THE BOUNDEDNESS BELOW OF 2 2 UPPER TRIANGULAR OPERATOR MATRICES In Sung Hwang and Woo Young Lee 1 When A LH and B LK are given we denote by M C an operator acting on the Hilbert space H K of the form M C := A C, where C LK, H. In this paper we characterize the boundedness below of M C. Our characterization is as follows: M C is bounded below for some C LK, H if and only if A is bounded below and αb βa if RB is closed; βa = if RB is not closed, where α and β denote the nullity and the deficiency, respectively. In addition, we show that if σ ap and σ d denote the approximate point spectrum and the defect spectrum, respectively, then the passage from σ A 0 ap to σ ap M C can be described as follows: σ A 0 ap = σ ap M C W for every C LK, H, where W lies in certain holes in σ ap A, which happen to be subsets of σ d A σ ap B. 1 Introduction The study of upper triangular operator matrices arises naturally from the following fact: if T is a Hilbert space operator and H is an invariant subspace for T then T has the following 2 2 upper triangular operator matrix representation: T = : H H H H, 0 and one way to study operators is to see them as entries of simpler operators. The upper triangular operator matrices more generally, block operator matrices have been studied 1 Supported in part by the KOSEF through the GARC at Seoul National University and the BSRI- 1998-015-D00028.
by numerous authors. This paper is concerned with the boundedness below of 2 2 upper triangular operator matrices. Let H and K be infinite dimensional separable Hilbert spaces, let LH, K denote the set of bounded linear operators from H to K, and abbreviate LH, H to LH. If A LH write σa for the spectrum of A. If A LH, K write NA for the null space of A; RA for the range of A; αa for the nullity of A, i.e., αa := dim NA; βa for the deficiency of A, i.e., βa := dim RA. Recall that an operator A LH, K is said to be bounded below if there exists k > 0 for which x k Ax for each x H. If A LH then the approximate point spectrum, σ ap A, and the defect spectrum, σ d A, of A are defined by σ ap A := {λ C : A λ is not bounded below}; σ d A := {λ C : A λ is not onto}. If S is a compact subset of C, write int S for the interior points of S; iso S for the isolated points of S; acc S for the accumulation points of S; S for the topological boundary of S. When A LH and B LK are given we denote by M C an operator acting on H K of the form M C := A C, where C LK, H. The invertibility, the spectrum and Weyl s theorem of M C were considered in [DJ],[HLL], and [Le]. In this paper we characterize the boundedness below of M C. Our characterization is as follows: Theorem 1. An 2 2 operator matrix M C := A C is bounded below for some C LK, H if and only if A is bounded below and { αb βa if RB is closed, βa = if RB is not closed. In Section 1 we give the proof of Theorem 1. In Section 2 we give a description of the passage from σ A 0 ap to σ ap M C. 1 Proof of Theorem 1 If T LH, K then the reduced minimum modulus of T is defined by cf. [Ap] { { } inf T x : dist x, NT = 1 if T 0 γt = 0 if T = 0. Thus γt > 0 if and only if T has closed non-zero range cf. [Ap],[Go]. If T LH is a non-zero operator then we can see [Ap] that γt = inf σ T \ {0}, where T denotes
T T 1 2. Thus we have that γt = γt. From the definition we can also see that if T is bounded below then x 1 γt T x for each x H. If M C := write A C 1 M C = I C A 0. Recall [Ha1, Theorem 3.3.2] that if S LK, H and T LH, K then 2 S, T bounded below = ST bounded below = T bounded below. Since is invertible for every C LK, H, applying 2 to 1 gives I C 3 A, B bounded below = M C bounded below = A bounded below. To prove Theorem 1 we establish an auxiliary lemma, which is a result of independent interest. Lemma 1. Let T LH and T 0. Then T satisfies one of the following two conditions: i There exists a unit vector x in NT such that T x = γt ; ii There exists an orthonormal sequence {x n } in NT such that T x n γt. In particular, if RT is not closed then T must satisfy the condition ii with γt = 0. Proof. Suppose T 0 and write α := γt = inf σ T \ {0}. Let E be the