On -Quasiclass A Operators

Throughout this paper let be a separable complex Hilbert space with inner product . Let denote the -algebra of all bounded linear operators on .

Let and let be an isolated point of . Here denotes the spectrum of . Then there exists a small enough positive number such that

(1.1)

Let

(1.2)

is called the Riesz idempotent with respect to , and it is well known that satisfies , , , and for all positive integers . Stampfli [1] proved that if is hyponormal (i.e., operators such that ), then

(1.3)

After that many authors extended this result to many other classes of operators. Chō and Tanahashi [2] proved that (1.3) holds if is either -hyponormal or log-hyponormal. In the case , the result was further shown by Tanahashi and Uchiyama [3] to hold for -quasihyponormal operators, by Tanahashi et al. [4] to hold for -quasihyponormal operators and by Uchiyama and Tanahashi [5] and Uchiyama [6] for class A and paranormal operators. Here an operator is called -hyponormal for if , and log-hyponormal if is invertible and . An operator is called -quasihyponormal if , where and is a positive integer; especially, when , , and , is called -quasihyponormal, -quasihyponormal, and quasihyponormal, respectively. And an operator is called paranormal if for all ; normaloid if for all positive integers . -hyponormal, log-hyponormal, -quasihyponormal, -quasihyponormal, and paranormal operators were introduced by Aluthge [7], Tanahashi [8], S. C. Arora and P. Arora [9], Kim [10], and Furuta [11, 12], respectively.

In order to discuss the relations between paranormal and -hyponormal and log-hyponormal operators, Furuta et al. [13] introduced a very interesting class of bounded linear Hilbert space operators: class A defined by , where which is called the absolute value of and they showed that class A is a subclass of paranormal and contains -hyponormal and log-hyponormal operators. Class A operators have been studied by many researchers, for example, [5, 14–19].

Recently Jeon and Kim [20] introduced quasiclass A (i.e., ) operators as an extension of the notion of class A operators, and they also proved that (1.3) holds for this class of operators when . It is interesting to study whether Stampli's result holds for other larger classes of operators.

In [21], Tanahashi et al. considered an extension of quasi-class A operators, similar in spirit to the extension of the notion of -quasihyponormality to -quasihyponormality, and prove that (1.3) holds for this class of operators in the case .

Definition 1.1.

is called a -quasiclass A operator for a positive integer if

(1.4)

Remark 1.2.

In [21], this class of operators is called quasi-class (A, ).

It is clear that the class of quasi-class A operators the class of k-quasiclass A operators and

(1.5)

We show that the inclusion relation (1.5) is strict, by an example which appeared in [20].

Example 1.3.

Given a bounded sequence of positive numbers , let be the unilateral weighted shift operator on with the canonical orthonormal basis by for all , that is,

(1.6)

Straightforward calculations show that is a -quasiclass A operator if and only if . So if and , then is a -quasiclass A operator, but not a -quasiclass A operator.

In this paper, firstly we consider some inequalities of -quasiclass A operators; secondly we prove that if is a -quasiclass A operator, then is isoloid and has finite ascent for all complex number ; at last we give a necessary and sufficient condition for to be a -quasiclass A operator when and are both non-zero operators.

In the following lemma, Tanahashi, Jeon, Kim, and Uchiyama studied the matrix representation of a -quasiclass A operator with respect to the direct sum of and its orthogonal complement.

Lemma 2.1 (see [21]).

Let be a -quasiclass A operator for a positive integer and let on be matrix expression. Assume that ran is not dense, then is a class A operator on and . Furthermore, .

Proof.

Consider the matrix representation of with respect to the decomposition : Let be the orthogonal projection of onto . Then . Since is a -quasiclass A operator, we have

(2.1)

Then

(2.2)

by Hansen's inequality [22]. On the other hand

(2.3)

Hence

(2.4)

That is, is a class A operator on .

For any ,

(2.5)

which implies .

Since , where is the union of the holes in which happen to be subset of by [23, Corollary 7], and and has no interior points, we have .

Theorem 2.2.

Let be a -quasiclass A operator for a positive integer . Then the following assertions hold.

for all and all positive integers .

If for some positive integer , then .

for all positive integers , where denotes the spectral radius of .

