1. Introduction
Throughout this paper, denotes an infinite dimensional complex Hilbert space with inner product and denotes the algebra of all bounded linear operators acting on . Spectrum, point spectrum, residual spectrum, continuous spectrum, and approximate spectrum of an operator T will be denoted by , , , , , respectively. The kernel and the range of an operator T will be denoted by kerT and ran(T) respectively.
For any operator , let , and consider the following standard definitions: normal if and T is hyponormal if (i.e., equivalently, if for every ).
An operator
T is said to be ∗-paranormal iff
for all
, or equivalently,
is ∗-paranormal iff
, for all
. The class of ∗-paranormal operators was introduced in [
1]. Another well-known generalization of ∗-paranormal operators are
-quasi-∗-paranormal operators defined as follows:
T is said to be
-quasi-∗-paranormal operator if
for all
and for non-negative integers
k and
n.
An operator
is said to be paranormal [
2] iff
The familiar Putnam-Fuglede theorem asserts that if
and
are normal operators and
for some
, then
(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
and
are normal operators and
for some
, then
.
Many authors extended this theorem for different non-normal classes of operators (see [
2,
4,
5,
6,
7,
8,
9,
10,
11,
12]).
In this paper, we shall generalize this theorem to certain -quasi-∗-paranormal operators.
The organization of the paper is as follows; in
Section 2, we give some properties for
-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
in the following cases:
- (i)
A and are n-∗-paranormal operators
- (ii)
A is a -quasi-∗-paranormal operator with reduced kernel and are n-∗-paranormal operator;
- (iii)
A is a n-∗-paranormal operator and are -quasi-∗-paranormal operator with reduced kernel (an operator T with reduced kernel means that its kernel is invariant under ).
These results extend those recently given in [
9,
13,
14] and as applications of our main theorems, we obtain the following:
if T is a -quasi-∗-paranormal operator with reduced kernel (resp. n-∗-paranormal operator or a n-quasi-∗-class A with reduced kernel), then T has a part in the class on a stable subspace and a compression quasi-affine transform to an isometry on the orthogonal complement of .
Next, we prove that if T is completely non-normal -quasi-∗-paranormal operator; for and verifying the defect operator is a Hilbert-Schmidt class, then .
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 [
16].
Definition 1. A Banach limit or a generalized limit is a bounded linear functional on (the Banach space of bounded complex sequences) which preserves the ordinary notion of convergence. That is if then .
Banach limit may be characterized as those continuous functional which satisfy the following conditions:
is positive, i.e., if for all then ;
, where ;
is shift-invariant, i.e., .
(see [
17]) for further details.
In the sequel we fix a generalized Banach limit
on
for a power-bounded operator
T;
, on the Hilbert space
. The following map is a bounded sesquilinear form
Since is bounded, then if and only if and so, this holds if and only if .
We denote by
the kernel of
, i.e.,
is said the stable subspace for
T. It is clear that
is an invariant subspace for any operator in the commutant of
T, i.e.,
is an hyperinvariant subspace. We recall the following definitions:
- (i)
A power-bounded operator T is said to be of class if the sequence does not converge to 0 for any non-zero vector x i.e., .
- (ii)
T is said to be strongly stable if and we write ;
- (iii)
T is of class if is of class ;
- (iv)
T is of class if .
It follows from Equation (
1) of the sesquilinear application
that there exists a positive operator
such that the equation
holds for all vectors
. The operator
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
holds for every Banach limit
g and
Furthermore, there exists an isometry
V on
such that
The concept of asymptotic limit and their generalizations play an important role in the hyperinvariant subspace problem [
16,
18]. Since
is a power-bounded operator whenever
T is, let
be the strong limit of
and let
be the associated isometry on
so that all the preceding properties hold for
.
Definition 2. Let , then
- (i)
the joint point spectrum, denoted by is the set - (ii)
the joint approximate point spectrum, denoted by is the set of scalars λ for which there exists a normalized sequence verifying
Notice that in general, ; however, the equality holds for the following operator classes: p-hyponormal or log-hyponormal, absolute-∗-k-paranormal.
Definition 3. is said to be -quasi-∗-paranormal operators if, for non-negative integers k and n, If
, it is clear that
T is
n-∗-paranormal operator [
8] and if
, then
T is ∗-paranormal [
1]. Also, if
,
T is
k quasi-hyponormal [
8] and if
,
T is
k-quasi-∗-hyponormal operator [
19].
3. Main Theorems
We are ready to show our main theorems.
Definition 4. An operator is said to have the (PF) property if for any operator and any isometry implies .
Lemma 6. [21] Let and . Then the following assertions are equivalent: - 1.
satisfy Fuglede-Putnam theorem;
- 2.
if for any operator , then reduces A, reduces B and , 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.