spectral measure on the Borel subsets of σ T such that T = z dez. There are two cases to consider. Case 1: α acc σ T \ {0}. In this case, there exists a strictly decreasing sequence {α n } of elements in σ T \ {0} such that α n α. Since the α n s are distinct, there exists a sequence {U n } of mutually disjoint open intervals such that α n U n for all n Z +. Define F n := U n σ T n Z +. Then the F n s are nonempty relatively open subsets of σ T. Thus EF n H = {0} for each n Z +. For each n Z +, choose a unit vector x n in EF n H. Since the F n s are mutually disjoint, it follows that {x n } is an orthonormal sequence. We will show that x n NT n Z +. If T is invertible then NT = N T = H, so evidently, x n NT. Now suppose T is not invertible. Since T is a normal operator, T is unitarily equivalent to a multiplication operator M φ. But since our argument below depends only on the inner product, we may assume without loss of generality that T is a multiplication operator. Let T := M φ. If F 0 := {0} then EF 0 is the multiplication by χ φ 1 0. Thus if f N T then φf = 0 and hence χφ 1 0f x = { 0 if fx = 0, fx if fx 0, which shows that EF 0 f = f. Therefore if f N T then for each n Z +, f, x n = EF 0 f, EF n x n = f, EF 0 F n x n = f, 0 = 0,
which shows that x n N T for all n Z +. It thus follows that x n NT. On the other hand, for each n 2, T x n 2 = T T x n, x n T T EFn H = r T T EFn H sup F n 2 sup Un 2 α 2 n 1, where r denotes the spectral radius. Therefore we have that α T x n α n 1 n 2, which implies that T x n α = γt. Case 2: α iso σ T \ {0}. Let L := E{α} and M := E σ T \ {α}. Then H can be decomposed as H = L M, where L and M are T -invariant subspaces, σ T L = {α} and σ T M = σ T \ {α}: more precisely, we can write T = α 0 0 T M : L M L M. But since T x = T x for all x H, it follows that for every unit vector x 0 in L, T x 0 = T x 0 = αx 0 = α. For the second assertion suppose γt = 0 and T 0. If T satisfies the condition i then there exists a unit vector x NT such that T x = 0, giving a contradiction. This shows that T must satisfy the condition ii. Proof of Theorem 1. We first claim that if A is bounded below and RB is closed, then 4 αb βa M C is bounded below for some C LK, H. To show this suppose αb βa. Since dim NB dim RA, there exists a isometry J : NB RA. Define an operator C : K H by C := J 0 NB RA : 0 0 NB. RA Then M C is one-one. Assume to the contrary that M C is not bounded below. Then there exists a sequence xn y n of unit vectors in H K for which A C xn Axn + Cy = n 0. y n By n Write y n := α n + β n for n Z +, where α n NB and β n NB. Since γb > 0 and By n 0, it follows that β n 0. Also by the definition of C, Cy n = Cα n + β n = Cα n 0 and hence α n 0. Therefore y n 0 and x n 1. But since Ax n 0, it follows that A is not bounded below, giving a contradiction. This proves that M C is bounded below.
Conversely, suppose M C is bounded below for some C LK, H. Write M C as in 1. Since I C and A 0 have closed ranges, it follows from an index theorem of R. Harte [Ha2],[Ha3] that N A 0 N I 0 = NM C R A 0 which implies that αb + βm C = βa + β I C RMC βm C β R I C, I C I C,. Since it follows that αb βa. This proves 4. We next claim that if A is bounded below and RB is not closed, then 5 βa = M C is bounded below for some C LK, H. To show this suppose βa =. Then with no restriction on RB, M C is bounded below for some C LK, H. To see this, observe dim RA =, so there exists an isomorphism C 0 : K RA. Define an operator C : K H by RA C := C 0 0 : K. RA Then a straightforward calculation shows that M C is one-one and Ax + Cy γm C = inf x 2 + y 2 =1 By inf Ax 2 + Cy 2 1 2 x 2 + y 2 =1 inf x 2 + y 2 =1 γa 2 x 2 + y 2 1 2 min {1, γa} > 0, which implies that M C is bounded below. For the converse, assume βa = N <. Since RB is not closed it follows from Lemma 1 that there exists an orthonormal sequence {y n } in NB such that By n 0. But since M C is bounded below we have inf A C x x 2 + y 2 =1 y = inf Ax+Cy x 2 + y 2 =1 By > 0. We now argue that there exist ϵ > 0 and a subsequence {y nk } of {y n } for which 6 dist RA, Cy nk > ϵ for all k Z +.
