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

Abstract

$T\in \mathcal{B}\left(\mathcal{H}\right)$ is said to be $(n,k)$-quasi-∗-paranormal operator if, for non-negative integers k and n, $\parallel T^{\ast }\left(T^{k}x\right){\parallel }^{(1+n})\le \parallel T^{(1+n)}\left(T^{k}x\right)\parallel \parallel T^k{x\parallel }^n$; for all $x\in \mathcal{H}$. In this paper, the asymmetric Putnam-Fuglede theorem for the pair $(A,B)$ of power-bounded operators is proved when (i) A and $B^{\ast }$ are n-∗-paranormal operators (ii) A is a $(n,k)$-quasi-∗-paranormal operator with reduced kernel and $B^{\ast }$ 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 $D_T$ is Hilbert-Schmidt class, then $T\in C_{10}$.

1. Introduction

Throughout this paper, $\mathcal{H}$ denotes an infinite dimensional complex Hilbert space with inner product $\langle ·,·\rangle$ and $\mathcal{B}\left(\mathcal{H}\right)$ denotes the algebra of all bounded linear operators acting on $\mathcal{H}$. Spectrum, point spectrum, residual spectrum, continuous spectrum, and approximate spectrum of an operator T will be denoted by $\sigma \left(T\right)$, ${\sigma }_p\left(T\right)$, ${\sigma }_r\left(T\right)$, ${\sigma }_c\left(T\right)$, ${\sigma }_a\left(T\right)$, respectively. The kernel and the range of an operator T will be denoted by kerT and ran(T) respectively.

For any operator $T\in \mathcal{B}\left(\mathcal{H}\right)$, let $\left|T\right|={\left(T^{\ast }T\right)}^{1/2}$, and consider the following standard definitions: normal if $T^{\ast }T=T{T}^{\ast }$ and T is hyponormal if $|T^{\ast }{|}^2\le {\left|T\right|}^2$ (i.e., equivalently, if $\parallel T^{\ast }x\parallel \le \parallel Tx\parallel$ for every $x\in \mathcal{H}$).

An operator T is said to be ∗-paranormal iff $\parallel T^{\ast }{x\parallel }^2\le \parallel T^{2}x\parallel \parallel x\parallel ,$ for all $x\in \mathcal{H}$, or equivalently, $T\in \mathcal{B}\left(\mathcal{H}\right)$ is ∗-paranormal iff $T^{\ast 2}{T}^2-2\lambda T{T}^{\ast }+{\lambda }^2\ge 0$, for all $\lambda >0$. 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

$$\parallel T^{\ast }\left(T^{k}x\right){\parallel }^{(1+n})\le \parallel T^{(1+n)}\left(T^{k}x\right)\parallel \parallel T^k{x\parallel }^n$$

for all $x\in \mathcal{H}$ and for non-negative integers k and n.

An operator $T\in \mathcal{B}\left(\mathcal{H}\right)$ is said to be paranormal [2] iff

$${\parallel Tx\parallel }^2\le \parallel T^{2}x\parallel \parallel x\parallel \mathrm{for}\mathrm{all}x\in \mathcal{H}.$$

The familiar Putnam-Fuglede theorem asserts that if $A\in \mathcal{B}\left(\mathcal{H}\right)$ and $B\in \mathcal{B}\left(\mathcal{H}\right)$ are normal operators and $AX=XB$ for some $X\in \mathcal{B}\left(\mathcal{H}\right)$, then $A^{\ast }X=X{B}^{\ast }$ (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 $A\in \mathcal{B}\left(\mathcal{H}\right)$ and $B\in \mathcal{B}\left(\mathcal{H}\right)$ are normal operators and $AX=X{B}^{\ast }$ for some $X\in \mathcal{B}\left(\mathcal{H}\right)$, then $A^{\ast }X=XB$.

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 $(n,k)$-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 $B^{\ast }$ are n-∗-paranormal operators

(ii)

A is a $(n,k)$-quasi-∗-paranormal operator with reduced kernel and $B^{\ast }$ are n-∗-paranormal operator;

(iii)

A is a n-∗-paranormal operator and $B^{\ast }$ are $(n,k)$-quasi-∗-paranormal operator with reduced kernel (an operator T with reduced kernel means that its kernel is invariant under $T^{\ast }$).

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 $(n,k)$-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 $C_{00}$ on a stable subspace ${\mathcal{H}}_0$ and a compression quasi-affine transform to an isometry on the orthogonal complement of ${\mathcal{H}}_0$.
• 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 $T\in C_{10}$.

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 [16].

Definition 1.

A Banach limit or a generalized limit is a bounded linear functional $glim$ on $l^{\infty }\left(\mathbb{N}\right)$ (the Banach space of bounded complex sequences) which preserves the ordinary notion of convergence. That is if $lim{x}_n=x$ then $glim\left(x_n\right)=x$.

Banach limit may be characterized as those continuous functional which satisfy the following conditions:

• $glim$ is positive, i.e., if $x_n\ge 0$ for all $n\in \mathbb{N}$ then $glim\left(x_n\right)\ge 0$;
• $glim\left(1\right)=1$, where $\left(1\right)=(1,1,1,\dots .)$;
• $glim$ is shift-invariant, i.e., $glim\left(x_n\right)=glim\left(x_{n+1}\right)$.

(see [17]) for further details.

In the sequel we fix a generalized Banach limit $glim$ on $l^{\infty }\left(\mathbb{N}\right)$ for a power-bounded operator T; $su{p}_n\parallel T^{n}x\parallel \le \infty$, on the Hilbert space $\mathcal{H}$. The following map is a bounded sesquilinear form

$${\varphi }_T(x,y)=gli{m}_n\langle T^{n}x,T^{n}y\rangle ;x,y\in \mathcal{H}$$

(1)

Since $\{\parallel T^n\parallel :n\ge 1\}$ is bounded, then $glim\parallel T^{n}x\parallel =0$ if and only if $in{f}_n\parallel T^{n}x\parallel =0$ and so, this holds if and only if $li{m}_n\parallel T^{n}x\parallel =0$.

