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OPERATORS WITH COMPACT IMAGINARY PART C. Foias, S. Hamid, C. Onica, and C. Pearcy∗† Abstract In this note we initiate a study of the old unsolved problem whether every T ∈ L(H) of the form T = H + iK with K compact has a nontrivial invariant subspace, using [6] as our main tool. In case K ≥ 0 we obtain some positive results. 1. Let H be a separable, infinite dimensional, complex Hilbert space. We write L(H) for the algebra bounded linear operators on H, K for the ideal of compact operators in L(H), and π for the quotient (Calkin) map L(H) → L(H)/K. For T in L(H) we will write, as usual, σ(T ), σe (T ), σle (T ), and kT ke , for the spectrum, essential (Calkin) spectrum, left essential spectrum, and essential (Calkin) norm of T , respectively, and C ∗ (S) for the unital C ∗ -algebra generated by a collection S ⊂ L(H). We remind the reader that the question whether every operator T in L(H) with compact imaginary part (i.e., T = H + iK with H, K selfadjoint and K ∈ K) has a nontrivial invariant subspace has been around for over fifty years, but, nevertheless, is still open. In fact, the major results in the area, by Livsic [9], Macaev [10], Schwartz [12] and Gohberg-Krein [7], are over forty-five years old. The purpose of this note is to use the little known but useful main theorem from [6] to make some progress on this old outstanding problem. For the reader’s convenience, we first state the main result from [6], which characterizes the set of all (positive semidefinite) Q that arise from the Aronszajn-Smith “invariant subspace procedure” applied to an arbitrary quasitriangular operator (cf. [11, Chapter 4]), in the form this theorem from [6] was used in [5] and [4], where it was also the main tool. Theorem 1.1. Let A be a unital, norm-separable, norm-closed subalgebra of L(H) and let Q ∈ L(H) be such that 0 Q 1. Then the following statements are equivalent: a) there exists a sequence {Qn }n∈N of projections in L(H) such that k(1 − Qn )AQn k → 0, A ∈ A, and {Qn } converges to Q in the weak operator topology (WOT). ∗ † 2000 Mathematics Subject Classification. Primary 47A15. Key words and phrases: invariant subspaces, compact imaginary part. 1 (1) b) With M := (QH)− (6= (0)) and N := ((1−Q)QH)− (6= (0)), there exist completely contractive algebra homomorphisms ϕ̃ : A → L(M) and θ̃ : π(A) → L(N ), with ϕ̃(1H ) = 1M and θ̃(π(1H )) = 1N , uniquely determined by the relations Q1/2 ϕ̃(A) = AQ1/2 |M , A ∈ A, (2) and (1 − Q)1/2 ϕ̃(A) = θ̃(π(A))(1 − Q)1/2 |M, A ∈ A. (3) Note that if for an arbitrary T in L(H), we write AT for the unital, norm-closed algebra generated by T , then every Q that arises from Theorem 1.1 (a) (with A = AT ) gives rise to its own algebra homomorphisms ϕ e and θe as in b) above. We also remind the reader that the Aronszajn-Smith procedure (cf. [1]) applied to a quasitriangular operator (cf. [8]) yielded the following (cf., e.g., [2] or [11, Ch. IV]). Theorem 1.2. Let T be an arbitrary quasitriangular operator in L(H). If 1 ≥ Q ≥ 0 arises as in Theorem 1.1 a), and Q(1 − Q) is not a quasiaffinity (i.e., either ker Q 6= (0) or ker(1 − Q) 6= (0)), then T has a nontrivial invariant subspace. As a consequence of Theorem 1.2, we denote by Q(H) the set of all (quasitriangular) T ∈ L(H) with the property that every Q satisfying a) and b) of Theorem 1.1 (with A = AT ) is such that Q(1 − Q) is a quasiaffinity. We also denote the set of all such Q (for a given T ) by QT , and we restrict