Proposition 2. Let be a power-bounded operator. A has the PF property if and only if where U is unitary and C is of class .
Proof. Since T is a power-bounded operator then there is a g-asymptotic limit associated with the operator which is a positive operator and has the form on the decomposition where is the stable subspace of .
Furthermore, there exists an isometry
V on
(the asymptotic isometry associated with
), satisfy Equation (
4), i.e.,
, where
, for all
. Hence,
It follows from the previous Lemma, for , that if T has the PF property (i.e., satisfy Fuglede-Putnam theorem) then reduces T and , are unitarily equivalent normal operators (we have ). Which means that is a unitary operator. Since is the stable subspace of and is of class then is of class . The reverse implications it follows immediately from the previous Lemma. □
Proposition 3. (P. Pagacz [9]) Every power-bounded n-∗-paranormal operator has the PF property. Theorem 1. Let be power-bounded n-∗-paranormal operators.
If for any , then .
Proof. Since B is n-∗-paranormal operator then by the Propositions 2 and 3, on the decomposition where is the stable subspace of . Setting .
It follows from
that
Since
A is
n-∗-paranormal operator and
U is unitary then by the Propositions 3, we get
We have
is invariant for
A and
is invariant for
. Hence, the operators
A,
and
can be written:
and
From Lemma 5,
is a (power)
n-∗-paranormal operator,
is of class
From the previous decompositions and Equation (
21), we get
, where
Y is an injective operator with dense range.
Hence,
(strongly). Since
is a (power)
n-∗-paranormal operator then, by Propositions 2 and 3, we deduce that
is not of class
. Hence,
. Therefore,
. Thus, from Equation (
23), we get
□
Theorem 2. Let be power-bounded -quasi-∗-paranormal operator with reduced kernel and be power-bounded n-∗-paranormal operator. If for some , then holds for all non-negative integers n and .
Proof. If
and
is reduced by its kernel, then by Corollary 1, we can write the operators
as follows
according to the decomposition.
, where N, M are normal operators and , are the subspaces spanned by the eigenspaces of and respectively, with .
Moreover, if
, then from
it follows that
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
we deduce that
is invariant for
A and
is invariant for
. Hence, the operators
A,
X and
B can be written:
and
From Lemma 5, is a (power) -quasi-∗-paranormal operator, is a (power-bounded) n-∗-paranormal operators and from Corollary 1, .
Also, implies where Y is injective with dense range.
From Lemma 5,
and
have the following matrices decompositions:
according to the decomposition
and
where
are power-bounded
n-∗-paranormal operators and
.
It is clear that
implies that
for any positive integer
k and therefore
. So
Y has the following matrix
where
is injective with dense range. Also, it follows from the equation
that
Since
and
are power-bounded
n-∗-paranormal operators, then from Theorem 1 we have
, and because of
Y is injective with dense range, we get
is an injective normal operator unitary equivalent to
. From Proposition 1, we get that
reduces
. Hence
and from
it follows that
. Since
Y has dense range then
. Therefore
which is a
-∗-paranormal operator. Finally, we deduce that
and then
and the proof is complete. □
Remark 1. By the same method, we can prove the dual version of Theorem 2. Indeed, let be a power-bounded n-∗-paranormal operator and be a power-bounded -quasi-∗-paranormal operator with reduced kernel. If for some , then holds for all non-negative integers n and .
Definition 5. An operator is said to be
- (i)
k-quasi-∗-class A if for non-negative integer k;
- (ii)
-quasi-paranormal operator iffor all and for non-negative integers
Lemma 7. We have the following proper inclusions:
- (i)
(k-quasi-∗-class A) ⊂ (k-quasi-∗-paranormal);
- (ii)
the class -quasi-∗-paranormal operator is normaloid, for .
Proof. 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
-quasi-∗-paranormal operators; for
is a subset of
n-paranormal one.
for all
Hence (n, 1)-quasi-∗-paranormal ⊂ (n + 1)-paranormal.
The case is similar. □
As a consequence, we get
Corollary 2. The asymmetric Fuglede-Putnam theorem holds for the pair of power-bounded operators in each of the following cases:
- 1.
is k-quasi-∗-class A operator with reduced kernel and is n-∗-paranormal operator;
- 2.
is n-∗-paranormal operator and is k-quasi-∗-class A operator with reduced kernel;
- 3.
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 ) in each of the following cases:
- (i)
n-∗-paranormal operator;
- (ii)
-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
[
13] and extended by Pagacz for
[
9]. The result (iii) generalizes that of Hoxha and Braha [
24] which was proved in the contraction operator case.
4. Application
Definition 6. A non-zero transform is said to be a quasi-invertible if it is injective and has dense range. is said to be a quasi-affine transform of if there exists a quasi-invertible intertwining R to T, i.e., .