We denote by ${\mathcal{H}}_0$ the kernel of ${\varphi }_T$, i.e.,

$${\mathcal{H}}_0=\{x\in \mathcal{H}:li{m}_n\parallel T^{n}x\parallel =0\}$$

(2)

${\mathcal{H}}_0$ is said the stable subspace for T. It is clear that ${\mathcal{H}}_0$ is an invariant subspace for any operator in the commutant of T, i.e., ${\mathcal{H}}_0$ 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 $\{\parallel T^{n}x\parallel :n\in N\}$ does not converge to 0 for any non-zero vector x i.e., ${\mathcal{H}}_0\left(T\right)=\left\{0\right\}$.

(ii)

T is said to be strongly stable if ${\mathcal{H}}_0\left(T\right)=\mathcal{H}$ and we write $T\in C_{0.}$;

(iii)

T is of class $C_{.j}:j=0,1$ if $T^{\ast }$ is of class $C_{j.};j=0,1$;

(iv)

T is of class $C_{ij}:i,j=0,1$ if $T\in C_{i.}\cap C_{.j}$.

It follows from Equation (1) of the sesquilinear application ${\varphi }_T$ that there exists a positive operator $A_{T,g}\in \mathcal{B}\left(\mathcal{H}\right)$ such that the equation ${\varphi }_T(x,y)=\langle A_{T,g}x,y\rangle$ holds for all vectors $x,y\in \mathcal{H}$. 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 $ker{A}_{T,g}={\mathcal{H}}_0$ holds for every Banach limit g and

$$gli{m}_n\parallel T^n{x\parallel }^2=\parallel A_{T,g}^{\frac{1}{2}}{x\parallel }^2={\parallel A_{T,g}^{\frac{1}{2}}Tx\parallel }^2$$

(3)

Furthermore, there exists an isometry V on $\overline{\mathrm{ran}\left(A\right)}$ such that

$$V{A}^{\frac{1}{2}}=A^{\frac{1}{2}}T.$$

(4)

The concept of asymptotic limit and their generalizations play an important role in the hyperinvariant subspace problem [16,18]. Since $T^{\ast }$ is a power-bounded operator whenever T is, let $A_{\ast }$ be the strong limit of $\{T^n{T}^{\ast n}:n\ge 1\}$ and let $V_{\ast }$ be the associated isometry on $\overline{\mathrm{ran}\left(A_{\ast }\right)}$ so that all the preceding properties hold for $T^{\ast },A_{\ast },V_{\ast }$.

Definition 2.

Let $T\in \mathcal{B}\left(\mathcal{H}\right)$, then

(i) 

the joint point spectrum, denoted by ${\sigma }_{jp}\left(T\right)$ is the set

$${\sigma }_{jp}\left(T\right)=\{\lambda \in \mathbb{C}:Tx=\lambda xand{T}^{\ast }x=\overline{\lambda }x\}.$$

(ii) 

the joint approximate point spectrum, denoted by ${\sigma }_{ja}\left(T\right)$ is the set of scalars λ for which there exists a normalized sequence $\left\{x_n\right\}\subset \mathcal{H}$ verifying

$$(T-\lambda )x_n\to 0 and {(T-\lambda )}^{\ast }{x}_n\to 0.$$

Notice that in general, ${\sigma }_{jp}\left(T\right)\subset {\sigma }_p\left(T\right)$; however, the equality holds for the following operator classes: p-hyponormal or log-hyponormal, absolute-∗-k-paranormal.

Definition 3.

$T\in \mathcal{B}\left(\mathcal{H}\right)$ is said to be $(n,k)$-quasi-∗-paranormal operators if, for non-negative integers k and n,

$$\parallel T^{\ast }\left(T^{k}x\right){\parallel }^{(1+n)}\le \parallel T^{(1+n)}\left(T^{k}x\right)\parallel \parallel T^k{x\parallel }^{n}forallx\in \mathcal{H}.$$

(5)

If $k=0$, it is clear that T is n-∗-paranormal operator [8] and if $n=1$, then T is ∗-paranormal [1]. Also, if $n=0$, T is k quasi-hyponormal [8] and if $n=1$, T is k-quasi-∗-hyponormal operator [19].

2. Properties of $(\mathit{n},\mathit{k})$-Quasi-∗-Hyponormal Operators

Lemma 1.

[19] If $T\in \mathcal{B}\left(\mathcal{H}\right)$, then T is $(n,k)$-quasi-∗-paranormal operator if and only if

$$T^{\ast k}{T}^{\ast (n+1)}{T}^{(n+1)}{T}^k-(n+1)t^n{T}^{\ast k}T{T}^{\ast }{T}^k+n{t}^{(n+1)}{T}^{\ast k}{T}^k\ge 0$$

(6)

for all $t>0$.

Lemma 2.

Let $T\in \mathcal{B}\left(\mathcal{H}\right)$. If T is a normal operator, then

$${\parallel Tx\parallel }^n\le \parallel T^{\left(n\right)}{x\parallel \parallel x\parallel }^n$$

(7)

for all $x\in \mathcal{H}$ and non-negative integer n.

Proof. 

We recall from [20] that if A is a positive operator on Hilbert space then

$${\langle Ax,x\rangle }^r\le \langle A^{r}x,x\rangle$$

(8)

for all $r>1$ and any unit vector x.

Let T be a normal operator and $n\ge 1$, then

$${\parallel Tx\parallel }^{2n}={\langle T^{\ast }Tx,x\rangle }^n$$

(9)

By the above inequality

$${\parallel Tx\parallel }^{2n}\le \langle {\left(T^{\ast }T\right)}^{n}x,x\rangle =\langle T^{\ast n}{T}^{n}x,x\rangle ={\parallel T^{n}x\parallel }^2$$

(10)

for all $n\ge 1$ and unit vector x. Hence, for $x=y/\parallel y\parallel ,y\ne 0$, we get our result. □

Lemma 3.