Indeed, assume to the contrary that dist RA, Cy n 0 as n. Thus there exists a sequence {x n } in H such that dist Ax n, Cy n 0. Let zn := w n := xn y n 1 y n. Then z n w n = 1 and A C xn y n zn w n = Azn +Cw n 1 x n and Bw n 0, giving a contradiction. This proves 6. There is no loss in simplifying the notation and assuming that 7 dist RA, Cy n > ϵ for all n Z +. Since βa = N, there exists an orthonormal basis {e 1,, e N } for RA. Let P m be the projection from H to {e m } for m = 1,, N, where denotes the closed linear span. If we let Cy n := α n + β n n Z +, where α n RA and β n RA, then by 7, β n > ϵ for all n Z +. Observe that n=1 1 n β n = and hence n=1 P m 0 1 n β ne iθ n = for some m 0 {1,, N} and for some θ n [0, 2π n Z +. Now if we write y := n=1 1 n y ne iθ n, then y 2 = n=1 1 n < and hence y K. But 2 Cy P m0 Cy = P m0 C 1 n y ne iθ n = P m0 1 n β ne iθ n =, n=1 giving a contradiction. Therefore we must have that βa =. This proves 5. Now Theorem 1 follows from 3, 4 and 5. n=1 The following corollary is immediate from Theorem 1. Corollary 1. For a given pair A, B of operators we have σ ap M C = σ ap A {λ C : RB λ is closed and βa λ < αb λ} C LK,H {λ C : RB λ is not closed and βa λ < }. The following is the dual statement of Corollary 1. Corollary 2. For a given pair A, B of operators we have σ d M C = σ d B {λ C : RA λ is closed and αb λ < βa λ} C LK,H {λ C : RA λ is not closed and αb λ < }. Combining Corollaries 1 and 2 gives:
Corollary 3 [DJ, Theorem 2]. For a given pair A, B of operators we have σm C = σ ap A σ d B {λ C : αb λ βa λ}. C LK,H Remark. In many applications, the entries of block operator matrices are unbounded operators. Section 1 deals only with the bounded case. We expect that an analogue of Theorem 1 holds for the unbounded case. 2 The passage from σ A 0 ap to σ ap M C In [HLL], it was shown that the passage from σ A 0 to σm C is accomplished by removing certain open subsets of σa σb from the former, that is, there is equality 8 σ A 0 = σm C W, where W is the union of certain of the holes in σm C which happen to be subsets of σa σb. However we need not expect the case for the approximate point spectrum see Examples 1 and 2 below. The passage from σ A 0 ap to σ ap M C is more delicate. Theorem 2. For a given pair A, B of operators we have that for every C LK, H, 9 η σ ap A σ ap B = η σ ap M C, where η denotes the polynomially-convex hull. More precisely, 10 σ A 0 ap = σ ap M C W, where W lies in certain holes in σ ap A, which happen to be subsets of σ d A σ ap B. Hence, in particular, r ap M C is a constant, and furthermore for every C LK, H, 11 r A C = r A 0 = r A 0 ap = r A C ap, where r and r ap denote the spectral radius and the approximate point spectral radius. Proof. First, observe that for a given pair A, B of operators we have that for every C LK, H, 12 σ ap A σ ap M C σ ap A σ ap B = σ A 0 ap : the first and the second inclusions follow from 3 and the last equality is obvious. We now claim that for every T LH, 13 η σt = η σ ap T.