attention in what follows to those T belonging to Q(H) with compact imaginary part. The following corollary of Theorem 1.1 is what will be needed below. Corollary 1.3. Suppose T = H + iK is in Q(H), with H, K nonzero and selfadjoint and K compact. Then T is quasitriangular, and if {Qn } is any sequence of finite-rank projections arising from Aronszajn-Smith procedure (cf. [2] or [11, Ch. IV]) satisfying (1) (with A = AT ) and converging in the WOT to an operator Q ∈ QT , then, in the notation of Theorem 1.1, M = N = H, and (2) and (3) become Q1/2 ϕ̃(A) = AQ1/2 , A ∈ AT , (4) and (1 − Q)1/2 ϕ̃(A) = θ̃(π(A))(1 − Q)1/2 , A ∈ AT . (5) We now introduce our first symmetrization of equations (4) and (5), with T as in Corollary 1.3, by setting A = T n , multiplying (4) on the left by (1 − Q)1/2 , (5) on the left by Q1/2 , and combining (4) and (5), yielding (1 − Q)T n Q = (Q − Q2 )1/2 θ̃(π(H))n (Q − Q2 )1/2 , n ∈ N. (6) Since the right-hand-side of (6) is selfadjoint for each n ∈ N, so is the left, and writing T n = Hn + iKn with Hn , Kn selfadjoint and Kn ∈ K, we obtain (1 − Q)(Hn + iKn )Q = Q(Hn − iKn )(1 − Q), 2 n ∈ N, (7) so QHn − Hn Q = i(Kn Q + QKn ) − 2i(QKn Q), n ∈ N, (8) and QT n − T n Q = QHn − Hn Q + i(QKn − Kn Q) = 2i(QKn − QKn Q), n ∈ N. (9) Next, suppose that T = H + iK is an arbitrary operator in Q(H) with H, K nonzero and selfadjoint, and K compact. For such T , we write RT for the unital, real, norm-closed algebra generated by T. It follows easily by induction that Im(R) ∈ K for every R ∈ RT . We now introduce our second symmetrization of equations (4) and (5). Lemma 1.5. Suppose T = H + iK belongs to Q(H), with H, K nonzero and selfadjoint, and K compact. Then, in the notation of Corollary 1.3, for every A = HA + iKA ∈ RT , with HA , KA selfadjoint and KA ∈ K, we have 1/2 e e Qθ(π(H (1 − Q)1/2 KA Q1/2 (1 − Q)1/2 , A )) − θ(π(HA ))Q = 2iQ A ∈ RT . (10) Proof. Upon multiplying (4) on the left by Q1/2 , and on both sides by (1 − Q)1/2 , then multiplying (5) on the left by Q, and combining (4) and (5) we obtain e Qθ(π(A))(1 − Q) = (1 − Q)1/2 Q1/2 AQ1/2 (1 − Q)1/2 , A ∈ AT . (11) Thus, in particular, since KA is compact for A ∈ RT , we have 1/2 1/2 e Qθ(π(H Q (HA + iKA )(1 − Q)1/2 Q1/2 , A ))(1 − Q) = (1 − Q) A ∈ RT (12) and taking the imaginary part of each side of (12) gives 1 e {Qθ(π(H A ))(1 − Q) 2i e − (1 − Q)θ(π(H A ))Q} 1 e e = {Qθ(π(H A )) − θ((π(HA ))Q}, 2i Q1/2 (1 − Q)1/2 KA (1 − Q)1/2 Q1/2 = (13) A ∈ RT , which yields (10). Note, in particular, a special case of (10) when A = T n := Hn + iKn with Hn , Kn selfadjoint and nonzero and Kn ∈ K: e e 2i(1 − Q)1/2 Q1/2 Kn (1 − Q)1/2 Q1/2 = Qθ(π(H n )) − θ(π(Hn ))Q n n e e = Qθ(π(H)) − θ(π(H)) Q e e = Q(θ(π(T )))n − θ((π(T )))n Q, Our investigation begins with the following lemma. 3 (14) n ∈ N. Lemma 1.6. Suppose T = H + iK ∈ Q(H), with H = H ∗ , K = K ∗ 6= 0, and K ∈ K, and there exists Q ∈ QT such that σp (Q) ∩ (0, 1) 6= ∅ – say Qe = λe with e 6= 0 and 0 < λ < 1. Then hIm(R)e, ei = 0, for all R ∈ RT . Proof. It suffices to show that hIm(Tn )e, ei = 0 for every n ∈ N. With Hn := Re(Tn ) and Kn := Im(Tn ), we have, using (9), 0 = λ hT n e, ei − hT n e, λei = h(QT n − T n Q)e, ei = 2i h(QKn − QKn Q)e, ei = 2i(λ hKn e, ei − λ2 hKn e, ei) = 2iλ(1 − λ) hKn e, ei , n ∈ N. But since 0 < λ < 1, this gives hKn e, ei = 0 for n ∈ N, as desired. (15) This next result is one of our main theorems. Theorem 1.7. Suppose T = H + iK ∈ Q(H), with H, K nonzero and selfadjoint, and K compact and positive