Let T be $(n,k)$-quasi-∗-paranormal operator. If $\overline{ran{T}^k}\ne \mathcal{H}$, then T has the following decomposition:

$$T=\left[\begin{array}{cc}{T}_1& S\\ 0& T_2\end{array}\right].$$

on $\mathcal{H}=\overline{ran{T}^k}\oplus {\left(ran{T}^k\right)}^{\perp }$, where $T_1$ is n-∗-paranormal operator, $T_2^k=0$ and $\sigma \left(T\right)=\sigma \left(T_1\right)\cup \left\{0\right\}.$

Proof. 

If $ran{T}^k\ne \mathcal{H}$, then $\mathcal{H}$ has the following non-trivial decomposition: $\mathcal{H}=\overline{ran{T}^k}\oplus {\left(ran{T}^k\right)}^{\perp }$. Also, it is clear that $\overline{ran{T}^k}$ is an invariant subspace for T such that $T_1{=T|}_{\overline{ran{T}^k}}$ is n-∗-paranormal operator and $T_2^k=0$; where $T_2=T_{|{\left(ran{T}^k\right)}^{\perp }}$. Hence, T has the triangular matrix form cited above. □

Proposition 1.

Let $T\in \mathcal{B}\left(\mathcal{H}\right)$.

1. 

If T is a power-bounded n-∗-paranormal operator and there is an invariant subspace $\mathcal{M}$ for which the restriction ${T|}_{\mathcal{M}}=N$ of T on $\mathcal{M}$ is a normal operator, then $\mathcal{M}$ reduces T and $N=U\oplus 0$ where U is unitary.

2. 

If T is a power-bounded $(n,k)$-quasi-∗-paranormal operator and there is an invariant subspace $\mathcal{M}$ for which the restriction ${T|}_{\mathcal{M}}=N$ of T on $\mathcal{M}$ is an injective normal operator then $\mathcal{M}$ reduces T and N is a unitary operator. In particular, if $\mathcal{M}=\overline{Ran{T}^k}$ and $T_1$ as in the previous Lemma, is normal operator, then $\overline{Ran{T}^k}$ reduces T.

Proof. 

• Let T be a power-bounded n-∗-paranormal operator and let us consider an invariant subspace $\mathcal{M}\subset \mathcal{H}$ for T such that ${T|}_{\mathcal{M}}=N$ is normal. The operator T has the following matrix decomposition

  $$T=\left[\begin{array}{cc}N& R\\ 0& \ast \end{array}\right]$$
  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 $N=U\oplus N_0$ where U is unitary and $N_0$ is of class $C_{00}$ for possible $N_0=0$. On the other hand, Since the operator T is n-∗-paranormal, it follows that

  $$\parallel R^{\ast }{x\parallel }^2+{\parallel Nx\parallel }^2=\parallel T^{\ast }{x\parallel }^2\le \parallel T^n{x\parallel }^{\frac{2}{n}}={\parallel N^{n}x\parallel }^{\frac{2}{n}},$$

  (11)

  for all unit vector $x\in \mathcal{M}$.
  Since the kernel of N reduces N, hence $N=N_1\oplus 0$.
  If $x\in kerN$ then from (7) we get $x\in ker{R}^{\ast }$. Thus

  $$R^{\ast }=0onkerN$$

  (12)
  For each unit vector $x\in \mathcal{M}\ominus kerN$ we have, $(\parallel R^{\ast }{x\parallel }^2{+1)}^n\le {\displaystyle \frac{1}{{\parallel Nx\parallel }^{2n}}}{\parallel N^{n}x\parallel }^2$
  From Lemma 1, we get for all $k\ge 1$, that

  $$(\parallel R^{\ast }{x\parallel }^2{+1)}^{nk}\le {\displaystyle \frac{1}{{\parallel Nx\parallel }^{2nk}}}\parallel N^n{x\parallel }^{2k}\le {\displaystyle \frac{1}{{\parallel Nx\parallel }^{2nk}}}{\parallel N^{nk}x\parallel }^2$$

  (13)
  If $N_0\ne 0$ then

  $$(\parallel R^{\ast }{x\parallel }^2{+1)}^{nk}\le {\displaystyle \frac{1}{{(\parallel Ux\parallel }^2+\parallel N_{0}x{{\parallel }^2)}^{nk}}}(\parallel U^{nk}{x\parallel }^2+\parallel N_0^{nk}x{\parallel }^2)$$

  (14)
  Since U is unitary and $\parallel x\parallel =1$ then

  $$(\parallel R^{\ast }{x\parallel }^2{+1)}^{nk}\le {\displaystyle \frac{1}{(1+\parallel N_{0}x{{\parallel }^2)}^{nk}}}(1+\parallel N_0^{nk}x{\parallel }^2)$$

  (15)
  Since $N_0\in C_{00}$ and $1+\parallel N_0{x\parallel }^2>1$ then for $k\to \infty$ we get $(\parallel R^{\ast }x{{\parallel }^2+1)}^{nk}\to 0$. However, $\parallel R^{\ast }{x\parallel }^2+1\ge 1$ which is a contradiction. Hence, $N_0=0$ and then $N=U$ on $\mathcal{M}\ominus kerN$.
  By substituting $N_0=0$ and $k=1$ in the inequality (11), we get $(\parallel R^{\ast }x{{\parallel }^2+1)}^n=1$. Therefore, $R^{\ast }x=0$ on $\mathcal{M}\ominus kerN$ and by (8), we have $R^{\ast }=0=R$.
• As in the case $\left(1\right)$, let us consider an invariant subspace $\mathcal{M}\subset \mathcal{H}$ for T such that ${T|}_{\mathcal{M}}=N$ is normal. The operator T has the following matrix

  $$T=\left[\begin{array}{cc}N& R\\ 0& \ast \end{array}\right],$$

  on the decomposition $\mathcal{H}=\mathcal{M}\oplus {\mathcal{M}}^{\perp }$, where N is an injective normal operator and then $\overline{ran{N}^k}=\mathcal{M}$. Hence, $\mathcal{M}\subseteq \overline{ran{N}^k}$. Indeed, we have $T^{k}x=N^{k}x$ for all $x\in \mathcal{M}$. Since N is normal then $ran{N}^k=ranN$ and by the assumption that N is injective, we get $\overline{ran{N}^k}=\mathcal{M}$.
  Since T is $(n,k)$-quasi-∗-paranormal operator, then

  $$\parallel T^{\ast }\left(T^{k}x\right){\parallel }^2\le \parallel T^{(n+1)}{T}^k{x\parallel }^{\frac{2}{(n+1)}}{\parallel T^{k}x\parallel }^{\frac{2n}{(n+1)}}$$

