L l is called a principal minor of L.
Definition 2.7 see 23 .Consider a linear transformation L : V → V. We say that L has the ultra P -property if for any Γ ∈ Aut K , every principal subtransformation of L Γ T LΓ has the P -property.

Main Results
In this section, we first give several sufficient conditions for the Lipschitz continuity of the solution mapping φ L in the SCLCP L, K, q .For the classical LCP and SDLCP, the Lipschitz continuity results have been studied in 9, 13, 14, 19 .Along this direction, we generalize them to general SCCP F, K, q case where a weaker condition, uniform Cartesian P -property, is used.Furthermore, we also establish relationship between the Lipschitz continuity of the solution mapping and the ultra P -property.
Theorem 3.1.Let F : V → V be a continuous linear or nonlinear transformation.If F has the uniform Cartesian P -property, then φ F is Lipschitz continuous.
Proof.Suppose that F has uniform Cartesian P -property.From 26, Theorem 6.2 , we know that for any q ∈ V, the problem 1.1 has a unique solution, that is, φ F q is a single point set.Thus, we let {x} φ F q 1 and {y} φ F q 2 for any q 1 , q 2 ∈ V.If x y, the inequality x − y ≤ κ q 1 − q 2 is obvious, where κ > 0. If x / y, from definition of uniform Cartesian P -property, there exists an index ν ∈ {1, . . ., m} such that

3.1
where the third equality follows from x ν , F x q 1 ν 0 y ν , F y q 2 ν because x and y are the solution of the problem 1.1 for q 1 , q 2 ∈ V, respectively.The second inequality is due to x ν , y ν , F x q 1 ν , and F y q 2 ν ∈ K ν .This implies that ρ x − y ≤ q 1 − q 2 .Letting κ 1/ρ gives x − y ≤ κ q 1 − q 2 .Hence, φ F is Lipschitzian.Remark 3.2.In Theorem 3.1, if the transformation F is linear, the condition of uniform Cartesian P -property reduces to the Cartesian P -property 26 .However, if we weaken the condition of uniform Cartesian P -property to the monotonicity for the linear transformation L, the conclusion of Theorem 3.1 is not true.The following example shows that the monotonicity property is not sufficient to conclude that the φ L is Lipschitz continuous with respect to q ∈ V.
Example 3.3.Let L : R 3 → R 3 be defined as It is obvious that L has the monotonicity property.It can be seen that SOL L, K 3 , e {0}, where K 3 ⊂ R 3 is a second-order cone, and e is identity element in Euclidean Jordan algebra R 3 .Moreover, it is easy to verify that α, 0, 0 T : 0 < α ∈ R ⊆ SOL L, K 3 , 0 .

3.3
It is an unbounded solution set.However, if the solution mapping φ L of SCLCP L, K 3 , 0 is Lipschitz continuous, then SOL L, K 3 , 0 must be a bounded set, which is clearly a contradiction.
Kong et al. 26 proved that the strong monotonicity implies the uniform Cartesian Pproperty whether the transformation F is linear or nonlinear.Moreover, when F L is linear transformation, by 21, Theorem 21 , if L is self-adjoint and has P -property, then L is strongly monotone.Hence, we have the following corollary.