To give a proof of Theorem 2.2, the following famous inequality is needful.

Lemma 2.3 (Hölder-McCarthy's inequality [24]).

Let . Then the following assertions hold.

for and all .

for and all .

by Hölder-McCarthy's inequality, we have

(2.7)

for is a -quasiclass A operator.

1. (2)

   If , it is obvious that . If , then by ( ). The rest of the proof is similar.

    
2. (3)

   We only need to prove the case , that is,

    

(2.8)

If for some , then by (2) and in this case . Hence (3) is clear. Therefore we may assume for all . Then

(2.9)

by ( ), and we have

(2.10)

Hence

(2.11)

By letting , we have

(2.12)

that is,

(2.13)

Lemma 2.4 (see [21]).

Let be a -quasiclass A operator for a positive integer . If and for some , then .

Proof.

We may assume that . Let be a span of . Then is an invariant subspace of and

(2.14)

Let be the orthogonal projection of onto . It suffices to show that in (2.14). Since is a -quasiclass A operator, and , we have

(2.15)

We remark

(2.16)

Then by Hansen's inequality and (2.15), we have

(2.17)

Hence we may write

(2.18)

We have

(2.19)

This implies and . On the other hand,

(2.20)

Hence and . Since is a -quasiclass A operator, by a simple calculation we have

(2.21)

Recall that if and only if and for some contraction . Thus we have . This completes the proof.

Lemma 2.5 (see [25]).

If satisfies for some complex number , then for any positive integer .

Proof.

It suffices to show by induction. We only need to show since is clear. In fact, if , then we have by hypothesis. So we have , that is, . Hence .

An operator is said to have finite ascent if for some positive integer .

Theorem 2.6.

Let be a -quasiclass A operator for a positive integer . Then has finite ascent for all complex number .

Proof.

We only need to show the case because the case holds by Lemmas 2.4 and 2.5.

In the case , we shall show that . It suffices to show that since is clear. Now assume that . We may assume since if , it is obvious that . By Hölder-McCarthy's inequality, we have

(2.22)

So we have , which implies . Therefore .

In the following lemma, Tanahashi, Jeon, Kim, and Uchiyama extended the result (1.3) to -quasiclass A operators in the case .

Lemma 2.7 (see [21]).

Let be a -quasiclass A operator for a positive integer . Let be an isolated point of and the Riesz idempotent for . Then the following assertions hold.

If , then is self-adjoint and

(2.23)

If , then .

An operator is said to be isoloid if every isolated point of is an eigenvalue of .

Theorem 2.8.

Let be a -quasiclass A operator for a positive integer . Then is isoloid.

Proof.

Let be an isolated point. If , by ( ) of Lemma 2.7, for . Therefore is an eigenvalue of . If , by ( ) of Lemma 2.7, for . So we have . Therefore is an eigenvalue of . This completes the proof.

Let denote the tensor product on the product space for nonzero , . The following theorem gives a necessary and sufficient condition for to be a -quasiclass A operator, which is an extension of [20, Theorem 4.2].

Theorem 2.9.

Let , be nonzero operators. Then is a -quasiclass A operator if and only if one of the following assertions holds

(1) or .

(2) and are -quasiclass A operators.

Proof.

It is clear that is a -quasiclass A operator if and only if

(2.24)

Therefore the sufficiency is clear.

To prove the necessary, suppose that is a -quasiclass A operator. Let , be arbitrary. Then we have

(2.25)

It suffices to prove that if ( ) does not hold, then ( ) holds. Suppose that and . To the contrary, assume that is not a -quasiclass A operator, then there exists such that

(2.26)

From (2.25) we have

(2.27)

that is,

(2.28)

for all . Therefore is a -quasiclass A operator. As the proof in Theorem 2.2 ( ), we have

(2.29)

So we have

(2.30)

for all by (2.28). Because is a -quasiclass A operator, from Lemma 2.1 we can write on , where is a class A operator (hence it is normaloid). By (2.30) we have

(2.31)

So we have

(2.32)

where equality holds since is normaloid.

This implies that . Since for all , we have . This contradicts the assumption . Hence must be a -quasiclass A operator. A similar argument shows that is also a -quasiclass A operator. The proof is complete.