  (16)
  Put $y=T^{k}x=N^{k}x\in \mathcal{M}=\overline{ran{N}^k}$ for all $x\in \mathcal{M}$, we get,

  $$(\parallel R^{\ast }{y\parallel }^2+{\parallel Ny\parallel }^2{)}^{n+1}\le \parallel N^{n+1}{y\parallel }^2{\parallel y\parallel }^{2n},$$

  (17)
  Since N is a contraction and $y=N^{k}x$ we get ${\parallel y\parallel }^{2n}\le 1$. Hence,

  $$(\parallel R^{\ast }{y\parallel }^2+{\parallel Ny\parallel }^2{)}^{n+1}\le {\parallel N^{n+1}y\parallel }^2,$$

  (18)

  for all $y\in \mathcal{M}=\overline{ran{N}^k}$ we get the desired result, by following the same steps as in the proof of the previous assertion (1).

 □

Lemma 4.

If $T\in \mathcal{B}\left(\mathcal{H}\right)$ is $(n,k)$-quasi-∗-paranormal operator, then $T^{\circ }$ is $(n,k)$-quasi-∗-paranormal operator.

Proof. 

The proof is a consequence of the Lemma 1 and the properties of the isometric ∗-homomorphism $\varphi$ of the Berberian technique.  □

Corollary 1.

Let T be a power-bounded $(n,k)$-quasi-∗-paranormal operator.

1. 

If $k=0$ (i.e., T is n-∗-paranormal) and $\lambda \in {\sigma }_p\left(T\right)$, then $ker(T-\lambda )$ reduces T. Also, if $k>0$ and $\lambda \in {\sigma }_p\left(T\right)-\left\{0\right\}$, then $ker(T-\lambda )$ reduces T.

2. 

If T is n-∗-paranormal and $Tx=\lambda x$ such that $x\ne 0$, then $T^{\ast }x=\overline{\lambda }x$ and ${\sigma }_p\left(T\right)={\sigma }_{jp}\left(T\right)$. The same result holds in case $k>0$ and $\lambda \ne 0$ and ${\sigma }_p\left(T\right)-\left\{0\right\}={\sigma }_{jp}\left(T\right)-\left\{0\right\}$.

3. 

If $\lambda \ne \mu$, then $ker(T-\lambda )\perp ker(T-\mu )$.

4. 

$T=N\oplus A$ on the decomposition $\mathcal{H}=\mathcal{M}\oplus {\mathcal{M}}^{\perp }$, where $\mathcal{M}$ is the subspace spanned by the eigenspaces of T, N is a normal operator and A is a power-bounded $(n,k)$-quasi-∗-paranormal operator with ${\sigma }_r\left(A^{\ast }\right)=\varnothing$. Moreover,

$$\sigma \left(A^{\ast }\right)\subseteq {\sigma }_p\left(A^{\ast }\right)\cup {\sigma }_c\left(A^{\ast }\right)\subseteq {\sigma }_a\left(A^{\ast }\right).$$

5. 

If T is n-∗-paranormal, then ${\sigma }_a\left(T\right)={\sigma }_{ja}\left(T\right)$; also, and if $k>0$, then ${\sigma }_a\left(T\right)-\left\{0\right\}={\sigma }_{ja}\left(T\right)-\left\{0\right\}$.

Proof. 

• The result follows immediately from Proposition 1 by taking $\mathcal{M}=ker(T-\lambda )$ and $N=\lambda I$ which is normal.
• It follows from item (1).
• If $Tx=\lambda x$ and $Ty=\mu y$ with $\lambda \ne \mu$, then

  $$\lambda \langle x,y\rangle =\langle Tx,y\rangle =\langle x,T^{\ast }y\rangle =\langle x,\overline{\mu }y\rangle =\mu \langle x,y\rangle$$

  implies $\langle x,y\rangle =0$.
• From items (1), (2), (3) and according to the decomposition $\mathcal{H}=\mathcal{M}\oplus {\left(\mathcal{M}\right)}^{\perp }$, where $\mathcal{M}$ is the subspace spanned by the eigenspaces, the operator T can be written

  $$T=\left[\begin{array}{cc}N& 0\\ 0& A\end{array}\right],$$

  where ${\displaystyle {N=T|}_{\mathcal{M}}}$ is a normal operator and A is a $(n,k)$-quasi-∗-paranormal operator. Since ${\sigma }_p\left(A\right)=\varnothing$ it yields $ker(A-\lambda )=\left\{0\right\}$, for all $\lambda \in \mathbb{C}$ and so

  $$ker{(A-\lambda )}^{\perp }={\left\{0\right\}}^{\perp } \mathrm{i}.\mathrm{e}., \overline{\mathrm{ran}(A^{\ast }-\overline{\lambda })}=\mathcal{K},$$

  where $\mathcal{K}$ is the initial space of A, i.e., $\mathcal{K}={\left({\oplus }_{{\lambda }_i\in {\sigma }_p\left(T\right)}ker(T-{\lambda }_i)\right)}^{\perp }.$
  Therefore, the residual spectrum of A is empty. From the decomposition of the spectrum, we get

  $$\sigma \left(A^{\ast }\right)={\sigma }_p\left(A^{\ast }\right)\cup {\sigma }_c\left(A^{\ast }\right)\subseteq {\sigma }_a\left(A^{\ast }\right).$$
• the last statement follows from Lemma 1 and the assertion (1).

