The Residual Spectrum and the Continuous Spectrum of Upper Triangular Operator Matrices

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Filomat 28:1 (2014, 65 71 DOI 10.2298/FIL1401065H Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat The Residual Spectrum and the Continuous Spectrum of Upper Triangular Operator Matrices Guojun Hai a, Alatancang Chen a a School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China Abstract. Let H and K be separable infinite dimensional Hilbert spaces. We denote by M C the 2 2 upper triangular operator matrix acting on H K of the form M C = ( A C 0 B. For given operators A B(H and B B(K, the sets σ r (M C and σ c (M C are characterized, where σ r ( and σ c ( denote the residual spectrum and the continuous spectrum, respectively. 1. Introduction The study of operator matrices arises naturally from the following fact: if X is a Hilbert space and we decompose X as a direct sum of two closed subspaces H and K, each bounded linear operator T : X X can be expressed as the operator matrix form ( A C T = D B with respect to the space decomposition X = H K, where A : H H, B : K K, C : K H and D : H K are bounded linear operators, respectively. In particular, if H is an invariant subspace for T then D = 0, and so T has an upper triangular operator matrix form. One way to study operators is to see them as being composed of simpler operators. The operator matrices have been studied by numerous authors [1 5, 7 11]. This paper is concerned with the residual spectrum and the continuous spectrum of operator matrices. In this paper, H and K are separable infinite dimensional Hilbert spaces. Let B(H, K denote the set of bounded linear operators from H into K. When H = K we write B(H, H = B(H. If T B(H, K, we use R(T, N(T and T to denote the range space, the null space and the adjoint of T. For a linear subspace M H, its closure and orthogonal complement are denoted by M and M. Write P M for the orthogonal projection onto M along M and T M for the restriction of T to M. If T B(H, K, write n(t for the nullity of T, i.e., n(t = dim N(T, and write d(t for the deficiency of T, i.e., d(t = dim R(T. 2010 Mathematics Subject Classification. Primary 47A10, 47A55 Keywords. the residual spectrum, the continuous spectrum, operator matrices Received: 09 April 2012; Accepted: 04 December 2013 Communicated by Dragana S. Cvetković - Ilić Research supported by the NSF of Inner Mongolia (No. 2011MS0104, NSF of China (No. 11371185, the Research Program of Sciences at Universities of Inner Mongolia Autonomous Region (No. NJZZ11011, SPH-IMU (No. Z20100116. Email address: 3695946@163.com (Guojun Hai

G. Hai, A. Chen / Filomat 28:1 (2014, 65 71 66 For T B(H, the resolvent set ρ(t and the spectrum σ(t of T are defined by ρ(t ={λ C : N(T λi = {0}, R(T λi = H}, σ(t ={λ C : T λi is not invertible}. Furthermore, the spectrum σ(t is classified by the point spectrum σ p (T, the residual spectrum σ r (T and the continuous spectrum σ c (T, and we define them by σ p (T = {λ C : N(T λi {0}}, σ r (T = {λ C : N(T λi = {0}, R(T λi H}, σ c (T = {λ C : N(T λi = {0}, R(T λi R(T λi = H}. It is easy to find (see [9, p. 92] that σ p (T, σ r (T and σ c (T are mutually disjoint and σ p (T σ r (T σ c (T = σ(t = C\ρ(T. Moreover, λ σ c (T if and only if λ σ c (T. When A B(H and B B(K are given operators, we denote by M C an operator on H K of the form for C B(K, H. The set ( A C 0 B σ(m C was determined in [4]. For the approximate point spectrum, the essential spectrum, the Weyl spectrum, the point spectrum, the continuous spectrum and the residual spectrum of M C, analogous results have been obtained in many literatures (see [1, 2, 5, 10]. The sets σ ae (M C and σ l (M C were discussed in [7, 8], where σ ae ( and σ l ( denote the essential C Inv(K,H C Inv(K,H approximate point spectrum and the left spectrum, and Inv(K, H denotes the set of all invertible operators from K into H. In this paper, the sets σ r (M C and σ c (M C are characterized. 2. Main results For the proof of our main results, we need some auxiliary lemmas. Lemma 2.1. Let A B(H and B B(K be given operators. Then n(m C n(a + n(b for every C B(K, H. Proof. For every C B(K, H, it is easy to see that N(M C = ( N(A 0 {( x y and hence one can see that n(m C n(a + n(b. : Ax + Cy = 0, x N(A, y N(B Lemma 2.2. Let A B(H and B B(K be given operators, and let R(A be closed. }, (1 (i If n(b > d(a, then M C is not injective for every C B(K, H. (ii If n(b = d(a <, then for every C B(K, H we have that M C is not injective or C 3 = P R(A C N(B : N(B R(A is invertible.

