On Finite Rank Perturbation of Diagonalizable Operators

Transcription

1 Functonal Analyss, Approxmaton and Computaton 6 (1) (2014), Publshed by Faculty of Scences and Mathematcs, Unversty of Nš, Serba Avalable at: On Fnte Rank Perturbaton of Dagonalzable Operators R Eskandar a, F Mrzapour a a Department of Mathematcs, Unversty of Zanjan, P O Box , Zanjan, Iran Abstract Let H be a Hlbert space In ths paper we gve a necessary and suffcent condton for a λ C to be an egenvalue of the lnear operator T = D + n =1 u v, where D s a dagonalzable operator and u, v H, = 1,, n 1 Introducton and Prelmnares Throughout ths paper, let H denote a separable (complex) Hlbert space, and B(H) the C -algebra of all bounded lnear operators on H We say that an operator D B(H) s dagonalzable f there exsts an orthonormal bass {e n } for H and a bounded sequence {λ n } such that D(e n ) = λ n e n for all n N For W B(H) we denote by W the set of all operators whch commute wth elements of W and set W = (W ) For any u, v H the rank one operator u v s defned by (u v)(x) = x, v u Let us recall that a norm-closed subspace M of H s called a nontrval hypernvarant subspace for T f {0} M H and t s an nvarant subspace for every operator S {T} We use the matrx representaton for bounded lnear operators on a separable Hlbert space; e, f T B(H) and {e n } s an orthonormal bass for a separable Hlbert space H, then an nfnte matrx (a j ) represents T when Tx = ( j a j x j )e j for all x = =1 x e H In ths case, we have a j 2 c and j a j 2 c for some c > 0, n ths case the matrx operator T = (a j ) s Hlbert-Schmdt operator f,j a j 2 < For more detals see [6, Theorem 621, Theorem 56] Our study motvated by the followng problem; Does every fnte rank perturbaton of a dagonalzable operator have a nontrval hypernvarant subspace? Ths problem has been consdered n several papers and solved n some specal cases [1-5] In [2] t was shown that f an operator T C1 has the form T = D + u v, where D s a dagonalzable operator and the Fourer coeffcents {α k } and {β k } of u and v wth respect to the orthonormal bass whch dagonalzes D satsfy k=1 ( α k β k 2 3 ) <, then T has a nontrval hypernvarant subspace In [1], t was shown that f T = D + n =1 u v, where D s a dagonalzable operator wth respect to the orthonormal bass {e n } and, moreover, the Fourer coeffcents of u, v belong to the space l 1, then T has a nontrval hypernvarant subspace It s clear that, f λ s an egenvalue of T, then N(T λ) s a nontrval hypernvarant subspace 2010 Mathematcs Subject Classfcaton Prmary 47A15; Secondary 47B07 Keywords Hypernvarant subspace; Matrx operator; Fnte rank perturbaton Receved: 7 January 2014; Accepted: 21 February 2014 Communcated by Dragan S Djordjevć Emal addresses: r eskandar@znuacr (R Eskandar), fmrza@znuacr (F Mrzapour)

2 R Eskandar, F Mrzapour / FAAC 6 (1) (2014), In [5] t was shown that f the operator T = D + u v s not a normal operator and for some n 0 N, α n0 = 0 or β n0 = 0(α n and β n are the Fourer coeffcents of u, v, respectvely), then T, the adjont of T, has an egenvalue Also t s shown that, f all Fourer coeffcent of u and v are non-zero and at least one egenvalue of D has multplcty larger than 1, then T has an egenvalue The followng theorem was proved n the same paper: Theorem 11 Let u, v H and D be a dagonalzable operator Suppose that T = D + u v and µ σ(d) Then µ ρ(t) f and only f the followng condtons are satsfed: () µ s an solated egenvalue of D, λ n0, of multplcty one, () β n0 = v, e n0 0 and α n0 = u, e n0 0 Our am n ths paper s to generalze ths theorem to the operator T = D + n k=1 u v where D s dagonalzable operator and u, v H for all = 1,, n 2 Man Results Throughout ths work, let H be a separable Hlbert space and D be a dagonalzable operator wth De n = λ n e n for all n = 1,, n In ths case the set of egenvalues of D s {λ n ; n N} Frst note that f {u 1, u 2,, u n } or {v 1, v 2,, v n } s lnearly dependent then T = D + n =1 u v = D + k =1 u v for some u, v n H and k < n, so wthout loss of generalty, we assume that {u 1,, u n } and {v 1,, v n } are lnearly ndependent Theorem 21 Let T = D + n =1 u v, λ σ(d) and u = k=1 α ke k, v = k=1 β ke k Then λ ρ(t) or λ ρ(t ) f and only f λ s an solated egenvalue of D wth multplcty m wth respect to egenvectors e n1, e n2,, e nm, such that α 1n1 α 1n2 α 1nm, α 2n1 α 2n2 α 2nm,, α nn1 α nn2 α nnm (1) and β 1n1 β 2n1 β nn1, β 1n2 β 2n2 β nn2,, β 1nm β 2nm β nnm (2) are lnearly ndependent Proof Suppose that λ s an solated egenvalue of D wth multplcty m and λ = λ n1 = = λ nm, such that (1) and (2) are lnearly ndependent We show that λ σ(t) Snce D λ and so T λ are Fredholm operators of ndex zero, t suffces to prove that λ σ p (T) Suppose that (T λ)x = 0 It follows that (D λ)x + x, v u = 0 =1 (3) Consderng n k -th component of the matrx representaton n (3), we get x, v α nk = 0, k = 1,, n =1