Corollary 3.4. Consider Euclidean Jordan algebra
ii self-adjoint and has P -property, or iii P -property and K is polyhedral, then φ L is Lipschitz continuous.
Remark 3.5.Even the transformation F is linear, the condition of uniform Cartesian Pproperty in Theorem 3.1 or strong monotonicity in Corollary 3.4 cannot be weakened to the GUS-property, otherwise the conclusion is not true.Example 4.2 will illustrate this point.
In the following theorem, we prove that if φ L is Lipschitz continuous, then L has the ultra P -property provided the linear transformation L is monotone.To establish another main result of this paper, the following lemmas play important roles.Lemma 3.6.a Suppose that φ L is Lipschitz continuous, and SOL L, K, q {0} for some q 0.Then, SOL L, K, q {0} for all q 0. b If SOL L, K, e {0} and if L has R 0 -property i.e., SOL L, K, 0 {0} , then L has Q-property.
c If φ L is Lipschitz continuous and L has Q-property, then for the every principal subtransformation L l of L, φ L l is the Lipschitz continuous with respect to any Jordan frame of V. b SOL L, K, q {0} for some q 0.
Proof.Part a is from 20, Lemma 6 , while part b is from 20, Lemma 1 .
Theorem 3.8.Let L : V → V be a linear transformation.Suppose L is monotone and the solution mapping φ L of SCLCP L, K, q is Lipschitz continuous.Then, a L has the ultra P -property; b L has the GUS-property.
Proof. a Consider any Jordan frame {e 1 , . . ., e r } of Euclidean Jordan algebra V and the principal subtransformation L l : L {e 1 ,...,e l } : V l → V l , where L Γ T LΓ for any Γ ∈ Aut K .Note that

3.4
Since L is monotone, it follows that the linear transformation L and L l are both monotone.Thus, we have SOL L, K, e {0} and SOL L l , K l , e l {0}, where K l and e l denote the symmetric cone and the identity element in V l , respectively.Furthermore, by direct calculation, it is not hard to prove that the solution mapping φ L of SCLCP L, K, q is Lipschitz continuous if and only if the solution mapping φ L of the corresponding SCLCP is Lipschitz continuous for the linear transformation L. Applying Lemma 3.6 a and b yields that L has Q-property.Then using Lemma 3.6 c , we obtain that the solution mapping φ L l of the corresponding SCLCP is Lipschitz continuous for the linear transformation L l .It follows from SOL L l , K l , e l {0} and Lemma 3.6 a again that L l has Q-property.This together with Lemma 3.7 says that the transformation L l is invertible.
Next, we want to prove that the transformation L l has P -property.Suppose that an element 0 / x ∈ V l operator commute with L l x and x • L l x 0. Since L l is monotone by the above analysis, we have 0 ≤ x, L l x x • L l x , e l ≤ 0, 3.5 which means that L l x • x, e l 0. Together with Property 1, it is easy to verify that L l x • x 0, and L l x and x have the same Jordan frame.Since L l x • x 0, we write where {f 1 , f 2 , . . ., f l } is a Jordan frame in V l , λ i / 0 i 1, . . ., k and 1 ≤ k ≤ l.Let Q k denote the projection operator from V l onto the eigenspace Let T k : Q k L l : W k → W k be the principal subtransformation of L l corresponding to {1, . . ., k}.From the definition of T k , it follows that T k x Q k L l x 0. By the same arguments as above, we know that T k has Q-property, and the solution mapping φ T k of the corresponding SCLCP is Lipschitz continuous for the transformation T k .Hence, from Lemma 3.7, we get that T k is invertible.This together with T k x 0 yields x 0, which gives a contradiction to x / 0. Therefore, we have proved that L has the ultra P -property.
b This is immediate by 23, Theorem 6.2 .
It was shown in 19, Theorem 2.2 that if L : S n → S n is monotone and φ L is Lipschitz continuous, then L has the P 2 -property.Note that P 2 -property in S n is equivalent to the ultra P -property in S n see 23 .Therefore, the result of Theorem 3.8 is a natural extension of 19, Theorem 2.2 to the setting of Euclidean Jordan algebra.

