Asymmetric Putnam-Fuglede Theorem for ( n , k )-Quasi-∗-Paranormal Operators

T ∈ B(H) is said to be (n, k)-quasi-∗-paranormal operator if, for non-negative integers k and n, ‖T∗(Tkx)‖(1+n) ≤ ‖T(1+n)(Tkx)‖‖Tkx‖n; for all x ∈ H. In this paper, the asymmetric Putnam-Fuglede theorem for the pair (A, B) of power-bounded operators is proved when (i) A and B∗ are n-∗-paranormal operators (ii) A is a (n, k)-quasi-∗-paranormal operator with reduced kernel and B∗ is n-∗-paranormal operator. The class of (n, k)-quasi-∗-paranormal operators properly contains the classes of n-∗-paranormal operators, (1, k)-quasi-∗-paranormal operators and k-quasi-∗-class A operators. As a consequence, it is showed that if T is a completely non-normal (n, k)-quasi-∗-paranormal operator for k = 0, 1 such that the defect operator DT is Hilbert-Schmidt class, then T ∈ C10.


Introduction
Throughout this paper, H denotes an infinite dimensional complex Hilbert space with inner product •, • and B(H) denotes the algebra of all bounded linear operators acting on H. Spectrum, point spectrum, residual spectrum, continuous spectrum, and approximate spectrum of an operator T will be denoted by σ(T), σ p (T), σ r (T), σ c (T), σ a (T), respectively.The kernel and the range of an operator T will be denoted by ker T and ran(T) respectively.
An operator T is said to be * -paranormal iff T * x 2 ≤ T 2 x x , for all x ∈ H, or equivalently, T ∈ B(H) is * -paranormal iff T * 2 T 2 − 2λTT * + λ 2 ≥ 0, for all λ > 0. The class of * -paranormal operators was introduced in [1].Another well-known generalization of * -paranormal operators are (n, k)-quasi- * -paranormal operators defined as follows: T is said to be (n, k)-quasi- * -paranormal operator if T * (T k x) (1+n ) ≤ T (1+n) (T k x) T k x n for all x ∈ H and for non-negative integers k and n.An operator T ∈ B(H) is said to be paranormal [2] iff Tx 2 ≤ T 2 x x for all x ∈ H.
The familiar Putnam-Fuglede theorem asserts that if A ∈ B(H) and B ∈ B(H) are normal operators and AX = XB for some X ∈ B(H), then A * X = XB * (see [3]).A simple example of two unilateral shifts shows that this theorem cannot be extended to the class of hyponormal operators.Let us write the Putnam-Fuglede theorem in an asymmetric form: if A ∈ B(H) and B ∈ B(H) are normal operators and AX = XB * for some X ∈ B(H), then A * X = XB.
In this paper, we shall generalize this theorem to certain (n, k)-quasi- * -paranormal operators.The organization of the paper is as follows; in Section 2, we give some properties for (n, k)-quasi- * -paranormal operators needed in the sequel.In Section 3, we present our main theorems to prove that the asymmetric Putnam-Fuglede theorem holds for some power-bounded operators A, B in the following cases: (i) A and B * are n- * -paranormal operators (ii) A is a (n, k)-quasi- * -paranormal operator with reduced kernel and B * are n- *paranormal operator; (iii) A is a n − * -paranormal operator and B * are (n, k)-quasi- * -paranormal operator with reduced kernel (an operator T with reduced kernel means that its kernel is invariant under T * ).