G. Hai, A. Chen / Filomat 28:1 (2014, 65 71 67 Proof. It follows from the closedness of R(A that M C has an operator matrix representation : H N(B N(B R(A R(A K and A 1 : H R(A is right invertible. Let A r 1 : R(A H be a right inverse of A 1. Then there is an invertible operator U = I H A r 1 C 1 A r 1 C 2 0 I N(B 0 0 0 I N(B : H N(B N(B H N(B N(B such that 0 C 3 C 4 U =. (2 (i If n(b > d(a, then for every C B(K, H we have that C 3 : N(B R(A is not injective. This implies that M C is not injective for every C B(K, H. (ii If n(b = d(a <, then for every C B(K, H we have that M C is not injective or M C is injective. When M C is injective, it follows from (2 that C 3 : N(B R(A is injective. Therefore C 3 : N(B R(A is invertible by n(b = d(a <. Lemma 2.3. Let T B(H, and let R(T be not closed. Then there is an infinite dimensional closed subspace M of R(T such that M R(T = {0}. Proof. Since R(T is not closed, it follows that R(T is an infinite dimensional closed subspace of H. Note that H is a separable infinite dimensional Hilbert space, and hence there exists an invertible operator J : R(T H. Then T = P R(T TJ : R(T R(T is a bounded linear operator and R(T = R( T. Now, applying [6, Lemma 2.1] to T we get that there is an infinite dimensional closed subspace M R(T such that M R( T = {0}, which implies that M R(T = {0}. Our main results are the following theorems. Theorem 2.4. Let A B(H and B B(K be given operators. Then σ r (M C = [{λ C : R(A λi is not closed, d(a λi + d(b λi > 0} {λ C : n(b λi d(a λi, n(b λi < d(a λi + d(b λi} {λ C : n(b λi = d(a λi = }]\σ p (A. Proof. First we prove that the right side of the above equality includes the left side. Suppose that there exists C B(K, H such that λ σ r (M C. Without loss of generality we assume that λ = 0. Then M C is injective and R(M C H K. Since M C is injective, it follows that A is injective. Now we consider two cases. Case I. Suppose that R(A is not closed. Since R(M C H K, it follows from Lemma 2.1 that n(a + n(b n(m > 0. Therefore d(a + d(b > 0. C Case II. Suppose that R(A is closed. Then by Lemma 2.2 (i and the injectivity of M C we can infer that n(b d(a. If n(b = then d(a =. Therefore n(b = d(a =. If n(b <, one can show that n(b < d(a + d(b. Assume to the contrary that n(b d(a + d(b. This, together with n(b d(a, implies that d(b = 0 and n(b = d(a <. From Lemma 2.2 (ii it follows that M C is not injective or

