On properties of k-quasi-class operators
© Li and Gao; licensee Springer. 2014
Received: 14 October 2013
Accepted: 10 February 2014
Published: 20 February 2014
Let n and k be positive integers; an operator is called a k-quasi-class operator if , which is a common generalization of class A and class operators. In this paper, firstly we prove some basic structural properties of this class of operators, showing that if T is a k-quasi-class operator, then the nonzero points of its point spectrum and joint point spectrum are identical, the eigen-spaces corresponding to distinct eigenvalues of T are mutually orthogonal, the nonzero points of its approximate point spectrum and joint approximate point spectrum are identical; secondly we consider the tensor products for k-quasi-class operators, giving a necessary and sufficient condition for to be a k-quasi-class operator when T and S are both nonzero operators.
Let ℋ be a separable complex Hilbert space and be the set of complex numbers. Let denote the -algebra of all bounded linear operators acting on ℋ. Recall that is called p-hyponormal for if ; when , T is called hyponormal. T is called paranormal if for all [2, 3]. T is called normaloid if for all (equivalently, , the spectral radius of T). In order to discuss the relations between paranormal and p-hyponormal and log-hyponormal operators (T is invertible and ), Furuta et al.  introduced a very interesting class of operators: class A defined by , where , which is called the absolute value of T and they showed that class A is a subclass of paranormal and contains p-hyponormal and log-hyponormal operators. Recently Yuan and Gao  introduced class (i.e., ) operators and n-paranormal operators (i.e., for every unit vector ) for some positive integer n. For more interesting properties on class and n-paranormal operators, see [6–8].
Let ℋ, be complex Hilbert spaces and the tensor product of ℋ, ; i.e., the completion of the algebraic tensor product of ℋ, with the inner product for , . Let and . denotes the tensor product of T and S; i.e., for , .
In this paper, firstly we prove some basic structural properties of this class of operators, showing that if T is a k-quasi-class operator, then the nonzero points of its point spectrum and joint point spectrum are identical, the eigen-spaces corresponding to distinct eigenvalues of T are mutually orthogonal, the nonzero points of its approximate point spectrum and joint approximate point spectrum are identical; secondly we consider the tensor products for k-quasi-class operators, giving a necessary and sufficient condition for to be a k-quasi-class operator when T and S are both nonzero operators.
2 The basic properties for k-quasi-class operators
In the following lemma, we study the matrix representation of a k-quasi-class operator with respect to the direct sum of and its orthogonal complement.
Lemma 2.1 Let be a k-quasi-class operator for positive integers n and k, and let on be matrix expression. Assume that is not dense, then is a class operator on and . Furthermore, .
That is, is a class operator on .
which implies .
Since , where is the union of the holes in , which happen to be a subset of by [, Corollary 7], , and has no interior points, we have .
for . □
In the following we give the relations between -quasiparanormal and k-quasi-class operators.
Theorem 2.2 Let T be a k-quasi-class operator for positive integers n and k. Then T is a -quasiparanormal operator.
To give a proof of Theorem 2.2, the following famous inequality is needed.
Lemma 2.3 (Hölder-McCarthy’s inequality )
for and all .
for and all .
hence T is a -quasiparanormal operator. □
Remark We give an example which is -quasiparanormal, but not k-quasi-class .
Example 2.4 Let . Then T is -quasiparanormal, but not k-quasi-class .
for all . Therefore T is -quasiparanormal by [, Lemma 2.2].
Theorem 2.5 Let be a k-quasi-class operator for positive integers k and n. If is an invariant subspace of T, then the restriction is also a k-quasi-class operator.
that is, is also a k-quasi-class operator. □
In the following, we shall show that if T is a k-quasi-class operator, then the nonzero points of its point spectrum and joint point spectrum are identical, the eigen-spaces corresponding to distinct eigenvalues of T are mutually orthogonal, the nonzero points of its approximate point spectrum and joint approximate point spectrum are identical.
Theorem 2.6 Let be a k-quasi-class operator for positive integers n and k. If and for some , then .
by . By (2.4) and (2.5), we have . This completes the proof. □
If , , and , then .
Without loss of generality, we assume . Then we have by Theorem 2.6.
Thus we have . Since , . □
Theorem 2.8 Let be a k-quasi-class operator for positive integers n and k. Then .
To prove Theorem 2.8, we need the following auxiliary results.
Lemma 2.9 (see )
for any in .
for any .
Lemma 2.10 (see )
Let be Berberian’s faithful ∗-representation. Then .
The proof is complete. □
If T satisfies for some complex λ, then for any positive integer n.
An operator is said to have finite ascent if for some positive integer n.
Theorem 2.12 Let be a k-quasi-class operator for positive integers n and k. Then has finite ascent for all complex number λ.
Proof By Theorem 2.2, we see that T is a -quasiparanormal operator. So has finite ascent for all complex number λ by [, Theorem 4.1]. □
3 Tensor products for k-quasi-class operators
Let denote the tensor product on the product space for nonzero and . The operation of taking tensor products preserves many properties of and , but by no means all of them. For example the normaloid property is invariant under tensor products, the spectraloid property is not (see [, pp.623 and 631]); and is normal if and only if T and S are normal [16, 17]; however, there exist paranormal operators and such that is not paranormal . Duggal  showed that for nonzero and , is p-hyponormal if and only if T, S are p-hyponormal. This result was extended to p-quasihyponormal operators, class A operators, ∗-class A operators, log-hyponormal operators and class operators (, ) in [20–23], respectively. The following theorem gives a necessary and sufficient condition for to be a k-quasi-class operator when T and S are both nonzero operators.
T and S are k-quasi-class operators.
Therefore the sufficiency is clear.
for all .
This implies that . Since for all , we have . This contradicts the assumption . Hence T must be a k-quasi-class operator. A similar argument shows that S is also a k-quasi-class operator. The proof is complete. □
This research is supported by the National Natural Science Foundation of China (11301155), (11271112), the Natural Science Foundation of the Department of Education, Henan Province (2011A110009), (13B110077), the Youth Science Foundation of Henan Normal University and the new teachers Science Foundation of Henan Normal University (No. qd12102).
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