A Special Linear Transformation
In this section, we specialize to a special linear transformation which is studied in the SCLCP setting, see 19, 23 .For a ∈ V, we consider the corresponding Lyapunov transformation L a .We will give several equivalent conditions regarding the ultra P -property of Lyapunov transformation L a .f L a has the ultra P -property; g L a has Q-property and the solution mapping φ L a of the SCLCP L a , K, q is Lipschitz continuous with respect to q ∈ V.
Proof. a ⇒ b For any 0 / x ∈ V , we have x, L a x x, a • x a, x 2 .Since a 0 and x 2 ∈ K, a, x 2 > 0 see b ⇒ h For any linear transformation, the strong monotonicity is equivalent to the strict monotonicity.Then, it follows from Corollary 3.4 that the solution mapping φ L a of the SCLCP L a , K, q is Lipschitz continuous with respect to q ∈ V.Moreover, it is true that the strong monotonicity implies Q-property for any linear transformation, see 21 .Hence, the conclusion of h is obtained.h ⇒ b Suppose that the solution mapping φ L a of the SCLCP L a , K, q is Lipschitz continuous with respect to q ∈ V, and L a has Q-property.Let {e 1 , . . ., e r } be a Jordan frame of V and x r i 1 λ i x e i .Note that x, L a x r i,j 1 λ i x λ j x e i , L a e j r i 1 λ 2 i x e i , L a e i .

4.1
Since L a has the Q-property and the solution map φ L a is Lipschitz continuous, L a e i , e i > 0 see 27, Theorem 3.1 .It follows from 4.1 that L a x , x > 0 for all 0 / x ∈ V. Therefore, the linear transformation L a has the strong monotonicity.The proof is complete.
In general, the above result may fail to hold.The following example shows that φ L is not Lipschitz continuous, but L has the GUS-property.Meanwhile, this example also shows that for Theorem 3.  It is easy to prove that A is positive stable and positive semidefinite, and L A is a linear transformation.From 10, Theorem 9 , we have that L A has GUS-property.On the other hand, since A is not a positive definite matrix, it follows from 19, Theorem 3.3 that φ L A is not Lipschitz continuous.

Concluding Remarks
In this paper, we have studied the Lipschitz continuity of the solution mapping for symmetric cone linear or nonlinear complementarity problems over Euclidean Jordan algebras and provided several sufficient conditions for the Lipschitz continuity of the solution mapping.
We have established the relationship between the Lipschitz continuity of the solution mapping and ultra P -property.Furthermore, for Lyapunov transformation, we have shown that the strong monotonicity property, the ultra P -property, GUS-property, the Lipschitz continuity of the solution mapping, and so forth are all equivalent to each other.

Proof. Please see 20 ,Lemma 3 . 7 .
Lemma 5 for part a , 20, Proposition 3 for part b , and 20, Lemma 4 for part c .If φ L is Lipschitz continuous and L has Q-property, then a the linear transformation L is invertible;

Theorem 4 . 1 .
For the Lyapunov transformation L a a ∈ V , the following statements are equivalent: a a 0; b L a is strongly monotone; c L a has (uniform) Cartesian P -property; d L a has GUS-property; e L a has P -property; 1 and Corollary 3.4, if weaken the condition of strong monotonicity to GUS-property, the conclusions of Theorem 3.1 and Corollary 3.4 are not true.

Example 4 . 2 .
Let V S 2 and K S 2 .For Lyapunov transformation defined by L A X : AX XA T .4.3 25, Proposition I.1.4 .Thus, L a has the strong monotonicity property.b ⇒ c It is straightforward by the definitions.The implication c ⇒ d follows from 26, Theorem 6.2 .d ⇒ e This follows from 21, Theorem 14 .e ⇒ a Suppose that the Lyapunov transformation L a has P -property.Let a r i 1 λ i a e i and I {i : λ i a ≤ 0}, where {e 1 , . . ., e r } is a Jordan frame of V. Note that a 0 if and only if I ∅.Suppose that I / ∅.Let x i∈I e i / 0.Then, x and L a x operator commute, and x • L a x i∈I λ i a e i 0. Therefore, by the P -property of L a , we have x 0 which leads to a 0. b ⇒ f It follows from 23, Theorem 6.1 .f ⇒ e It is obvious.