THE STABILITY OF PROPERTY ( gw ) UNDER COMPACT PERTURBATION
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THE STABILITY OF PROPERTY (gw) UNDER COMPACT PERTURBATION M. H. M. Rashid · T. Prasad
Received: 5 March 2013 / Revised: 18 April 2013 / Accepted: 22 April 2013 / Published online: 17 July 2014 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2014
Abstract Let H be a complex separable infinite dimensional Hilbert space. In this paper, a necessary and sufficient condition is given for an operator T on H to satisfy that f (T ) obeys property (gw) for each function f analytic on some neighborhood of σ (T ). Also, we investigate the stability of property (gw) under (small) compact perturbations. Keywords Property (w) · Property (gw) · Compact perturbation Mathematics Subject Classifications (2010) Primary 47A10 · 47A80 · Secondary 47A53
1 Introduction Let B(H) denote the algebra of all bounded linear operators T acting on a separable Hilbert space H and K(H) the ideal of compact operators in B(H). If the range T (H) of T ∈ B(H) is closed and α(T ) = dim (T −1 (0)) < ∞ (resp., β(T ) = dim (H \ T (H)) < ∞) then T is an upper semi-Fredholm (resp., lower semi-Fredholm) operator. Let SF+ (H ) (resp., SF− (H)) denote the semigroup of upper semi-Fredholm (resp., lower semi-Fredholm) operators on H. An operator T ∈ B(H) is said to be semi-Fredholm, T ∈ SF , if T ∈ SF+ (H) ∪ SF− (H) and Fredholm if T ∈ SF+ (H) ∩ SF− (H). If T is semi-Fredholm then the index of T is defined by: ind (T ) = α(T ) − β(T ).
M. H. M. Rashid () Department of Mathematics, Faculty of Science, Mu’tah University, P.O.Box(7), Al-Karak, Jordan e-mail: malik [email protected] T. Prasad Department of Science and Humanities, Ahalia School of Engineering and Technology, Palakkad, 678557, Kerala, India e-mail: [email protected]
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M. H. M. RASHID, T. PRASAD
It is well known that if T is semi-Fredholm and K ∈ K(H), then T + K is also semiFredholm and ind (T + K) = ind T . The classes of upper semi-Weyl operators W+ (X) and lower semi-Weyl operators W− (X) are defined by:
W+ (H) = {T ∈ B(H) : T ∈ SF+ (H) and ind (T ) ≤ 0} and W− (H) = {T ∈ B(H) : T ∈ SF− (H) and ind (T ) ≥ 0}. Let a := a(T ) be the ascent of an operator T ; i.e., the smallest nonnegative integer p such that T −p = T −(p+1) . If such integer does not exist, we put a(T ) = ∞. Analogously, let d := d(T ) be the descent of an operator T ; i.e., the smallest nonnegative integer q such that T q H = T q+1 (H), and if such integer does not exist we put d(T ) = ∞. It is well known that if a(T ) and d(T ) are both finite then a(T ) = d(T ) [16, Proposition 38.3]. Moreover, 0 < a(T − λI ) = d(T − λI ) < ∞ precisely when λ is a pole of the resolvent of T , see Heuser [16, Proposition 50.2]. A bounded linear operator T acting on a Banach space H is Weyl, T ∈ W , if T ∈ W+ (H) ∩ W− (H) and Browder, T ∈ B , if T is Fredholm of finite ascent and descent. Let C denote the set of complex numbers and let σ (T ) denote the usual spectrum of T . The Wolf spectrum σSF (T ), Weyl spectrum σw (T ) and Browder spectrum σb (T ) of T are
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