semidefinite. If there exists a Q ∈ QT such that σp (Q) ∩ (0, 1) 6= ∅, then T has a nontrivial reducing subspace M ⊂ ker K such that T |M is selfadjoint. Consequently, T has a nontrivial hyperinvariant subspace (n.h.s.). Proof. Suppose Q ∈ QT satisfies Qe = λe with e 6= 0 and 0 < λ < 1. Then from Lemma 1.6 we obtain that Kn := Im(Tn ) satisfies hKn e, ei = 0 for all n ∈ N. In particularW(n = 1), hKe, ei = 0, and since K ≥ 0, Ke = 0. We next show that (0) 6= M := {T n e : n ∈ N0 } ⊂ ker K (6= H). First, since Ke = 0, 0 = hK3 e, ei = (HKH + KH 2 + (H 2 − K 2 )K)e, e = hHKHe, ei = hKHe, Hei . Thus KHe = 0 and KT e = KHe + iK 2 e = 0. Now suppose, by induction, that we have shown that KHj e = 0 and Kj e = 0 for j = 1, · · · , n. We wish to show that KHn+1 e = 0 and Kn+1 e = 0. To this end, since Kn e = 0 by the induction hypothesis, we write Kn+1 e = (KHn + HKn )e = 0. (16) Hence, we obtain via (16), 0 = hK2n+3 e, ei = h(Kn+1 HHn+1 + Hn+1 KHn+1 + Hn+1 HKn+1 − Kn+1 KKn+1 )e, ei = h(KHn+1 e, Hn+1 ei , and therefore, KHn+1 e = 0 as desired. Thus Kn e = 0 and KT n e = 0 for n ∈ N, so M ⊂ ker K. Hence M ⊃ T M = (H + iK)M = HM, and similarly, T ∗ M = HM ⊂ M, which shows that M reduces H and T and that T |M = H|M . If T |M is not a scalar multiple of 1M , then T |M has a nontrivial hyperinvariant subspace, and thus so does T by [3]. On the other hand, if T |M is a (real) scalar multiple of 1M , then T has 4 a nontrivial eigenspace, which cannot be all of H since K 6= 0, and this eigenspace is hyperinvariant for T. Corollary 1.8. Suppose T = H + iK ∈ Q(H), with H, K nonzero, K ∈ K, and K ≥ 0. If T is purely accretive (i.e., T has no nontrivial reducing subspace on which it is selfadjoint), then for every Q ∈ QT , σp (Q) = ∅. Corollary 1.9. Suppose T = H +iK ∈ Q(H), with H, K nonzero and selfadjoint, and K positive semidefinite and compact, and there exists Q ∈ QT such that σe (Q) is a singleton {λ} with 0 < λ < 1. Then T has a nontrivial hyperinvariant subspace. Proof. It is obvious that Q = λ1H + J where J = J ∗ is compact. If J has only finitely many nonzero eigenvalues then ker J must be infinite dimensional, so λ is an eigenvalue of Q, and the result follows from Theorem 1.7. On the other hand, if J has infinitely many nonzero eigenvalues, they must converge to 0, so Q has eigenvalues arbitrarily close to λ, and the result follows as before. Our next result shows that under certain conditions, with T ∈ Q(H), an associated e Q ∈ QT , and Tb := θ(π(T )), if Tbf = rf with r ∈ R and f 6= 0, then T f = rf also. Theorem 1.10. Suppose T = H + iK ∈ Q(H), with H, K nonzero, K ∈ K, and K ≥ 0. Suppose also that there exists Q ∈ QT such that the corresponding e Tb := θ(π(T )) satisfies Tbf = rf for some unit vector f and r ∈ R. Then, with M := ∨n∈N0 Qn f, T M ⊂ M and T |M = r1M . In particular, T f = rf, and the eigenspace of T corresponding to the eigenvalue r is a n.h.s. for T . Proof. We write R := Q1/2 (1 − Q)1/2 , note that R is a quasiaffinity, and apply (14) with n = 1 to get b f i − hQf, Hf b i = r hQf, f i − hQf, rf i = 0. 