G. Hai, A. Chen / Filomat 28:1 (2014, 65 71 68 C 3 : P R(A C N(B : N(B R(A is invertible for every C B(K, H. When C 3 is invertible, we derive from d(b = 0 that R(M C = H K. Indeed, since R(A is closed, it follows that M C has an operator matrix representation : H N(B N(B R(A R(A K and A 1 : H R(A is right invertible. Let A r 1 : R(A H be a right inverse of A 1. Then there is an invertible operator V = I H A r 1 C 1 A r 1 C 2 + A r 1 C 1C 1 3 C 4 0 I N(B C 1 3 C 4 0 0 I N(B : H N(B N(B H N(B N(B such that 0 C 3 C 4 V =. Note that d(b = 0, and hence R(M C V = H K. This, together with the invertibility of V, shows that R(M C = H K. This is a contradiction. Next we show that the converse inclusion is also true. Without loss of generality, assume that λ = 0. We consider three cases: Case I. Suppose that A is injective, R(A is not closed and d(a + d(b > 0. Since R(A is not closed, it follows from Lemma 2.3 that there exists an infinite dimensional closed subspace M R(A such that M R(A = {0}. Hence there is an operator C B(K, H such that N(B = N(C and R(C M. Then clearly R(C R(A = {0}. This, together with (1 and the injectivity of A, shows that M C is injective. On the other hand, since M C has an operator matrix representation A 1 C 1 0 0 0 0 : H N(B N(B R(A R(A K, it follows from d(a + d(b > 0 that R(M C H K. Hence 0 σ r (M C. Case II. Suppose that A is injective, n(b d(a and n(b < d(a + d(b. Since n(b d(a, there is an operator C B(K, H such that N(C = N(B and R(C R(A. Then M C has an operator matrix representation : H N(B N(B R(A R(A K. (3 Note that A is injective and N(C = N(B, and hence M C is injective. On the other hand, if d(b > 0 then clearly R(M C H K; if d(b = 0, it follows from d(a + d(b > n(b that d(a > n(b, and hence R(C 3 R(A, which implies that R(M C H K. Therefore 0 σ r (M C. Case III. Suppose that A is injective and n(b = d(a =. Since n(b = d(a =, there exists C B(K, H such that N(C = N(B, R(C R(A and R(C R(A. Then M C has an operator matrix representation (3. In a similar way as above, we obtain that M C is injective. Also, since R(C 3 R(A, it follows that R(M C H K. Thus 0 σ r (M C. For the continuous spectrum, we also have the following result.

G. Hai, A. Chen / Filomat 28:1 (2014, 65 71 69 Theorem 2.5. Let A B(H and B B(K be given operators. Then σ c (M C =[{λ C : R(A is not closed, d(a λi n(b λi} {λ C : R(B is not closed, d(a λi n(b λi} {λ C : d(a λi = n(b λi = }]\[σ p (A σ p (B ]. Proof. We first show that the right side of the above equality includes the left side. Suppose that there exists C B(K, H such that λ σ c (M C. Without loss of generality, assume that λ = 0. Then M C is injective and R(M C R(M C = H K. Since M C is injective, it follows that A is injective. On the other hand, note that R(M C = H K implies that M C is injective, and hence B is injective. Therefore 0 σ p (A σ p (B. We consider two cases. Case I. Suppose that R(A is not closed or R(B is not closed. If R(B is not closed, then R(A is not closed or d(a n(b. In fact, if we assume that R(A is closed and d(a < n(b, then M C is not injective for every C B(K, H by Lemma 2.2 (i, which is a contradiction. If R(A is not closed, we can also show by applying the above argument to the adjoint M that R(B is not closed or n(b d(a. C Case II. Suppose that R(A and R(B are closed. Then M C as an operator from H N(B N(B into R(A R(A K has the following operator matrix representation. Note that A and B are injective, and hence A 1 : H R(A and B 1 : N(B K are invertible. Then there exist invertible operators I A 1 1 C 1 A 1 1 C 2 U = 0 I 0 : H N(B N(B H N(B N(B 0 0 I and I 0 0 V = 0 I C 4 B 1 1 0 0 I : R(A R(A K R(A R(A K such that VM C U =. Since R(M C is not closed, it follows from the invertibility of U and V that R(VM C U is not closed. This, together with the closedness of R(A and R(B, shows that R(C 3 is not closed. Thus d(a = n(b =. Next we show that the converse inclusion is also true. Without loss of generality, assume that λ = 0. We consider three cases: Case I. Suppose that A is injective, R(B = K, R(A is not closed and d(a n(b. Since R(A is not closed, it follows from Lemma 2.3 that there exists an infinite dimensional closed subspace M R(A such that M R(A = {0}. On the other hand, note that d(a n(b, and hence there is a closed subspace G 2 N(B such that dim G 2 = d(a. Let N(B = G 1 G 2. By dim G 2 = d(a and dim M = one can show that there exist bounded linear operators C 2 : G 2 R(A and C 1 : G 1 M such that C 2 is invertible and C 1 is injective. Define C B(K, H by ( C1 0 0 C = : G 0 C 2 0 1 G 2 N(B R(A R(A.

