On Properties of Class A ( n ) and n-Paranormal Operators

and Applied Analysis 3


Introduction
Let H be a separable complex Hilbert space and let C be the set of complex numbers.Let (H) denote the  * -algebra of all bounded linear operators acting on H.If  ∈ (H), we will write ker  and ran for the null space and range of , respectively.Also let () = dim ker , () = dim ker  * and let (),   () denote the spectrum, point spectrum of .Let  = () be the ascent of , that is, the smallest nonnegative integer  such that ker   = ker  +1 .If such integer does not exist, we put () = ∞.Analogously, let  = () be the descent of , that is, the smallest nonnegative integer  such that ran  = ran +1 , and if such integer does not exist, we put () = ∞.An operator  ∈ (H) is called upper (lower, resp.)semi-Fredholm if ran is closed and () < ∞ (() < ∞, resp.).If  ∈ (H) is either an upper semi-Fredholm operator or a lower semi-Fredholm operator, then  is called a semi-Fredholm operator, and the index of a semi-Fredholm operator  ∈ (H), denoted by ind(), is given by the integer ind() = () − ().If both () and () are finite, then  is called a Fredholm operator.An operator  ∈ (H) is called Weyl if it is Fredholm of index zero and Browder if it is Fredholm of finite ascent and descent.The essential spectrum   (), the Weyl spectrum   (), and the Browder spectrum   () of  ∈ (H) are defined by   () = { ∈ C :  −  is not Fredholm},   () = { ∈ C :  −  is not Weyl}, and   () = { ∈ C :  −  is not Browder}.
Let H, K be complex Hilbert spaces and H ⊗ K the tensor product of H, K, that is, the completion of the algebraic tensor product of H, K with the inner product A contraction is an operator  such that ‖‖ ≤ 1; equivalently, ‖‖ ≤ ‖‖ for every  ∈ H.A contraction  is said to be a proper contraction if ‖‖ < ‖‖ for every nonzero  ∈ H.A strict contraction is an operator  such that ‖‖ < 1.A strict contraction is a proper contraction, but a proper contraction is not necessarily a strict contraction, although the concepts of strict and proper contraction coincide for compact operators.A contraction  is of class  0. if ‖  ‖ → 0 when  → ∞ for every  ∈ H (i.e.,  is a strongly stable contraction) and it is said to be of class  1. if lim  → ∞ ‖  ‖ > 0 for every nonzero  ∈ H. Classes  .0 and  .1 are defined by considering  * instead of  and we define the class   for ,  = 0, 1 by   =  .⋂  . .An isometry is a contraction for which ‖‖ = ‖‖ for every  ∈ H.
In general, the following implications hold: In this paper, firstly we consider the tensor products for class () operators, giving a necessary and sufficient condition for  ⊗  to be a class () operator when  and  are both nonzero operators; secondly we consider the properties for -paranormal operators, showing that a paranormal contraction is the direct sum of a unitary and a  .0completely nonunitary contraction.

Tensor Products for Class 𝐴(𝑛) Operators
Let  ⊗  denote the tensor product on the product space H ⊗ K for nonzero  ∈ (H) and  ∈ (K).The operation of taking tensor products  ⊗  preserves many properties of  ∈ (H) and  ∈ (K), but it was not always this way.For example, the normaloid property is invariant under tensor products, the spectraloid property is not (see [9, pp. 623 and 631]), and  ⊗  is normal if and only if  and  are normal [10,11]; however, there exist paranormal operators  ∈ (H) and  ∈ (K) such that  ⊗  is not paranormal [12].Duggal [13] showed that for nonzero  ∈ (H) and  ∈ (K),  ⊗  is -hyponormal if and only if ,  are -hyponormal.This result was extended to -quasihyponormal operators, class  operators, log-hyponormal operators, and class (, ) [14][15][16], respectively.The following theorem gives a necessary and sufficient condition for  ⊗  to be a class () operator when  and  are both nonzero operators.
On the contrary, assume that  is not a class () operator; then there exists  0 ∈ H such that From (3), we have for all  ∈ K; that is, for all  ∈ K. Therefore,  is a class () operator.We have By ( 9) and ( 11), we have This implies that  = 0.This contradicts the assumption  ̸ = 0. Hence  must be a class () operator.A similar argument shows that  is also a class () operator.The proof is complete.