These results extend those recently given in [9,13,14] and as applications of our main theorems, we obtain the following: 1. if T is a (n, k)-quasi- * -paranormal operator with reduced kernel (resp.n- * -paranormal operator or a nquasi- * -class A with reduced kernel), then T has a part in the class C 00 on a stable subspace H 0 and a compression quasi-affine transform to an isometry on the orthogonal complement of H 0 . 2. Next, we prove that if T is completely non-normal (n, k)-quasi- * -paranormal operator; for k = 0, 1 and verifying the defect operator D T is a Hilbert-Schmidt class, then T ∈ C 10 .
This generalizes the results given by Takahashi and Uchiyama [15] for completely non-normal hyponormal contraction operators and those given by Duggal, Jeonb, Kim [13] for the case of completely non-normal * -paranormal contraction operators.
Let us recall some facts about the construction of the limit isometric operator or the g-asymptotic limit associated with a power-bounded operator T (see [16]).Definition 1.A Banach limit or a generalized limit is a bounded linear functional glim on l ∞ (N) (the Banach space of bounded complex sequences) which preserves the ordinary notion of convergence.That is if limx n = x then glim(x n ) = x.
Banach limit may be characterized as those continuous functional which satisfy the following conditions: ).
In the sequel we fix a generalized Banach limit glim on l ∞ (N) for a power-bounded operator T; sup n T n x ≤ ∞, on the Hilbert space H.The following map is a bounded sesquilinear form Since { T n : n ≥ 1} is bounded, then glim T n x = 0 if and only if in f n T n x = 0 and so, this holds if and only if lim n T n x = 0.
We denote by H 0 the kernel of φ T , i.e., H 0 is said the stable subspace for T. It is clear that H 0 is an invariant subspace for any operator in the commutant of T, i.e., H 0 is an hyperinvariant subspace.We recall the following definitions: (i) A power-bounded operator T is said to be of class C 1. if the sequence { T n x : n ∈ N} does not converge to 0 for any non-zero vector x i.e., H 0 (T) = {0}.(ii) T is said to be strongly stable if H 0 (T) = H and we write It follows from Equation (1) of the sesquilinear application φ T that there exists a positive operator A T,g ∈ B(H) such that the equation φ T (x, y) = A T,g x, y holds for all vectors x, y ∈ H.The operator A T,g is said the g-asymptotic limit of T which is usually depends on the particular choice of the generalized limit g.It is well known that ker A T,g = H 0 holds for every Banach limit g and Furthermore, there exists an isometry V on ran(A) such that The concept of asymptotic limit and their generalizations play an important role in the hyperinvariant subspace problem [16,18].Since T * is a power-bounded operator whenever T is, let A * be the strong limit of {T n T * n : n ≥ 1} and let V * be the associated isometry on ran(A * ) so that all the preceding properties hold for T * , A * , V * .(ii) the joint approximate point spectrum, denoted by σ ja (T) is the set of scalars λ for which there exists a normalized sequence {x n } ⊂ H verifying Notice that in general, σ jp (T) ⊂ σ p (T); however, the equality holds for the following operator classes: p-hyponormal or log-hyponormal, absolute- * -k-paranormal.