Indeed since int σ ap T int σt and σt σ ap T, we have that σt σ ap T, which implies that the passage from σ ap T to σt is filling in certain holes in σ ap T, proving 13. Now suppose λ σ ap A σ ap B \ σ ap M C. Thus by 12, λ σ ap B \ σ ap A. Since M C λ is bounded below it follows from Theorem 1 that if RB λ is not closed then βa λ =, and if instead RB λ is closed then βa λ αb λ > 0, where the last inequality comes from the fact that B λ is not one-one since B λ is not bounded below. Therefore λ σ d A. On the other hand, λ should be in one of the holes in σ ap A: for if this were not so then by 13, A λ would be invertible, a contradiction. This proves 9 and 10. The equality 11 follows at once from 9 and 13. Recall [Pe, Definition 4.8] that an operator A LH is quasitriangular if there exists a sequence {P n } n=1 of projections of finite rank in LH that converges strongly to the identity and satisfies P n AP n AP n 0. Also recall that an operator A LH is called left-fredholm if A has closed range and αa < and right-fredholm if A has closed range and βa <. If A is both left- and right-fredholm, we call it Fredholm. The index, ind A, of a left- or right-fredholm operator A is defined by ind A = αa βa. If A LH then the left essential spectrum, σ + e A, the right essential spectrum, σ e A, and the essential spectrum, σ e A, of A are defined by σ + e A = {λ C : A λ is not left-fredholm}; σ e A = {λ C : A λ is not right-fredholm}; σ e A = {λ C : A λ is not Fredholm}. Recall [Pe, Definition 1.22] that the spectral picture of an operator A LH, denoted SPA, is the structure consisting of the set σ e A, the collection of holes and pseudoholes in σ e A, and the indices associated with these holes and pseudoholes, where a hole in σ e A is a nonempty bounded component of C \ σ e A and a pseudohole in σ e A is a nonempty component of σ e A \ σ + e A or of σ e A \ σ e A. From the work of Apostol, Foias and Voiculescu [Pe, Theorem 1.31], we have that A is quasitriangular if and only if the spectral picture of A contains no hole or pseudohole associated with a negative number. We now have: Corollary 4. If A is a quasitriangular operator e.g., A is either compact or cohyponormal then for every B LK and C LK, H, σ ap M C = σ ap A σ ap B. Proof. The inclusion is the second inclusion in 12. For the reverse inclusion suppose λ σ ap A σ ap B. If λ σ ap A σ ap B \ σ ap M C then by Theorem 2, λ σ d A σ ap B and A λ is bounded below. But since A is quasitriangular, we have that βa λ αa λ = 0. Therefore A λ is invertible, a contradiction. We conclude with three examples. We first recall the definition of Toeplitz operators on the Hardy space H 2 T of the unit circle T = D in the complex plane. Recall that the
Hilbert space L 2 T has a canonical orthonormal basis given by the trigonometric functions e n z = z n, for all n Z, and that the Hardy space H 2 T is the closed linear span of {e n : n = 0, 1,...}. Write CT for the set of all continuous complex valued functions on T and H T := L H 2. If P denotes the orthogonal projection from L 2 T to H 2 T, then for every φ L T the operator T φ defined by T φ g := P φg g H 2 T is called the Toeplitz operator with symbol φ. For the basic theory of Toeplitz operators, see [Do1], [Do2], [GGK1], [GGK2], and [Ni]. Example 1. One might expect that in Theorem 2, W is the union of certain of the holes in σ ap M C together with the closure of some isolated points of σ ap B. But this is not the case. To see this, let φ H be an inner function i.e., φ = 1 a.e. with dim φh 2 = e.g., φz = exp z+λ z λ with λ = 1, let ψ be any function in CT with ψ < 1, and let J be an isometry from H 2 to φh 2. Define Tφ J M J :=. 