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Journal of Mathematics
Volume 2013 (2013), Article ID 865396, 5 pages
Essentially -Hankel Operators
1Department of Mathematics, University of Delhi, Delhi 110007, India
2Department of Mathematics, Hans Raj College, University of Delhi, Delhi 110007, India
Received 21 August 2012; Accepted 18 October 2012
Academic Editor: Ka F. C. Yiu
Copyright © 2013 S. C. Arora and Jyoti Bhola. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The notion of essentially -Hankel operators is introduced on the space . In addition to the discussion of some algebraic and topological properties of the set , the set of all essentially -Hankel operators on , it is shown that an essentially Toeplitz Rhaly operator with determining sequence is in if and only if
The notion of Toeplitz operators was introduced by Toeplitz  in the year 1911. Hankel operators are the formal companions of Toeplitz operators. It is well known that Toeplitz and Hankel operators are characterized as solutions to the operator equations and , respectively, where denotes the unilateral forward shift on . The solutions of the operator equation (for an arbitrary complex number were described by Sun in the year 1984 . In the year 2002, Avendaño  introduced the notion of -Hankel operators as those operators which satisfy the operator equation . In a different direction, Avendaño  studied the notion of Hankel operators in reference to the Calkin algebra and introduced the notion of essentially Hankel operators on . Motivated by these developments, in this paper, we introduce and study the notion of essentially -Hankel operators on the space .
For a fixed complex number , we denote the set of all essentially -Hankel operators on by . The set is shown to be a norm-closed vector subspace of containing no essentially invertible operator. It is shown that for a general , is neither an algebra of operators on nor a self-adjoint set. Although the set contains no nonzero Toeplitz operators, it turns out to be invariant under multiplication by Toeplitz operators. It turns out that is an algebra without identity, where denotes the class of all essentially Toeplitz operators on . In particular, for purely imaginary , is a -algebra. We also show that if , contains no noncompact Rhaly operators.
We begin with the following
Definition 1 (see ). A bounded linear operator on is said to be an essentially Toeplitz operator if it satisfies for some compact operator on . The set of all essentially Toeplitz operators is denoted by .
Definition 2 (see ). A bounded linear operator on is said to be an essentially Hankel operator if it satisfies for some compact operator on . The set of all essentially Hankel operators on is denoted by . For more details, one can refer to .
Definition 3 (see ). is said to be a -Hankel operator if it satisfies .
Clearly, a 0-Hankel operator is just a Hankel operator.
Definition 4 (see ). Given a sequence of scalars, the Rhaly matrix (terraced matrix)  is defined as This means that Rhaly matrices are lower triangular matrices with constant row segments. It is known that  if is bounded, then the Rhaly matrix represents a bounded linear operator on the space (identified with ) and .
We now introduce the notion of essentially -Hankel operators on the space as follows
Definition 5. For a fixed complex number , a bounded linear operator on is said to be an essentially -Hankel operator if it satisfies the operator equation
for some compact operator on , denoting the unilateral forward shift on .
We denote the set of all essentially -Hankel operators on by . Some basic facts and observations about which follow from the definition itself are as follows: (i), where denotes the ideal of all compact operators on ,(ii)since the zero operator on is compact, every -Hankel operator is in ,(iii) if , then , where is the set of all essentially Hankel operators on introduced by Avendaño ,(iv)if is a Hankel operator, then . Therefore, every Hankel operator is in .(v)we see that the operator cannot be compact on , for any complex number . Therefore , for any , where denotes the identity operator on ,(vi)for , if and only if (vii)from the definition itself, it is clear that if is a -Hankel operator and is a compact operator on , then is in . That is, compact perturbations of -Hankel operators are in . It was shown by Avendaño that the reverse implication is not true for the case . Avendaño  proved that the Cesaro operator (i.e., the Rhaly operator corresponding to the sequence ) whose matrix with respect to the standard orthonormal basis is given by is in but is not expressible in 0-Hankel plus compact form,(viii) is a norm-closed vector subspace of , the space of all bounded linear operators on .
Proof. For in and , we have where . Therefore, . Also, if in , where each is in , then in . As each and is a uniformly closed subspace of , it follows that . Thus, . Hence, the conclusionis clear.
We see that if , then On taking adjoints on both the sides, we get Therefore, .
Thus, is a self-adjoint set. We show that for a general complex number , this is not the case. That is, is not a self-adjoint set in general. For this we begin with the following.