Lemma 2. Let T ∈ B(H). If T is a normal operator, then
Tx n ≤ T (n) x x n (7) for all x ∈ H and non-negative integer n Proof.We recall from [20] that if A is a positive operator on Hilbert space then for all r > 1 and any unit vector x.
Let T be a normal operator and n ≥ 1, then By the above inequality for all n ≥ 1 and unit vector x.Hence, for x = y/ y , y = 0, we get our result.
Lemma 3. Let T be (n, k)-quasi- * -paranormal operator.If ran T k = H, then T has the following decomposition: Proof.If ran T k = H, then H has the following non-trivial decomposition: H = ran T k ⊕ (ran T k ) ⊥ .Also, it is clear that ran T k is an invariant subspace for T such that T 1 = T| ran T k is n- * -paranormal operator and T k 2 = 0; where T 2 = T |(ran T k ) ⊥ .Hence, T has the triangular matrix form cited above.

Proposition 1. Let T ∈ B(H).
1.If T is a power-bounded n- * -paranormal operator and there is an invariant subspace M for which the restriction T| M = N of T on M is a normal operator, then M reduces T and N = U ⊕ 0 where U is unitary.2. If T is a power-bounded (n, k)-quasi- * -paranormal operator and there is an invariant subspace M for which the restriction T| M = N of T on M is an injective normal operator then M reduces T and N is a unitary operator.In particular, if M = RanT k and T 1 as in the previous Lemma, is normal operator, then RanT k reduces T. Proof.
1. Let T be a power-bounded n- * -paranormal operator and let us consider an invariant subspace M ⊂ H for T such that T| M = N is normal.The operator T has the following matrix decomposition On the one hand, we have that N is a power-bounded normal operator, since N is normaloid it follows that N is a contraction.It is well known that N = U ⊕ N 0 where U is unitary and N 0 is of class C 00 for possible N 0 = 0. On the other hand, Since the operator for all unit vector x ∈ M.
Since the kernel of N reduces N, hence N = N 1 ⊕ 0.
If x ∈ kerN then from (7) we get x ∈ kerR * .Thus For each unit vector x ∈ M kerN we have, ( From Lemma 1, we get for all k ≥ 1, that If N 0 = 0 then Since U is unitary and x = 1 then Since N 0 ∈ C 00 and 1 + N 0 x 2 > 1 then for k → ∞ we get ( R * x 2 + 1) nk → 0. However, R * x 2 + 1 ≥ 1 which is a contradiction.Hence, N 0 = 0 and then N = U on M kerN.
By substituting N 0 = 0 and k = 1 in the inequality (11), we get ( R * x 2 + 1) n = 1.Therefore, R * x = 0 on M kerN and by (8), we have R * = 0 = R. 2. As in the case (1), let us consider an invariant subspace M ⊂ H for T such that T| M = N is normal.The operator T has the following matrix , where N is an injective normal operator and then ranN k = M. Hence, M ⊆ ranN k .Indeed, we have T k x = N k x for all x ∈ M. Since N is normal then ranN k = ranN and by the assumption that N is injective, we get ranN k = M.
Since T is (n, k)-quasi- * -paranormal operator, then Since N is a contraction and y = N k x we get y 2n ≤ 1.Hence, for all y ∈ M = ranN k we get the desired result, by following the same steps as in the proof of the previous assertion (1).
Proof.The proof is a consequence of the Lemma 1 and the properties of the isometric *-homomorphism φ of the Berberian technique.
Therefore, the residual spectrum of A is empty.From the decomposition of the spectrum, we get 5. the last statement follows from Lemma 1 and the assertion (1).
Lemma 5.If T is (n, k)-quasi- * -paranormal and M is an invariant subspace for T, Then T| M is (n, k)-quasi- * -paranormal.
Proof.According to the decomposition H = M ⊕ M ⊥ , then T can be written (19) and for all x ∈ M we have that T k x = A k x and Hence, A is (n, k)-quasi- * -paranormal.