0 T ψ Note that T φ is a non normal isometry and hence σ ap T φ = T. Since RT φ RJ, it x x x follows that M J y y for all y H 2 H 2, which says that M J is bounded below. Observe xn γm J = inf M J y n 1. xn =1 y n Thus by [Go, Theorem V.1.6], we have that for all λ < 1 γm J, i M J λ is semi-fredholm; ii αm J λ αm J = 0, which implies that M J λ is bounded below for all λ < 1. But since σt ψ is contained in the polynomially convex hull of the range of ψ, it follows from our assumption that σ ap T ψ D. Thus by Theorem 2 we have that σ ap M J = T. Note that σ ap T ψ has Tφ 0 disappeared in the passage from σ ap 0 T ψ to σ ap M J. Example 2. We need not expect a general information for removing in the passage from σ A 0 ap to σ ap M C. To see this, let T φ, T ψ, and J be given as in Example 1. Also let ζ be a function in CT such that σ ap T ζ is a compact subset σ of σ ap T ψ. We define, on H 2 H 2, A := T φ T φ, B := T ψ T ζ, C := J 0 and in turn T φ 0 J 0 0 T M C := φ 0 0 0 0 T ψ 0. 0 0 0 T ζ A straightforward calculation shows σ A 0 ap = σ ap A σ ap B = T σ ap T ψ.
On the other hand, M C is unitarily equivalent to the operator J Tφ J Tφ 0 0 T ψ 0 T ζ. Tφ By Example 1 above, σ ap 0 T ψ = T. It therefore follows that Tφ J Tφ 0 σ ap M C = σ ap 0 T σap ψ 0 T ζ = T σ. Example 3. One might conjecture that if M C is bounded below then RB is closed. But this is not the case. For example, in Example 1, take a function ψ CT whose range includes 0, and consider M C. References [Ap] C. Apostol, The reduced minimum modulus, Michigan Math. J. 32 1985, 279 294. [Do1] R.G. Douglas, Banach Algebra Techniques in Operator Theory, Academic press, New York, 1972. [Do2] R.G. Douglas, Banach Algebra Techniques in the Theory of Toeplitz Operators, CBMS 15, Providence: AMS, 1973. [DJ] H.K. Du and P. Jin, Perturbation of spectrums of 2 2 operator matrices, Proc. Amer. Math. Soc. 121 1994, 761 776. [GGK1] I. Gohberg, S. Goldberg and M.A. Kaashoek, Classes of Linear Operators, Vol I, OT 49, Birkhäuser, Basel, 1990. [GGK2] I. Gohberg, S. Goldberg and M.A. Kaashoek, Classes of Linear Operators, Vol II, OT 63, Birkhäuser, Basel, 1993. [Go] S. Goldberg, Unbounded Linear Operators, McGraw-Hill, New York, 1966. [HLL] J.K. Han, H.Y. Lee and W.Y. Lee, Invertible completions of 2 2 upper triangular operator matrices, Proc. Amer. Math. Soc. 128 2000, 119 123. [Ha1] R.E. Harte, Invertibility and Singularity for Bounded Linear Operators, Dekker, New York, 1988. [Ha2] R.E. Harte, The ghost of an index theorem, Proc. Amer. Math. Soc. 106 1989, 1031 1033. [Ha3] R.E. Harte, The ghost of an index theorem II preprint 1999. [Le] W.Y. Lee, Weyl s theorem for operator matrices, Int. Eq. Op. Th. 32 1998, 319 331. [Ni] N.K. Nikolskii, Treatise on the Shift Operator, Springer, New York, 1986. [Pe] C.M. Pearcy, Some Recent Developments in Operator Theory, CBMS 36, Providence:AMS, 1978. Department of Mathematics Sungkyunkwan University Suwon 440-746, Korea E-mail: In Sung Hwang ishwang@math.skku.ac.kr Woo Young Lee wylee@yurim.skku.ac.kr 1991 Mathematics Subject Classification. Primary 47A10, 47A55