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On -class A contractions

Abstract

A Hilbert space operator T belongs to -class A if | T 2 | | T | 2 0. The famous Fuglede-Putnam theorem is as follows: the operator equation AX=XB implies A X=X B when A and B are normal operators. In this paper, firstly we prove that if T is a contraction of -class A operators, then either T has a nontrivial invariant subspace or T is a proper contraction and the nonnegative operator D=| T 2 | | T | 2 is a strongly stable contraction; secondly, we show that if X is a Hilbert-Schmidt operator, A and ( B ) 1 are -class A operators such that AX=XB, then A X=X B .

MSC:47B20, 47A63.

1 Introduction

Let be a complex Hilbert space and let be the set of complex numbers. Let B(H) denote the C -algebra of all bounded linear operators acting on . For operators TB(H), we shall write kerT and ranT for the null space and the range of T, respectively. Also, let σ(T) denote the spectrum of T.

Recall that TB(H) is called p-hyponormal for p>0 if ( T T ) p ( T T ) p 0 [1]; when p=1, T is called hyponormal. And T is called paranormal if T x 2 T 2 xx for all xH [2, 3]. And T is called normaloid if T n = T n for all nN (equivalently, T=r(T), the spectral radius of T). In order to discuss the relations between paranormal and p-hyponormal and log-hyponormal operators (T is invertible and log T TlogT T ), Furuta et al. [4] introduced a very interesting class of operators: class A defined by | T 2 | | T | 2 0, where |T|= ( T T ) 1 2 which is called the absolute value of T; and they showed that the class A is a subclass of paranormal and contains p-hyponormal and log-hyponormal operators. Recently Duggal et al. [5] introduced -class A operators (i.e., | T 2 | | T | 2 0) and -paranormal operators (i.e., T x 2 T 2 xx for all xH); and they proved that a -class A operator is a generalization of hyponormal operator and -class A operators form a subclass of the class of -paranormal operators.

A contraction is an operator T such that T1. A contraction T is said to be a proper contraction if Tx<x for every nonzero xH. A strict contraction is an operator T such that T<1. A strict contraction is a proper contraction, but a proper contraction is not necessary a strict contraction, although the concepts of strict and proper contractions coincide for compact operators. A contraction T is of class C 0 if T n x0 when n for every xH (i.e., T is a strongly stable contraction) and it is said to be of class C 1 if lim n T n x>0 for every nonzero xH. Classes C 0 and C 1 are defined by considering T instead of T, and we define the class C α β for α,β=0,1 by C α β = C α C β . An isometry is a contraction for which Tx=x for every xH.

In this paper, firstly we prove that if T is a contraction of -class A operators, then either T has a nontrivial invariant subspace or T is a proper contraction and the nonnegative operator D=| T 2 | | T | 2 is a strongly stable contraction; secondly, we show that if X is a Hilbert-Schmidt operator, A and ( B ) 1 are -class A operators such that AX=XB, then A X=X B .

2 On -class A contractions

Theorem 2.1 If T is a contraction of -class A operators, then the nonnegative operator D=| T 2 | | T | 2 is a contraction whose power sequence { D n } converges strongly to a projection P, and T P=0.

Proof Suppose that T is a contraction of -class A operators, then D=| T 2 | | T | 2 0. Let R= D 1 2 . Then for every xH,

D n + 1 x , x = R n + 1 x 2 = D R n x , R n x = | T 2 | R n x , R n x | T | 2 R n x , R n x = | T 2 | 1 2 R n x 2 | T | R n x 2 R n x 2 T R n x 2 R n x 2 = D n x , x .

Thus R (and so D) is a contraction and { D n } is a decreasing sequence of nonnegative contractions. Hence { D n } converges strongly to a projection P. Moreover,

n = 0 m T R n x 2 n = 0 m ( R n x 2 R n + 1 x 2 ) = x 2 R m + 1 x 2 x 2

for all nonnegative integers m and every xH. Therefore T R n x0 as n, hence we have

T Px= T lim n D n x= lim n T R 2 n x=0

for every xH. So that T P=0. □

Theorem 2.2 Let T be a contraction of -class A operators. If T has no nontrivial invariant subspace, then

  1. (i)

    T is a proper contraction;

  2. (ii)

    the nonnegative operator D=| T 2 | | T | 2 is a strongly stable contraction (so that D C 00 ).

Proof (i) Suppose that T is a -class A operator, then | T | 2 | T 2 |. We have

T x 2 = | T | 2 x , x | T 2 | x , x | T 2 | x x= T 2 x x

for every xH. By [6] Theorem 3.6, we have that

T Tx= T 2 xif and only ifTx=Tx.