Main Theorems
We are ready to show our main theorems.Definition 4.An operator T ∈ B(H) is said to have the (PF) property if TX = XV * for any operator X ∈ B(K, H) and any isometry V ∈ B(K) implies T * X = XV.Lemma 6. [21] Let A ∈ B(K) and B ∈ B(H).Then the following assertions are equivalent: 1. A, B satisfy Fuglede-Putnam theorem; 2. if AX = XB for any operator X ∈ B(H, K), then ran(X) reduces A, (kerX) ⊥ reduces B and A| ran(X) , B| (kerX) ⊥ are unitarily equivalent normal operators.
The following result was given by Duggal-Kubrusl [22] in the contractive case and by Pagacz [9] in the general case but our proof seems more direct, simpler and gives more explicit decomposition than Pagacz's proof.Proof.Since T is a power-bounded operator then there is a g-asymptotic limit A * associated with the operator T * which is a positive operator and has the form Furthermore, there exists an isometry V on ranA * = H 0 ⊥ (the asymptotic isometry associated with A * ), satisfy Equation (4), i.e., VX = XT * , where Xh = R Proof.Since B is n- * -paranormal operator then by the Propositions 2 and 3, B = U ⊕ B 0 on the decomposition H = H 0 ⊕ H 0 ⊥ where H 0 is the stable subspace of Since A is n- * -paranormal operator and U is unitary then by the Propositions 3, we get We have ran(X 1 ) is invariant for A and ker X 1 is invariant for B 0 .Hence, the operators A, X 1 and B 0 can be written: From Lemma 5, A 1 is a (power) n- * -paranormal operator, B * 01 is of class C 0. From the previous decompositions and Equation (21), we get A 1 Y = YB * 01 , where Y is an injective operator with dense range. Thus, Hence, A n 1 Yh = YB * n 01 h ≤ Y B * n 01 h → 0 (strongly).Since A 1 is a (power) n- * -paranormal operator then, by Propositions 2 and 3, we deduce that A 1 is not of class C 0. .Hence, Y = 0. Therefore, X 1 = 0. Thus, from Equation ( 23), we get Theorem 2. Let A be power-bounded (n, k)-quasi- * -paranormal operator with reduced kernel and B be power-bounded n- * -paranormal operator.If AX = X B * for some X ∈ B(H), then A * X = X B holds for all non-negative integers n and k > 0.
Proof.If σ p (A) − {0} = ∅ = σ p (B) − {0} and A is reduced by its kernel, then by Corollary 1, we can write the operators A, B as follows according to the decomposition.
where N, M are normal operators and M, K are the subspaces spanned by the eigenspaces of A and B respectively, with σ r (A To prove the adjoint version of this system it is enough to prove the earlier equation because the first three equations are particular cases of it.Instead consider the following decomposition: From the equation AX = XB * we deduce that ran(X) is invariant for A and ker X is invariant for B * .Hence, the operators A, X and B can be written: and From Lemma 5, A 1 is a (power) (n, k)-quasi- * -paranormal operator, B 1 is a (power-bounded) n- * -paranormal operators and from Corollary 1, 1 where Y is injective with dense range.From Lemma 5, A 1 and B 1 have the following matrices decompositions: ⊥ where A 11 , B * 11 are power-bounded n- * -paranormal operators and where Y 1 : ran(B k 1 ) → ran(A k 1 ) is injective with dense range.Also, it follows from the equation Remark 1.By the same method, we can prove the dual version of Theorem 2. Indeed, let A be a power-bounded n- * -paranormal operator and B be a power-bounded (n, k)-quasi- * -paranormal operator with reduced kernel.
If AX = X B * for some X ∈ B(H), then A * X = X B holds for all non-negative integers n and k > 0.
Definition 5.An operator T ∈ B(H) is said to be for all x ∈ H and for non-negative integers n, k.
We give a proof of (ii) which seems direct and simpler than given in [19] Istratescu and Istratescu [23] have proved that n-paranormal operators are normaloid.Thus, for proving (ii) it suffices to show that the class (n, k)-quasi- * -paranormal operators; for k = 0, 1 is a subset of n-paranormal one.
The case k = 0 is similar.
As a consequence, we get Corollary 2. The asymmetric Fuglede-Putnam theorem holds for the pair of power-bounded operators (A, B) in each of the following cases: 1.A is k-quasi- * -class A operator with reduced kernel and B is n- * -paranormal operator; 2. A is n- * -paranormal operator and B is k-quasi- * -class A operator with reduced kernel; 3. A, B ∈ B(H) are k-quasi- * -class A operators with 0 not in their approximate spectrum.
As an application of Theorems 1, 2, Corollary 2 and Pagacz's Theorem [9], we get the following: Corollary 3. Let T be a power-bounded operator, then T has the Wold-type decomposition (i.e., T is a direct sum of a unitary operator and an operator of class C .0 ) in each of the following cases: (n, k)-quasi- * -paranormal operator with reduced kernel; (iii) k-quasi- * -class A operator with reduced kernel.
We note that (i) was proved by Duggal in case n = 1 [13] and extended by Pagacz for n ≥ 1 [9].The result (iii) generalizes that of Hoxha and Braha [24] which was proved in the contraction operator case.