On 𝑛-Paranormal Operators
An operator  ∈ (H) is said to have the single valued extension property (SVEP) at  ∈ C if, for every open neighborhood G of , the only function  ∈ (G) such that ( − )() = 0 on  is 0 ∈ (G), where (G) means the space of all analytic functions on .When  has SVEP at each  ∈ C, say that  has SVEP.
In the following, we consider the properties of paranormal operators.References [17,18] showed that paranormal contractions and * -paranormal contractions in (H) are the direct sum of a unitary and a  .0contraction.In the following theorem, we extend this result to -paranormal operators.
Theorem 2 (see [19]).Let  be a contraction of -paranormal operators for a positive integer .Then  is the direct sum of a unitary and a  .0completely nonunitary contraction.
Proof.If  is a contraction, then the sequence {   *  } is a decreasing sequence of self-adjoint operators, converging strongly to a contraction.Let  = (lim  → ∞    *  ) 1/2 . is self-adjoint and 0 ≤  ≤  and  2  * =  2 .By [20] we have that there exists an isometry : ran() → ran() such that  =  * on ran() and ‖  ‖ → ‖‖ for every  ∈ ran(). can be extended to a bounded linear operator on H; we still denote it by .Let   =   ,  ∈ N ∪ {0}.Then for all nonnegative integers , So we have, for all  ≤ ,     =  − .The sequence {‖  ‖} is a bounded above increasing sequence.In the following, we will prove that if  is -paranormal for a positive integer , then  is a projection.Firstly we prove that {  } is a constant sequence.Suppose that  is a -paranormal operator for a positive integer .Then, for all  ≥ 1 and nonzero  ∈ ran(), Putting   = ‖  ‖ − ‖ −1 ‖, we have that where   ≥ 0 and   → 0 as  → ∞.Suppose that there exists an integer  ≥ 1 such that   > 0; then  +1 ≥ (  /) > 0, and we have that   ≥ (  /) > 0, for all  >  by an induction argument.This is contradictory with the fact that   → 0 as  → ∞.Consequently, we have that   = 0 for all , which implies that ‖ −1 ‖ = ‖  ‖ for all  ≥ 1.This means that for all  ∈ ran()‖  ‖ = ‖‖ = ‖‖.So we have that  2 =  on ran(), and so  =  on ran().Therefore, we have that  = (  0 0 0 ) on H = ran() ⊕ ker().Hence  is a projection.By [21], we have that if  is a projection, then  has a decomposition: where   is unitary and the completely nonunitary part   of  is the direct sum of backward unilateral shift  * and a  .0-contraction  0 .We will prove that  * is missing from the direct sum.It is well known that an operator  =  1 ⊕  2 has SVEP at a point  if and only if  1 and  2 have SVEP at the point .Since -paranormal operators have SVEP by [6,Corollary 3.4], it follows that if  * is present in the direct sum of , then it has SVEP.This contradicts the fact that the backward unilateral shift does not have SVEP anywhere on its spectrum except for the boundary point of its spectrum.Therefore,  =   ⊕  0 .The proof is complete.
In the following, we give a sufficient condition for a paranormal contraction to be proper.Uchiyama [23] showed that if  is paranormal and () = 0, then  is compact and normal.Now we extend this result to -paranormal operators.
where  00 () is the set of all isolated points which are eigenvalues of  with finite multiplicities.This implies that ()\() is a finite set or a countable infinite set with 0 as its only accumulation point.