3 and thus x, v 1 α 1n1 α 1nm + + x, v n R Eskandar, F Mrzapour / FAAC 6 (1) (2014), α nn1 α nnm = 0 (4) By hypothess, for every = 1,, n, we have x, v = 0 Now, (3) mples that (D λ)x = 0, so x = m k=1 c ke nk for some scalars c k Therefore, by (3), we have c k β nk u = 0 Snce {u 1,, u n } s a lnearly ndependent set, hence, takng complex conjugates c 1 β 1n1 β 2n1 β nn1 k + + c m β 1nm β 2nm β nnm = 0 Ths yelds that c k = 0, for k = 1,, n, and therefore x = 0 Conversely, assume that the second statement of theorem does not hold Thus we have three cases: (1) λ s not egenvalue of D; (2) λ s an egenvalue of D, but t s not solated; (3) λ s an egenvalue of D wth multplcty m and at least one of (1) and (2) are lnearly dependent If the cases (1) and (2) hold, then there exsts a sequence (λ jk ) of egenvalues of D such that converges to λ Snce we have (T λ)e jk = (λ jk λ)e jk + e jk, v u, =1 (T λ)e jk λ jk λ + e jk, v u, whch converges to zero and shows that λ σ(t) If the case (3) s vald, and we assume that (2) s lnearly dependent and λ = λ n1 = λ n2 = = λ nm, then (T λ)e nk = e nk, v u = β nk u (5) On the other hand, by usng the fact that (2) s lnearly dependent, there exst scalars c k not smultaneously zero such that c k β nk = 0, = 1,, m k Therefore (T λ)(c 1 e n1 + + c m e nm ) = 0 Hence λ s an egenvalues of T Smlarly we can show that λ s an egenvalue of T whenever (1) s lnearly dependent Remark 22 If λ satsfes n the hypothess of the above theorem, then N(T λ) s a nontrval hypernvarant subspace of T The above theorem s a generalzaton of Propostons 21, 22 and 23 n [5], for the fnte rank perturbaton of dagonalzable operators Let W be the set of scalars λ, such that u and v belong to Im(D λ) Im(D λ), for every = 1,, n Assume that u H and u = k=1 α ke k and λ W Then (D λ) 1 : Im(D λ) H s well-defned and u Im(D λ) f and only f α k 2 k=1 λ k < λ 2 Now we have the followng theorem =1

4 R Eskandar, F Mrzapour / FAAC 6 (1) (2014), Theorem 23 Suppose that λ W Then λ σ p (T) f and only f 1 + (D λ) 1 u 1, v 1 (D λ) 1 u n, v 1 A(λ) = (D λ) 1 u 1, v n 1 + (D λ) 1 u n, v n s nvertble (6) Proof Frst, assume A(λ) s nvertble and there exsts x 0 such that (T λ)x = 0 Thus we have (D λ)(x + x, v 1 (D λ) 1 u x, v n (D λ) 1 u n ) = 0 Ths mples that x + x, v 1 (D λ) 1 u x, v n (D λ) 1 u n = 0 whch at least one of the x, v s not zero By the nner product wth v ( = 1,, n), we get (1 + (D λ) 1 u 1, v 1 ) x, v (D λ) 1 u n, v 1 x, v n = 0 (D λ) 1 u 1, v n x, v (1 + (D λ) 1 u n, v n ) x, v n = 0 (7) Ths s a contradcton to the nvertblty of matrx (6) Hence N(T λ) = 0 Now suppose that the matrx (6) s not nvertble Then there s a nonzero vector x such that x, v 1, x, v 2,, x, v n s a soluton of the homogeneous systems of equatons (7) Hence y = n =1 x, v (D λ) 1 u s nonzero and (I + n =1 (D λ) 1 u v )y = 0, whence (T λ)y = 0 We defne the lnear operator ϕ : M n (B(H)) H by ϕ(a j ) =,j a j, we also set B(λ) = ((D λ) 1 u (D λ) 1 v j ) and K(λ) = ϕ(a 1 (λ)b(λ)), where A(λ) was ntroduced n Theorem 23, for every λ W Corollary 24 For any f n Im(D λ) and λ W we have Proof Let f Im(D λ) and A 1 (λ) = (a j ), then (T λ) [ (D λ) 1 K(λ) ] f = f (T λ) [ (D λ) 1 K(λ) ] f = f a j f, (D λ) 1 v j u + f, (D λ) 1 v l u l,j = f a j f, (D λ) 1 v j u +,j j=1 =1 f, (D λ) 1 v l u l a j (D λ) 1 u, v l f, (D λ) 1 v j u l = f a j f, (D λ) 1 v j u + = f,j j=1 f, (D λ) 1 v l u l (δlj a lj ) f, (D λ) 1 v j ul K(λ) f, v l u l

5 R Eskandar, F Mrzapour / FAAC 6 (1) (2014), Acknowledgments The authors would lke to sncerely thank from the referee for several useful comments References [1] Q Fang and J Xa Invarant subspaces for certan fnte-rank perturbatons of dagonal operator, J Funct Anal, 263 (2012), [2] C Foas, IB Jung, E Ko, C Pearcy Spectral decomposablty of rank-one perturbatons of normal operators, J Math Anal Appl, 375 (2011), [3] C Foas, IB Jung, E Ko, C Pearcy On rank-one perturbatons of normal operaors, J Funct Anal, 253 (2007), [4] S Hamd, C Onca and C Pearcy On the Hypernvarant Subspace Problem II, Indan Unv Math J, Vol 54(3) (2005), [5] EJ Ionascu, Rank-one perturbatons of dagonal operators, Integr equ oper theory, 39 (2001), [6] J Wedmann, Lnear Operators n Hlbert Spaces, Sprnger-Verlag, New York, 1980