 □

Lemma 5.

If T is $(n,k)$-quasi-∗-paranormal and $\mathcal{M}$ is an invariant subspace for T, Then ${T|}_{\mathcal{M}}$ is $(n,k)$-quasi-∗-paranormal.

Proof. 

According to the decomposition $\mathcal{H}=\mathcal{M}\oplus {\mathcal{M}}^{\perp }$, then T can be written

$$T=\left[\begin{array}{cc}A& C\\ 0& B\end{array}\right]$$

where ${A=T|}_{\mathcal{M}}$. Since T is $(n,k)$-quasi-∗-paranormal, then

$$\parallel T^{\ast }\left(T^{k}x\right){\parallel }^2\le \parallel T^{(n+1)}{T}^k{x\parallel }^{\frac{2}{(n+1)}}{\parallel T^{k}x\parallel }^{\frac{2n}{(n+1)}}$$

(19)

and for all $x\in \mathcal{M}$ we have that $T^{k}x=A^{k}x$ and

$$\begin{array}{ccc}\hfill \parallel A^{\ast }\left(A^{k}x\right){\parallel }^2& \le & \parallel T^{\ast }\left(T^{k}x\right){\parallel }^2\hfill \\ & \le & \parallel T^{(n+1)}{T}^k{x\parallel }^{\frac{2}{(n+1)}}{\parallel T^{k}x\parallel }^{\frac{2n}{(n+1)}}\left(\mathrm{from}\mathrm{inequality}\left(19\right)\right)\hfill \\ & =& \parallel A^{(n+1)}{A}^k{x\parallel }^{\frac{2}{(n+1)}}{\parallel A^{k}x\parallel }^{\frac{2n}{(n+1)}}\hfill \end{array}$$

Hence, A is $(n,k)$-quasi-∗-paranormal. □

3. Main Theorems

We are ready to show our main theorems.

Definition 4.

An operator $T\in \mathcal{B}\left(\mathcal{H}\right)$ is said to have the (PF) property if $TX=X{V}^{\ast }$ for any operator $X\in \mathcal{B}(\mathcal{K},\mathcal{H})$ and any isometry $V\in \mathcal{B}\left(\mathcal{K}\right)$ implies $T^{\ast }X=XV$.

Lemma 6.

[21] Let $A\in \mathcal{B}\left(\mathcal{K}\right)$ and $B\in \mathcal{B}\left(\mathcal{H}\right)$. Then the following assertions are equivalent:

1. 

$A,B$ satisfy Fuglede-Putnam theorem;

2. 

if $AX=XB$ for any operator $X\in \mathcal{B}(\mathcal{H},\mathcal{K})$, then $\overline{\mathrm{ran}\left(X\right)}$ reduces A, ${(kerX)}^{\perp }$ reduces B and ${A|}_{\overline{\mathrm{ran}\left(X\right)}}$, ${B|}_{{(kerX)}^{\perp }}$ 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 $T\in \mathcal{B}\left(\mathcal{H}\right)$ be a power-bounded operator. A has the PF property if and only if $A=U\oplus C$ where U is unitary and C is of class $C_{.0}$.

Proof. 

Since T is a power-bounded operator then there is a g-asymptotic limit $A_{\ast }$ associated with the operator $T^{\ast }$ which is a positive operator and has the form $A_{\ast }=0\oplus A_1$ on the decomposition $\mathcal{H}={\mathcal{H}}_0\oplus {{\mathcal{H}}_0}^{\perp }$ where ${\mathcal{H}}_0=ker{A}_{\ast }$ is the stable subspace of $T^{\ast }$.

Furthermore, there exists an isometry V on $\overline{\mathrm{ran}{A}_{\ast }}={{\mathcal{H}}_0}^{\perp }$ (the asymptotic isometry associated with $A^{\ast }$), satisfy Equation (4), i.e., $VX=X{T}^{\ast }$, where $Xh=R^{\frac{1}{2}}h$, for all $h\in \mathcal{H}$. Hence,

$$TX=X{V}^{\ast }$$

(20)

It follows from the previous Lemma, for $T=A,B=V^{\ast }$, that if T has the PF property (i.e., $T,V^{\ast }$ satisfy Fuglede-Putnam theorem) then $\overline{\mathrm{ran}X}={{\mathcal{H}}_0}^{\perp }$ reduces T and ${T|}_{\overline{\mathrm{ran}\left(X\right)}}$, $V^{\ast }$ are unitarily equivalent normal operators (we have ${(kerX)}^{\perp }={{\mathcal{H}}_0}^{\perp }$). Which means that ${T|}_{\overline{\mathrm{ran}\left(X\right)}}$ is a unitary operator. Since ${\mathcal{H}}_0$ is the stable subspace of $T^{\ast }$ and $T^{\ast }{|}_{{\mathcal{H}}_0}$ is of class $C_{.0}$ then ${T|}_{{\mathcal{H}}_0}$ is of class $C_{.0}$. 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 $A\in \mathcal{B}\left(\mathcal{K}\right),B\in \mathcal{B}\left(\mathcal{H}\right)$ be power-bounded n-∗-paranormal operators.

If $AX=X{B}^{\ast }$ for any $X\in \mathcal{B}(\mathcal{H},\mathcal{K})$, then $A^{\ast }X=XB$.

Proof. 

Since B is n-∗-paranormal operator then by the Propositions 2 and 3, $B=U\oplus B_0$ on the decomposition $\mathcal{H}={\mathcal{H}}_0\oplus {{\mathcal{H}}_0}^{\perp }$ where ${\mathcal{H}}_0$ is the stable subspace of $B^{\ast }$. Setting $X=[X_1,X_2]\in \mathcal{B}({\mathcal{H}}_0\oplus {{\mathcal{H}}_0}^{\perp },K)$.