Application
Definition 6.A non-zero transform X ∈ B(K, H) is said to be a quasi-invertible if it is injective and has dense range.T ∈ B(H) is said to be a quasi-affine transform of R ∈ B(K) if there exists a quasi-invertible X ∈ B(K, H) intertwining R to T, i.e., TX = XR.Proposition 4. If a power-bounded operator is of class C 1. , then it is a quasi-affine transform of an isometry.
Proof.If T is a power-bounded operator on H of class C 1. , then it follows by the above remarks that ker A T,g = ker A From Equation (3), T is a quasi-affine transform of an isometry V on ran(A T,g ).
We note here that the previous was given by Duggal, Kubrusly [13] in the contractive case.We give the Kerchy's Lemma [16] which was first proven by Sz-Nagy and Foias [18] for contractions and by Kerchy for power-bounded operators.Lemma 8. (Kerchy) If T is a power-bounded operator on H, then T has the following matrix form: , where H 0 is the stable subspace of T, T 0 ∈ C 0. and T 1 ∈ C 1. .

Remark 2.
Since the spectral radius of a power-bounded operators is not greater then 1, then the power-bounded normaloid operators are contractions.Hence, by the Lemma 7, (n, k)-quasi- * -paranormal operators (in particular k-quasi- * -class A operator if k = 0, 1) and k-paranormal operators are contractions.
A contraction T on a separable Hilbert space H is said to be a completely non-unitary if it has no non-trivial unitary direct summand.T is said to be of class C 0 , written T ∈ C 0 if ψ( f ) = f (T) = 0; for some non-zero function f , where ψ is a weak*weak continuous homomorphism from the Hardy space H ∞ (D) on the open unit disc D to the weakly closed subalgebra of B(H) generated by T, that is an extension of the usual functional calculus.This is the H ∞ -functional calculus developed by Sz-Nagy and Foias [18].It is well known that each contraction of class C 0 is of class C 00 and the converse is given by Takahashi and Uchiyama (Theorem 1, [15]), under the assumption that the defect operator As a consequence of our main results, we have that if T is a power-bounded and completely non-unitary n- * -paranormal operator (resp.be a (n, k)-quasi- * -paranormal operator or a k-quasi- * -class A operator with reduced kernels), then T has part (its restriction on the invariant subspace H 0 ) in C 0 and its compression on H 0 ⊥ is quasi-affine transform of an isometry.
Proposition 5. Let T be a power-bounded and completely non-unitary n- * -paranormal operator (resp.be a (n, k)-quasi- * -paranormal operator or a k-quasi- * -class A operator with reduced kernels).Then T has the following triangular matrix on the decomposition H = H 0 ⊕ H 0 ⊥ , where H 0 is the stable subspace of T and T 1 is quasi-affine transform of an isometry.
Proof.Since T is completely non-unitary, then it follows from Corollary 3 that T is of class C .0 .Since the C .0 property is invariant under the restriction to an invariant subspace, therefore by the Kerchy's Lemma, we get the desired triangular matrix form (25) of T on the decomposition H = H 0 ⊕ H 0 ⊥ and the assertions (i) and (ii) follow immediately.(iii) follows from Kerchy's Lemma and Proposition 4.
Proposition 6.Let T be a power-bounded and completely non-unitary n- * -paranormal operator (resp.be a (n, k)-quasi- * -paranormal operator or a k-quasi- * -class A operator with reduced kernels).If T is a contraction with the above matrix form (25) such that the defect operator D T = (I − T * T) 1 2 is of Hilbert-Schmidt class.Then, T 0 ∈ C 0 and σ p (T) is at most countable.
Furthermore, the following assertions are equivalent: Proof.Since T 0 is a contraction such that the defect operator D T 0 is of Hilbert-Schmidt class, i.e., tr(I − T Since the point spectrum of a completely non-unitary does not intersect with the unite circle and σ p (T 1 ) is empty, then σ p (T) lies in σ(T 0 ).However, T 0 ∈ C 0 , that is the spectrum of T 0 does not fill the unit disc.Hence, σ p (T) is at most countable.
Remark 3. The assertions (i), (ii) and (iii) above are proven in [15] for all contraction in C .0 such that the defect operator D T = (I − T * T) 1 2 is of Hilbert-Schmidt class.
Proposition 7. If T is a power-bounded n- * -paranormal operators (resp.the (n, 1)-quasi- * -paranormal operators with reduced kernel) such that its spectrum lies in the unit circle T, then T is a unitary operator.
Proof.We have that our classes cited in the Proposition are invariant under multiplication by non-zero scalar and are contractions normaloid by Remark 3 and Lemma 5. Therefore, by following the proof given by Duggal [13], we obtain the desired result.
It is well known that a contraction normal operator is a direct sum of a unitary operator and un operator of class C 00 .So the natural question is what happen for a non-normal operators?Takahashi and Uchiyama [15] proved that a completely non-normal hyponormal operator such that the defect operator D T is of Hilbert-Schmidt class, is of class C 10 and Duggal, Jeonb, Kim [13] extended this result under the same assumptions to the case * -paranormal operators.In the following, we generalize this result in more general classes.Proof.From Proposition 4, T has the following triangular matrix on the decomposition H = H 0 ⊕ H 0 ⊥ where H 0 is the stable subspace of T, T 0 ∈ C 00 and T 1 ∈ C 10 .
Therefore, by Proposition 5, D T 0 is of Hilbert-Schmidt class and T 0 ∈ C 0 with the form where σ(A) = σ p (A) ⊂ D and σ(B) ⊂ T (where D is the open unit disc).From Corollary 3 and the fact that T 0 is completely non-normal, it follows that σ p (T 0 ) is empty and yields σ(T 0 ) = σ(B) ⊂ T. Therefore, by the Proposition 7, T 0 is unitary; a contradiction.This shows that T 0 is absent.Finally, we conclude that T = T 1 ∈ C 10 .