2i hRKRf, f i = hQHf, Thus, since RKR ≥ 0, f ∈ ker RKR. Next, note that by similar reasoning, we get b − HQ b 3 )f, f i 0 = h(Q3 H 2 b − HQ) b + Q(QH b − HQ)Q b b − HQ)Q b = h{Q2 (QH + (QH }f, f i = 0 + hRKRQf, Qf i + 0. Thus Qf ∈ ker RKR also, and by an easy induction argument using the formula b − HQ b n= Qn H n X j=1 j−1 b − HQ)Q b Qn−j (QH = n X Qn−j (RKR)Qj−1 , n ∈ N, (17) j=1 and the fact that b − HQ b n )f, f i = hQn rf, f i − hQn f, rf i = 0, h(Qn H n ∈ N, we obtain by induction that RKRQn f = 0, 5 n ∈ N0 , (18) b− i.e., that M := ∨n∈N0 Qn f ⊂ ker RKR. Therefore, from (17) and (18) we get (Qn H b n )f = 0, n ∈ N, and thus HQ b n f = Qn Hf b = rQn f, n ∈ N, i.e., HM b HQ ⊂ M and b b H|M = r1M. Moreover, (11) applied to A = T yields RT R = QH(1 − Q), n ∈ N0 , and since obviously M is reducing for Q, 1 − Q, and R, we obtain b − Q)g = Qr(1 − Q)g = rR2 g, RT Rg = QH(1 g ∈ M, and since R is a quasiaffinity, T Rg = rRg, g ∈ M. Thus, since (RM)− = M, T M ⊂M and T |M = r1M . In particular, T f = rf, and the eigenspace Er of T associated with r is a n.h.s. for T (since K 6= 0, M ⊂ E r 6= H). Theorems 1.7 and 1.10 can easily be seen to be equivalent (by use of the Cayley transform) to the following, where T is the unit circle in C. Theorem 1.11. Suppose T is a contraction in Q(H) such that 0 6= 1 − T ∗ T ∈ K and σ(T ) ∩ T 6= T, and let Q ∈ QT . Then the following hold : a) If σp (T ) ∩ (0, 1) 6= ∅, then T has a nontrivial reducing subspace M such that T |M is unitary. Consequently, T has a n.h.s. e b) If there exists Q ∈ QT such that the corresponding Tb = θ(π(T )) satisfies Tbf = rf for some nonzero f in H, then T f = rf as well, and the eigenspace corresponding to λ is a n.h.s. for T . The above results lead to many interesting problems. Here are two: Problem 1.12. What is the analog of Theorems 1.7 and 1.10 without the hypothesis that K ≥ 0? Problem 1.13. If H is given to be diagonalizable or, more generally, normal, what result can be proved? References [1] N. Aronszajn and K. Smith, Invariant subspaces of completely continuous operators, Ann. of Math. 60(1954), 345-350. [2] D. Deckard, R. G. Douglas, and C. Pearcy, On invariant subspaces of quasitriangular operators, Amer. J. Math. 91(1969), 637-647. [3] R.G. Douglas and C. Pearcy, Hyperinvariant subspaces and transitive algebras, Michigan Math. J. 19(1972), 1-12. [4] C. Foias, S. Hamid, C. Onica, and C. Pearcy, Inequalities for some positive functionals on operator algebras, Acta Sci. Math. (Szeged) 74(2008), 239-244. [5] C. Foias, I. B. Jung, E. Ko, and C. Pearcy, Operators that admit a moment sequence, Israel J. Math., 145(2005), 83-91. 6 [6] C. Foias, C. Pasnicu, and D. Voiculescu, Weak limits of almost invariant projections, J. Operator Theory, 2(1979), 79-93. [7] I. C. Gohberg and M. G. Krein, The theory of Volterra operators on Hilbert space and its applications, Moscow 1967. [8] P. Halmos, Quasitriangular operators, Acta Sci. Math. (Szeged) 29(1968), 283293. [9] M. S. Livsic, On the spectral resolution of linear nonselfadjoint operators, Amer. Math. Soc. Transl. 5(1957), 67-114. [10] V. I. Macaev, On a class of completely continuous operators, Soviet Math. Dokl. 2(1961), 972-975. [11] C. Pearcy, Some recent developments in operator theory, CBMS Regional Conference Series in Mathematics 36, AMS, Providence, R.I., 1978. [12] J. Schwartz, Subdiagonalization of operators in Hilbert space with compact imaginary part, Comm. Pure Appl. Math., 15 (1962), 159-172. Ciprian Foias Department of Mathematics Texas A&M University College Station, TX 77843, USA Sami M. Hamid Department of Mathematics University of North Florida Jacksonville, FL 32224, USA E-mail : [email protected] Constantin Onica Department of Mathematics The University of Texas - Pan American Edinburg, TX 78539, USA E-mail : [email protected] Carl Pearcy Department of Mathematics Texas A&M University College Station, TX 77843, USA E-mail : [email protected] 7