Put U={xH:Tx=Tx}=ker( | T | 2 T 2 ), which is a subspace of . In the following, we shall show that U is an invariant subspace of T. For every xU, we have

T ( T x ) 2 T 2 T x 2 = T 4 x 2 = T 2 x 2 = T T x 2 T 2 T x T x = T 2 T x T x ,
(2.1)

where the second inequality holds since T is a -class A operator. So, we have that

T 4 x 2 T 2 T x Tx,

that is, T 3 x T 2 Tx. Hence we have

T 3 x= T 2 T x .
(2.2)

By (2.2), we have

T 3 x= T 2 T x T T ( T x ) ,
(2.3)

that is, T 2 xT(Tx). Hence

T 2 x= T ( T x ) .
(2.4)

Hence by (2.1) and (2.4), we have

T 2 T x Tx= T ( T x ) 2 .

So, we have that T(Tx)=TTx. That is, U is an invariant subspace of T. Now suppose T is a contraction of -class A operators. If T is a strict contraction, then it is trivially a proper contraction. If T is not a strict contraction (i.e., T=1) and T has no nontrivial invariant subspace, then U={xH:Tx=x}={0} (actually, if U=H, then T is an isometry, and isometries have nontrivial invariant subspaces). Thus, for every nonzero xH, Tx<x, so T is a proper contraction.

  1. (ii)

    Let T be a contraction of -class A operators. By Theorem 2.1 we have D is a contraction, { D n } converges strongly to a projection P, and T P=0. So, PT=0. Suppose T has no nontrivial invariant subspace. Since kerP is a nonzero invariant subspace for T whenever PT=0 and T0, it follows that kerP=H. Hence P=0 and so D n converges strongly to 0, that is, D=| T 2 | | T | 2 is a strongly stable contraction. D is self-adjoint, so that D C 00 . □

Since a self-adjoint operator T is a proper contraction if and only if T is a C 00 -contraction, we have the following corollary by Theorem 2.2.

Corollary 2.3 Let T be a contraction of -class A operators. If T has no nontrivial invariant subspace, then both T and the nonnegative operator D=| T 2 | | T | 2 are proper contractions.

3 The Fuglede-Putnam theorem for -class A operators

The famous Fuglede-Putnam theorem is as follows [3, 7, 8].

Theorem 3.1 Let A and B be normal operators and X be an operator such that AX=XB, then A X=X B .

The Fuglede-Putnam theorem was first proved in the case A=B by Fuglede [7] and then a proof in the general case was given by Putnam [8]. Berberian [9] proved that the Fuglede theorem was actually equivalent to that of Putnam by a nice operator matrix derivation trick. Rosenblum [10] gave an elegant and simple proof of the Fuglede-Putnam theorem by using Liouville’s theorem. There were various generalizations of the Fuglede-Putnam theorem to nonnormal operators; we only cite [1114]. For example, Radjabalipour [13] showed that the Fuglede-Putnam theorem holds for hyponormal operators; Uchiyama and Tanahashi [14] showed that the Fuglede-Putnam theorem holds for p-hyponormal and log-hyponormal operators. If let XB(H) be Hilbert-Schmidt class, Mecheri and Uchiyama [15] showed that normality in the Fuglede-Putnam theorem can be replaced by A and B class A operators. In this paper, we show that if X is a Hilbert-Schmidt operator, A and ( B ) 1 are -class A operators such that AX=XB, then A X=X B .

Let C 2 (H) denote the Hilbert-Schmidt class. For each pair of operators A,BB(H), there is an operator Γ A , B defined on C 2 (H) via the formula Γ A , B (X)=AXB in [11]. Obviously, Γ A , B AB. The adjoint of Γ A , B is given by the formula Γ A , B (X)= A X B ; see details [11].

Let AB denote the tensor product on the product space HH for non-zero A,BB(H). In [5], Duggal et al. give a necessary and sufficient condition for AB to be a -class A operator.

Lemma 3.2 (see [5])

Let A,BB(H) be non-zero operators. Then AB belongs to -class A operators if and only if A and B belong to -class A operators.

Theorem 3.3 Let A and BB(H). Then Γ A , B is a -class A operator on C 2 (H) if and only if A and B belong to -class A operators.

Proof The unitary operator U: C 2 (H)HH by a map ( x y ) xy induces the -isomorphism Ψ:B( C 2 (H))B(HH) by a map XUX U . Then we can obtain Ψ( Γ A , B )=A B ; see details [16]. This completes the proof by Lemma 3.2. □

Lemma 3.4 (see [17])

Let TB(H) be a -class A operator. If λ0 and (Tλ)x=0 for some xH, then ( T λ ) x=0.

Now we are ready to extend the Fuglede-Putnam theorem to -class A operators.

Theorem 3.5 Let A and ( B ) 1 be -class A operators. If AX=XB for X C 2 (H), then A X=X B .