Discussions and Further Studies
The following Putnam-Fuglede theorem is very well known: Theorem 4. (Putnam-Fuglede Theorem) [4,5].Assume that A, B ∈ B(H) are normal operators.If AX = XB for some X ∈ B(H), then A * X = XB * .

Proposition 2 .
Let T ∈ B(H) be a power-bounded operator.A has the PF property if and only if A = U ⊕ C where U is unitary and C is of class C .0 .

1 2 hProposition 3 .Theorem 1 .
, for all h ∈ H. Hence,TX = XV *(20)It follows from the previous Lemma, for T = A, B = V * , that if T has the PF property (i.e., T, V * satisfy Fuglede-Putnam theorem) then ranX = H 0 ⊥ reduces T and T| ran(X) , V * are unitarily equivalent normal operators (we have (kerX) ⊥ = H 0 ⊥ ).Which means that T| ran(X) is a unitary operator.Since H 0 is the stable subspace of T * and T * | H 0 is of class C .0 then T| H 0 is of class C .0 .The reverse implications it follows immediately from the previous Lemma.(P.Pagacz[9]) Every power-bounded n- * -paranormal operator has the PF property.Let A ∈ B(K), B ∈ B(H) be power-bounded n- * -paranormal operators.If AX = XB * for any X ∈ B(H, K), then A * X = XB.

Theorem 3 . 1 2
If T is a completely non-normal n- * -paranormal operators (resp.the (n, 1)-quasi- * -paranormal operators with reduced kernel) such that the defect operator D T = (I − T * T) is of Hilbert-Schmidt class, then T ∈ C 10 .
11is an injective normal operator unitary equivalent to B 11 .From Proposition 1, we get that ran(A k 1 ) reduces A 1 .Hence R = 0 and from A 1 Y = YB * 1 it follows that Y * 2 A * 12 = 0. Since Y has dense range then A 12 = 0. Therefore A 1 = A 11 ⊕ 0 which is a (n)- * -paranormal operator.Finally, we deduce that A * 1 Y = YB 1 and then A * X = XB and the proof is complete.
k 11 .Since A 11 and B * 11 are power-bounded n- * -paranormal operators, then from Theorem 1 we have A * k 11 Y 1 = Y 1 B k 11 , and because of Y is injective with dense range, we get A