It follows from $AX=X{B}^{\ast }$ that

$$A{X}_1=X_1{B}_0^{\ast }$$

(21)

$$A{X}_2=X_2{U}^{\ast }$$

(22)

Since A is n-∗-paranormal operator and U is unitary then by the Propositions 3, we get

$$A^{\ast }{X}_2=X_{2}U$$

(23)

We have $\overline{\mathrm{ran}\left(X_1\right)}$ is invariant for A and $ker{X}_1$ is invariant for $B_0^{\ast }$. Hence, the operators A, $X_1$ and $B_0$ can be written:

$$A=\left[\begin{array}{cc}{A}_1& S\\ 0& A_2\end{array}\right],X_1=\left[\begin{array}{cc}Y& 0\\ 0& 0\end{array}\right]$$

and

$$B_0=\left[\begin{array}{cc}{B}_{01}& R\\ 0& B_{02}\end{array}\right]$$

From Lemma 5, $A_1$ is a (power) n-∗-paranormal operator, $B_{01}^{\ast }$ is of class $C_{0.}$ From the previous decompositions and Equation (21), we get $A_{1}Y=Y{B}_{01}^{\ast }$, where Y is an injective operator with dense range.

Thus,

$$A_1^{n}Yh=Y{B}_{01}^{\ast n}h$$

Hence, $\parallel A_1^{n}Yh\parallel =\parallel Y{B}_{01}^{\ast n}h\parallel \le \parallel Y\parallel \parallel B_{01}^{\ast n}h\parallel \to 0$ (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

$$A^{\ast }X=[0,A^{\ast }{X}_2]=[0,X_{2}U]=[0,X_2](B_0\oplus U)=XB.$$

(24)

 □

Theorem 2.

Let $\mathcal{A}$ be power-bounded $(n,k)$-quasi-∗-paranormal operator with reduced kernel and $\mathcal{B}$ be power-bounded n-∗-paranormal operator. If $\mathcal{A}\mathcal{X}=\mathcal{X}{\mathcal{B}}^{\ast }$ for some $\mathcal{X}\in \mathcal{B}\left(\mathcal{H}\right)$, then ${\mathcal{A}}^{\ast }\mathcal{X}=\mathcal{X}\mathcal{B}$ holds for all non-negative integers n and $k>0$.

Proof. 

If ${\sigma }_p\left(\mathcal{A}\right)-\left\{0\right\}\ne \varnothing \ne {\sigma }_p\left(\mathcal{B}\right)-\left\{0\right\}$ and $\mathcal{A}$ is reduced by its kernel, then by Corollary 1, we can write the operators $\mathcal{A},\mathcal{B}$ as follows

$$\mathcal{A}=\left[\begin{array}{cc}N& 0\\ 0& A\end{array}\right]\mathrm{and}\mathcal{B}=\left[\begin{array}{cc}M& 0\\ 0& B\end{array}\right]$$

according to the decomposition.

$\mathcal{H}=\mathcal{M}\oplus {\mathcal{M}}^{\perp }=\mathcal{K}\oplus {\mathcal{K}}^{\perp }$, where N, M are normal operators and $\mathcal{M}$, $\mathcal{K}$ are the subspaces spanned by the eigenspaces of $\mathcal{A}$ and $\mathcal{B}$ respectively, with ${\sigma }_r\left({\mathcal{A}}^{\ast }\right)={\sigma }_r\left({\mathcal{B}}^{\ast }\right)=\varnothing$.

Moreover, if $\mathcal{X}=\left[\begin{array}{cc}{X}_1& X_2\\ X_3& X\end{array}\right]$, then from $\mathcal{A}\mathcal{X}=\mathcal{X}{\mathcal{B}}^{\ast }$ it follows that

$$\left\{\begin{array}{c}N{X}_1=X_1{M}^{\ast }\\ N{X}_2=X_2{B}^{\ast }\\ A{X}_3=X_3{M}^{\ast }\\ AX=X{B}^{\ast }\end{array}\right.$$

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:

$$\begin{array}{ccc}\hfill {\mathcal{H}}_1& =& \overline{\mathrm{ran}\left(X\right)}\oplus {\overline{\mathrm{ran}\left(X\right)}}^{\perp }\mathrm{with}{\mathcal{H}}_1={\mathcal{M}}^{\perp }\hfill \\ \hfill {\mathcal{H}}_2& =& \overline{\mathrm{ran}\left(X^{\ast }\right)}\oplus kerX\mathrm{with}{\mathcal{H}}_2={\mathcal{K}}^{\perp }\hfill \end{array}$$

From the equation $AX=X{B}^{\ast }$ we deduce that $\overline{\mathrm{ran}\left(X\right)}$ is invariant for A and $kerX$ is invariant for $B^{\ast }$. Hence, the operators A, X and B can be written:

$$A=\left[\begin{array}{cc}{A}_1& S\\ 0& A_2\end{array}\right]\in B\left({\mathcal{H}}_1\right),X=\left[\begin{array}{cc}Y& 0\\ 0& 0\end{array}\right]\in B\left({\mathcal{H}}_2,{\mathcal{H}}_1\right)$$

and

$$B=\left[\begin{array}{cc}{B}_1& R\\ 0& B_2\end{array}\right]\in B\left({\mathcal{H}}_2\right).$$

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, ${\sigma }_r\left(A_1^{\ast }\right)={\sigma }_r\left(B_1^{\ast }\right)=\varnothing$.

Also, $AX=X{B}^{\ast }$ implies $A_{1}Y=Y{B}_1^{\ast }$ where Y is injective with dense range.