Theorem 6. If and are distinct complex numbers, then
Proof. Let . Then, where . Subtracting the previous two equations we have where . This implies that . Therefore, . Reverse inclusion is obvious by the definition.
Theorem 7. If , then , where .
Proof. Let . Then, for some compact operator on . Taking adjoints on both sides of (13), we obtain where . Thus, . This means that , where .
Corollary 8. A necessary condition for a noncompact operator in , , to be self-adjoint is that is purely imaginary.
Proof. Let , where is a noncompact operator on . Then, , where . Now, if , then . As is non-compact, we have . That is, . That is, is purely imaginary.
Theorem 9. Let . Then, , where denotes the essential spectrum of the operator .
Proof. Let . Then,
where is a compact operator on .
Case (i). : In this case, satisfies where . Clearly, cannot be Fredholm, for if is a Fredholm operator of index , then and are Fredholm operators with indices and , respectively, leading to , which is absurd. Therefore, .
Case (ii). : In this case, if is essentially invertible, then is a compact operator on . This leads to the essential similarly of and , which is a contradiction as , where denotes the unit circle in the complex plane. So, in this case also.
In the next theorem, we show that there is no nonzero Toeplitz operator in . For this we need the following lemmas.
Lemma 10. A nonzero Toeplitz operator cannot be a -Hankel operator.
Proof. Let be a nonzero Toeplitz operator. Then, If possible, suppose that is -Hankel also. Then, From (17) and (18), it follows that That is, for all . This means that is finite dimensional and hence a compact operator on . But nonzero Toeplitz operators are never compact. Thus, we have a contradiction and the conclusion follows.
Lemma 11. If is a nonzero Toeplitz operator, then so is , denoting the unilateral forward shift on .
Using the previous two lemmas, we now prove that the set contains no nonzero Toeplitz operator.
Theorem 12. , where denotes the set of all Toeplitz operators on .
Proof. Let . Then, for some compact operator on . Since is a Toeplitz operator, is also a Toeplitz operator on . It follows that is a Toeplitz operator on . As a nonzero Toeplitz operator cannot be compact, we must have . That is, is a -Hankel operator on . Now, using Lemma 10, we get that . Hence,
In the next theorem, we show that the set is invariant under multiplication by Toeplitz operators.
Theorem 13. If and is any Toeplitz operator on , then and both are in .
Proof. Let be a Toeplitz operator on the space . Then, we have Since is essentially unitary, we have the commutator of with , and that with is compact on . Now, let . Then, (i)consider Therefore, (ii)Consider Therefore, Hence, the conclusion is clear.
More generally, we have the following.
Theorem 14. If and , then , .
Proof. Let and . Then,
Also, Therefore, .
We mention here that the previous result was proved for the special case by Avendaño . It is easy to see that is a -algebra. This fact together with the previous theorem gives us that is an algebra of operators on as shown in the following.
Theorem 15. is an algebra of operators on with no identity.
Proof. Since is a -algebra and is a vector subspace of , it follows that is a vector subspace of . Also, if , then using Theorem 14, we get that . Thus, is an algebra of operators on . As , the theorem is proved.
Remark 16. We have seen that if is purely imaginary, then is a self-adjoint set. Therefore, for purely imaginary , turns to be a -algebra.
Theorem 17. If , then for some compact operator on .
Proof. Let . Then,
for some compact operator on .
If , then . From (32) it follows that . That is, . This implies that , where . Thus, , where . Hence, the result is clear.
Corollary 18. If , then for some compact operator on .
Proof. Let . Then, , where . Applying Theorem 17 to , we get for some compact operator on . This means that Therefore, where . So, the conclusion follows.
Remark 19. It is known that [3, 7] kernel of a -Hankel operator is an invariant subspace of , and closure of the range of a -Hankel operator is an invariant subspace of . We mention here that these facts about -Hankel operators follow as deductions to Theorem 17 and Corollary 18. For if is -Hankel then in Theorem 17 and Corollary 18 leading to and , respectively.
It is known that a Rhaly operator is essentially Toeplitz if and only if it is essentially Hankel. That is, if and only if . We show that this is not the case if . In fact, an essentially Toeplitz Rhaly operator is in if and only if it is compact. Precisely, we have the following.
Theorem 20. Let be a Rhaly operator with determining sequence . Then if and only if .
The support of UGC Research Grant no.8-1(2)/2010(MRP/NRCB) for carrying out the research work is gratefully acknowledged.
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