Then M C has an operator matrix representation A 1 C 1 0 0 0 0 C 2 0 0 G. Hai, A. Chen / Filomat 28:1 (2014, 65 71 70 : H G 1 G 2 N(B R(A R(A K. Clearly, A 1, B 1, C 1 and C 2 are injective, and so by R(A M = {0} one can see that M C is injective. On the other hand, from R(B = K and R(C 2 = R(A we have that R(M C = H K. Now R(M C H K follows from the fact that R(A is not closed. Therefore 0 σ c (M C. Case II. Suppose that A is injective, R(B = K, R(B is not closed and d(a n(b. Then clearly B is injective, R(A = H, R(B is not closed and n(a d(b. In a similar way with above, we can show that there exists C B(K, H such that 0 σ c (M C, which implies that 0 σ c(m C. Case III. Suppose that A is injective, R(B = K and n(b = d(a =. Since n(b = d(a =, there is an operator C 3 : N(B R(A such that N(C 3 = {0} and R(C 3 R(C 3 = R(A. Define C B(K, H by C = ( 0 0 C 3 0 : N(B N(B R(A R(A. Then M C has an operator matrix representation : H N(B N(B R(A R(A K. Clearly, M C is injective and R(M C R(M C = H K, which implies that 0 σ c (M C. Finally, an example, to illustrate our results, is given. Example 2.6. Let H = K = l 2. Define operators A B(l 2 and B B(l 2 by Ax = (x 1, 0, x 2, 0, x 3, 0,, Bx = (x 1, x 3, x 5, x 7, x 9,, where x = (x 1, x 2, l 2. Clearly, A is left invertible and B is right invertible. Moreover, it is not hard to show that d(a = n(b =. By Theorems 2.4 and 2.5, there exist C 1 B(l 2 and C 2 B(l 2 such that 0 σ r (M C1 and 0 σ c (M C2. In fact, define C 1 B(l 2 and C 2 B(l 2 by C 1 x = (0, 0, 0, x 2, 0, x 4, 0, x 6, 0,, C 2 x = (0, x 2, 0, 1 2 x 4, 0, 1 3 x 6, 0,, where x = (x 1, x 2, l 2. Then a straightforward calculation shows that 0 σ r (M C1 and 0 σ c (M C2. References [1] X. Cao, M. Guo, B. Meng, Semi-Fredholm spectrum and Weyl s theorem for operator matrices, Acta Math. Sin. (Engl. Ser. 22 (2006 169 178. [2] D. S. Djordjević, Perturbations of spectra of operator matrices, J. Operator Theory 48 (2002 467-486 [3] S. V. Djordjević, H. Zguitti, Essential point spectra of operator matrices through local spectral theory, J. Math. Anal. Appl. 338 (2008 285 291. [4] H. Du, J. Pan, Perturbation of spectrums of 2 2 operator matrices, Proc. Amer. Math. Soc. 121 (1994 761 766. [5] I. S. Hwang, W. Y. Lee, The boundedness below of 2 2 upper triangular operator matrices, Integral Equations Operator Theory 39 (2001 267 276. [6] Y. JI, Quasitriangular + small compact = strongly irreducible, Trans. Amer. Math. Soc. 351 (1999 4657 4673.

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