From Lemma 5, $A_1$ and $B_1$ have the following matrices decompositions:

$$A_1=\left[\begin{array}{cc}{A}_{11}& R\\ 0& A_{12}\end{array}\right]\mathrm{and}{B}_1=B_{11}\oplus 0$$

according to the decomposition

$$\overline{\mathrm{ran}\left(X\right)}=\overline{\mathrm{ran}\left(A_1^k\right)}\oplus {\overline{\mathrm{ran}\left(A_1^k\right)}}^{\perp }$$

and

$$\overline{\mathrm{ran}\left(X^{\ast }\right)}=\overline{\mathrm{ran}\left(B_1^k\right)}\oplus {\overline{\mathrm{ran}\left(B_1^k\right)}}^{\perp }$$

where $A_{11},B_{11}^{\ast }$ are power-bounded n-∗-paranormal operators and $A_{22}^k=0$.

It is clear that $A_{1}Y=Y{B}_1^{\ast }$ implies that $A_1^{k}Y=Y{B}_1^{\ast k}$ for any positive integer k and therefore $\overline{Y(\overline{\mathrm{ran}\left(B^k\right)})}=\overline{\mathrm{ran}{A}_1^k}$. So Y has the following matrix

$$Y=\left[\begin{array}{cc}{Y}_1& D\\ 0& Y_2\end{array}\right]$$

where $Y_1:\overline{ran\left(B_1^k\right)}\to \overline{ran\left(A_1^k\right)}$ is injective with dense range. Also, it follows from the equation $A_{1}Y=Y{B}_1^{\ast }$ that $A_{11}^k{Y}_1=Y_1{B}_{11}^{\ast k}.$ Since $A_{11}$ and $B_{11}^{\ast }$ are power-bounded n-∗-paranormal operators, then from Theorem 1 we have $A_{11}^{\ast k}{Y}_1=Y_1{B}_{11}^k$, and because of Y is injective with dense range, we get $A_{11}$ is an injective normal operator unitary equivalent to $B_{11}$. From Proposition 1, we get that $\overline{ran\left(A_1^k\right)}$ reduces $A_1$. Hence $R=0$ and from $A_{1}Y=Y{B}_1^{\ast }$ it follows that $Y_2^{\ast }{A}_{12}^{\ast }=0$. Since Y has dense range then $A_{12}=0$. Therefore $A_1=A_{11}\oplus 0$ which is a $\left(n\right)$-∗-paranormal operator. Finally, we deduce that $A_1^{\ast }Y=Y{B}_1$ and then $A^{\ast }X=XB$ and the proof is complete. □

Remark 1.

By the same method, we can prove the dual version of Theorem 2. Indeed, let $\mathcal{A}$ be a power-bounded n-∗-paranormal operator and $\mathcal{B}$ be a power-bounded $(n,k)$-quasi-∗-paranormal operator with reduced kernel. If $\mathcal{A}\mathcal{X}=\mathcal{X}{\mathcal{B}}^{\ast }$ for some $\mathcal{X}\in \mathcal{B}\left(\mathcal{H}\right)$, then ${\mathcal{A}}^{\ast }\mathcal{X}=\mathcal{X}\mathcal{B}$ holds for all non-negative integers n and $k>0$.

Definition 5.

An operator $T\in \mathcal{B}\left(\mathcal{H}\right)$ is said to be

(i) 

k-quasi-∗-class A if $T^{\ast k}|T^2|T^k\ge T^{\ast k}{\left|T^{\ast }\right|}^2{T}^k$ for non-negative integer k;

(ii) 

$(n,k)$-quasi-paranormal operator if

$$\parallel T\left(T^{k}x\right){\parallel }^{(1+n})\le \parallel T^{(1+n)}\left(T^{k}x\right)\parallel \parallel T^k{x\parallel }^n$$

for all $x\in \mathcal{H}$ and for non-negative integers $n,k.$

Lemma 7.

We have the following proper inclusions:

(i) 

(k-quasi-∗-class A) ⊂ (k-quasi-∗-paranormal);

(ii) 

the class $(n,k)$-quasi-∗-paranormal operator is normaloid, for $k=0,1$.

Proof. 

For (i) see [19].

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.

$$\begin{array}{ccc}\hfill {\parallel T\left(Tx\right)\parallel }^{2(n+1)}& =& {\langle T^{2}x,T^{2}x\rangle }^{n+1}\hfill \\ & =& {\langle T^{\ast }{T}^{2}x,Tx\rangle }^{n+1}\hfill \\ & \le & \parallel T^{\ast }{T}^2{x\parallel }^{n+1}{\parallel Tx\parallel }^{n+1}\hfill \\ & \le & |T^{n+1}{T}^{2}x\parallel \parallel T^2{\parallel }^n{\parallel Tx\parallel }^{n+1}(T\mathrm{is}(n,1)-\mathrm{quasi}-\ast -\mathrm{paranormal})\hfill \\ & =& |T^{n+2}Tx\parallel \parallel T^2{\parallel }^n{\parallel Tx\parallel }^{n+1}\hfill \end{array}$$

for all $x\in \mathcal{H}.$

Hence (n, 1)-quasi-∗-paranormal ⊂ (n + 1)-paranormal.

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 $(\mathcal{A},\mathcal{B})$ in each of the following cases:

1. 

$\mathcal{A}$ is k-quasi-∗-class A operator with reduced kernel and $\mathcal{B}$ is n-∗-paranormal operator;

2. 

$\mathcal{A}$ is n-∗-paranormal operator and $\mathcal{B}$ is k-quasi-∗-class A operator with reduced kernel;

3. 