Proposition 4. If a power-bounded operator is of class , then it is a quasi-affine transform of an isometry.
Proof. If
T is a power-bounded operator on
of class
, then it follows by the above remarks that
. Since
A is a positive operator, then
From Equation (
3),
T is a quasi-affine transform of an isometry
V on
. □
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 , then T has the following matrix form:on the decomposition , where is the stable subspace of T, and . 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, -quasi-∗-paranormal operators (in particular k-quasi-∗-class A operator if ) and k-paranormal operators are contractions.
A contraction
T on a separable Hilbert space
is said to be a completely non-unitary if it has no non-trivial unitary direct summand.
T is said to be of class
, written
if
; for some non-zero function
f, where
is a weak*weak continuous homomorphism from the Hardy space
on the open unit disc
D to the weakly closed subalgebra of
generated by
T, that is an extension of the usual functional calculus. This is the
-functional calculus developed by Sz-Nagy and Foias [
18]. It is well known that each contraction of class
is of class
and the converse is given by Takahashi and Uchiyama (Theorem 1, [
15]), under the assumption that the defect operator
is of Hilbert-Schmidt class.
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 -quasi-∗-paranormal operator or a k-quasi-∗-class A operator with reduced kernels), then T has part (its restriction on the invariant subspace ) in and its compression on 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 -quasi-∗-paranormal operator or a k-quasi-∗-class A operator with reduced kernels). Then T has the following triangular matrixon the decomposition , where is the stable subspace of T and - (i)
;
- (ii)
;
- (iii)
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
. Since the
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
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 -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 is of Hilbert-Schmidt class. Then, and is at most countable. Furthermore, the following assertions are equivalent:
- (i)
;
- (ii)
;
- (iii)
.
Proof. Since
is a contraction such that the defect operator
is of Hilbert-Schmidt class, i.e.,
, by Theorem 1 in [
15],
is of class
.
Since the point spectrum of a completely non-unitary does not intersect with the unite circle and is empty, then lies in . However, , that is the spectrum of does not fill the unit disc. Hence, is at most countable. □
Remark 3. The assertions (i), (ii) and (iii) above are proven in [15] for all contraction in such that the defect operator is of Hilbert-Schmidt class. Proposition 7. If T is a power-bounded n-∗-paranormal operators (resp. the -quasi-∗-paranormal operators with reduced kernel) such that its spectrum lies in the unit circle , 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
. 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
is of Hilbert-Schmidt class, is of class
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.
Theorem 3. If T is a completely non-normal n-∗-paranormal operators (resp. the -quasi-∗-paranormal operators with reduced kernel) such that the defect operator is of Hilbert-Schmidt class, then .
Proof. From Proposition 4,
T has the following triangular matrix
on the decomposition
where
is the stable subspace of
and
. Therefore, by Proposition 5,
is of Hilbert-Schmidt class and
with the form
where
and
(where
is the open unit disc). From Corollary 3 and the fact that
is completely non-normal, it follows that
is empty and yields
. Therefore, by the Proposition 7,
is unitary; a contradiction. This shows that
is absent. Finally, we conclude that
. □
5. Discussion and Further Studies
The following Putnam-Fuglede theorem is very well known:
Theorem 4. (Putnam-Fuglede Theorem) [4,5]. Assume that are normal operators. If for some , then .
There are many generalizations of this theorem to several classes of operators (see [
3,
4,
5,
7,
8,
10,
16,
21,
25,
26,
27]) etc. In 1978, S.K Berberian [
28] showed that the Putnam-Fuglede theorem holds when A and B* are hyponormal and
X is a Hilbert-Schmidt operator. Radjapalipour [
3] showed that Putnam-Fuglede theorem remains valid for hyponormal operators. In 2002, Uchiyama and Tanahashi [
25] proved that Putnam-Fuglede theorem still holds for
p-hyponormal and log-hyponormal operators. Bachir and Lombarkia [
5] gave the extension of Putnam-Fuglede Theorem for
w-hyponormal and class
. Recently, Mecheri and Uchiyama [
7] extended Putnam-Fuglede to class
operators. In this paper, we generalize the Putnam-Fuglede theorem to a large class of operators, say
-quasi-∗-paranormal operators. These results extend those given in [
8,
14,
17,
20].
As application of our main theorems, we obtain:
Characterization of -quasi-∗-paranormal operators with reduced kernel.
Characterization of completely non-normal -quasi-∗-paranormal operators. These generalizes the results given by
- (i)
Tanahashi and Uchiyama [
15] for completely non-normal hyponormal contraction operator.
- (ii)
Duggal, Jeon, and Kim [
13] completely non-normal ∗-paranormal contraction operator.