Proof Let Γ be defined on C 2 (H) by ΓY=AY B 1 . Since A and ( B 1 ) = ( B ) 1 are -class A operators, we have that Γ is a -class A operator on C 2 (H) by Theorem 3.3. Moreover, we have ΓX=AX B 1 =X because of AX=XB. Hence X is an eigenvector of Γ. By Lemma 3.4 we have Γ X= A X ( B 1 ) =X, that is, A X=X B . The proof is complete. □

References

  1. Aluthge A: On p -hyponormal operators for 0<p<1 . Integral Equ. Oper. Theory 1990, 13: 307–315. 10.1007/BF01199886

    Article  MathSciNet  MATH  Google Scholar 

  2. Furuta T: On the class of paranormal operators. Proc. Jpn. Acad. 1967, 43: 594–598. 10.3792/pja/1195521514

    Article  MathSciNet  MATH  Google Scholar 

  3. Furuta T: Invitation to Linear Operators. Taylor & Francis, London; 2001.

    Book  MATH  Google Scholar 

  4. Furuta T, Ito M, Yamazaki T: A subclass of paranormal operators including class of log-hyponormal and several classes. Sci. Math. 1998, 1(3):389–403.

    MathSciNet  MATH  Google Scholar 

  5. Duggal BP, Jeon IH, Kim IH: On -paranormal contractions and properties for -class A operators. Linear Algebra Appl. 2012, 436: 954–962. 10.1016/j.laa.2011.06.002

    Article  MathSciNet  MATH  Google Scholar 

  6. Kubrusly CS, Levan N: Proper contractions and invariant subspace. Int. J. Math. Math. Sci. 2001, 28: 223–230. 10.1155/S0161171201006287

    Article  MathSciNet  MATH  Google Scholar 

  7. Fuglede B: A commutativity theorem for normal operators. Proc. Natl. Acad. Sci. USA 1950, 36: 35–40. 10.1073/pnas.36.1.35

    Article  MathSciNet  MATH  Google Scholar 

  8. Putnam CR: On normal operators in Hilbert space. Am. J. Math. 1951, 73: 357–362. 10.2307/2372180

    Article  MathSciNet  MATH  Google Scholar 

  9. Berberian SK: Note on a theorem of Fuglede and Putnam. Proc. Am. Math. Soc. 1959, 10: 175–182. 10.1090/S0002-9939-1959-0107826-9

    Article  MathSciNet  MATH  Google Scholar 

  10. Rosenblum M: On a theorem of Fuglede and Putnam. J. Lond. Math. Soc. 1958, 33: 376–377.

    Article  MathSciNet  MATH  Google Scholar 

  11. Berberian SK: Extensions of a theorem of Fuglede-Putnam. Proc. Am. Math. Soc. 1978, 71: 113–114. 10.1090/S0002-9939-1978-0487554-2

    Article  MathSciNet  MATH  Google Scholar 

  12. Furuta T: On relaxation of normality in the of Fuglede-Putnam theorem. Proc. Am. Math. Soc. 1979, 77: 324–328. 10.1090/S0002-9939-1979-0545590-2

    Article  MathSciNet  MATH  Google Scholar 

  13. Radjabalipour M: An extension of Fuglede-Putnam theorem for hyponormal operators. Math. Z. 1987, 194: 117–120. 10.1007/BF01168010

    Article  MathSciNet  MATH  Google Scholar 

  14. Uchiyama A, Tanahashi K: Fuglede-Putnam’s theorem for p -hyponormal or log-hyponormal operators. Glasg. Math. J. 2002, 44: 397–410. 10.1017/S0017089502030057

    Article  MathSciNet  MATH  Google Scholar 

  15. Mecheri S, Uchiyama A: An extension of the Fuglede-Putnam’s theorem to class A operators. Math. Inequal. Appl. 2010, 13(1):57–61.

    MathSciNet  MATH  Google Scholar 

  16. Brown A, Pearcy C: Spectra of tensor products of operators. Proc. Am. Math. Soc. 1966, 17: 162–166. 10.1090/S0002-9939-1966-0188786-5

    Article  MathSciNet  MATH  Google Scholar 

  17. Mecheri S: Isolated points of spectrum of k -quasi--class A operators. Stud. Math. 2012, 208: 87–96. 10.4064/sm208-1-6

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (11071188); the Natural Science Foundation of the Department of Education, Henan Province (2011A110009), Project of Science and Technology Department of Henan province (122300410375) and Key Scientific and Technological Project of Henan Province (122102210132).

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Correspondence to Fugen Gao.

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Gao, F., Li, X. On -class A contractions. J Inequal Appl 2013, 239 (2013). https://doi.org/10.1186/1029-242X-2013-239

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Keywords

  • -class A operators
  • contraction operators
  • the Fuglede-Putnam theorem