$\mathcal{A},\mathcal{B}\in \mathcal{B}\left(\mathcal{H}\right)$ 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:

(i) 

n-∗-paranormal operator;

(ii) 

$(n,k)$-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 $n\ge 1$ [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 $X\in \mathcal{B}(\mathcal{K},\mathcal{H})$ is said to be a quasi-invertible if it is injective and has dense range. $T\in \mathcal{B}\left(\mathcal{H}\right)$ is said to be a quasi-affine transform of $R\in \mathcal{B}\left(\mathcal{K}\right)$ if there exists a quasi-invertible $X\in \mathcal{B}(\mathcal{K},\mathcal{H})$ 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 $\mathcal{H}$ of class $C_{1.}$, then it follows by the above remarks that $ker{A}_{T,g}=ker{A}_{T,g}^{\frac{1}{2}}={\mathcal{H}}_0=\left\{0\right\}$. Since A is a positive operator, then

$$\overline{\mathrm{ran}\left(A_{T,g}\right)}=\overline{\mathrm{ran}(A_{T,g}^{\frac{1}{2}})}=\mathcal{H}.$$

From Equation (3), T is a quasi-affine transform of an isometry V on $\overline{\mathrm{ran}\left(A_{T,g}\right)}$. □

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 $\mathcal{H}$, then T has the following matrix form:

$$T=\left[\begin{array}{cc}{T}_0& D\\ 0& T_1\end{array}\right]$$

on the decomposition $\mathcal{H}={\mathcal{H}}_0\oplus {{\mathcal{H}}_0}^{\perp }$, where ${\mathcal{H}}_0$ is the stable subspace of T, $T_0\in C_{0.}$ and $T_1\in C_{1.}$.

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 $\mathcal{H}$ 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 $T\in C_0$ if $\psi \left(f\right)=f\left(T\right)=0$; for some non-zero function f, where $\psi$ is a weak*weak continuous homomorphism from the Hardy space ${\mathcal{H}}^{\infty }\left(D\right)$ on the open unit disc D to the weakly closed subalgebra of $\mathcal{B}\left(\mathcal{H}\right)$ generated by T, that is an extension of the usual functional calculus. This is the ${\mathcal{H}}^{\infty }$-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 $D_T={(I-T^{\ast }T)}^{\frac{1}{2}}$ 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 $(n,k)$-quasi-∗-paranormal operator or a k-quasi-∗-class A operator with reduced kernels), then T has part (its restriction on the invariant subspace ${\mathcal{H}}_0$) in $C_0$ and its compression on ${{\mathcal{H}}_0}^{\perp }$ 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

$$T=\left[\begin{array}{cc}{T}_0& D\\ 0& T_1\end{array}\right]$$

(25)

on the decomposition $\mathcal{H}={\mathcal{H}}_0\oplus {{\mathcal{H}}_0}^{\perp }$, where ${\mathcal{H}}_0$ is the stable subspace of T and

(i) 

$T_0\in C_{00}$;

(ii) 

$T_1\in C_{10}$;

(iii) 

$T_1$ 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 $\mathcal{H}={\mathcal{H}}_0\oplus {{\mathcal{H}}_0}^{\perp }$ 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 $D_T={(I-T^{\ast }T)}^{\frac{1}{2}}$ is of Hilbert-Schmidt class. Then, $T_0\in C_0$ and ${\sigma }_p\left(T\right)$ is at most countable.

Furthermore, the following assertions are equivalent:

(i) 

$T\in C_{0.}$;

(ii) 

$T\in C_0$;

(iii) 

$ind\left(T\right)=0$.

Proof. 

Since $T_0$ is a contraction such that the defect operator $D_{T_0}$ is of Hilbert-Schmidt class, i.e., $tr(I-T_0^{\ast }{T}_0)<\infty$, by Theorem 1 in [15], $T_0$ is of class $C_0$.

Since the point spectrum of a completely non-unitary does not intersect with the unite circle and ${\sigma }_p\left(T_1\right)$ is empty, then ${\sigma }_p\left(T\right)$ lies in $\sigma \left(T_0\right)$. However, $T_0\in C_0$, that is the spectrum of $T_0$ does not fill the unit disc. Hence, ${\sigma }_p\left(T\right)$ 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 $D_T={(I-T^{\ast }T)}^{\frac{1}{2}}$ 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 $\mathbb{T}$, 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.

Theorem 3.

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 $D_T={(I-T^{\ast }T)}^{\frac{1}{2}}$ is of Hilbert-Schmidt class, then $T\in C_{10}$.

Proof. 

From Proposition 4, T has the following triangular matrix

$$T=\left[\begin{array}{cc}{T}_0& D\\ 0& T_1\end{array}\right]$$

(26)

on the decomposition $\mathcal{H}={\mathcal{H}}_0\oplus {{\mathcal{H}}_0}^{\perp }$ where ${\mathcal{H}}_0$ is the stable subspace of $T,T_0\in C_{00}$ and $T_1\in C_{10}$. Therefore, by Proposition 5, $D_{T_0}$ is of Hilbert-Schmidt class and $T_0\in C_0$ with the form

$$T_0=\left[\begin{array}{cc}A& D\\ 0& B\end{array}\right]$$

(27)

where $\sigma \left(A\right)={\sigma }_p\left(A\right)\subset \mathbb{D}$ and $\sigma \left(B\right)\subset \mathbb{T}$ (where $\mathbb{D}$ is the open unit disc). From Corollary 3 and the fact that $T_0$ is completely non-normal, it follows that ${\sigma }_p\left(T_0\right)$ is empty and yields $\sigma \left(T_0\right)=\sigma \left(B\right)\subset \mathbb{T}$. Therefore, by the Proposition 7, $T_0$ is unitary; a contradiction. This shows that $T_0$ is absent. Finally, we conclude that $T=T_1\in C_{10}$. □

5. Discussion and Further Studies

The following Putnam-Fuglede theorem is very well known:

Theorem 4.

(Putnam-Fuglede Theorem) [4,5].

Assume that $A,B\in B\left(\mathcal{H}\right)$ are normal operators. If $AX=XB$ for some $X\in B\left(\mathcal{H}\right)$, then $A^{\ast }X=X{B}^{\ast }$.

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 $\left(Y\right)$. Recently, Mecheri and Uchiyama [7] extended Putnam-Fuglede to class $\mathcal{A}$ operators. In this paper, we generalize the Putnam-Fuglede theorem to a large class of operators, say $(n,k)$-quasi-∗-paranormal operators. These results extend those given in [8,14,17,20].

As application of our main theorems, we obtain:

• Characterization of $(n,k)$-quasi-∗-paranormal operators with reduced kernel.
• Characterization of completely non-normal $(n,k)$-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.