Numerical Ranges of Powers of Operators Citation Shiu, Elias Sai Wan (1975) Numerical Ranges of Powers of Operators. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/344c-kv97. https://resolver.caltech.edu/CaltechTHESIS:08242021-184834279 Abstract We study the relations between a Hilbert space operator and the numerical ranges of its powers in this thesis. Let β(ℋ) denote the set of bounded linear operators on a complex Hilbert space. For T ∈ β(ℋ), let σ(T) and W(T) denote its spectrum and numerical range, respectively. The following are proved using von Neumann's theory of spectral sets: (i) σ(T) ⊂ (γ,∞) with γ > 0 and if T is not self-adjoint, then there is an index N such that {z ∈ ℂ : |z| ≤ γ n } ⊂ W(T n ) whenever n ≥ N (ii) T n is accretive, n = 1, 2, ..., k, if and only if the closed sector {z ∈ ℂ : |Arg z| ≤ π/2k} ⋃ {0} is spectral for T. In this case ∥ImTx∥ ≤ tan(π/2k) ∥ReTx∥ for each x ∈ ℋ. (i) remains valid if we replace T n by T n D, where D is a surjective operator commuting with T. Various situations in which the commutativity assumption is relaxed are examined. A theorem for finite dimensional matrices by C. R. Johnson is generalized to the operator case: If ∉ Cl(W(T n )), n = 1, 2, 3, ..., then an odd power of T is normal. Furthermore, if T is a convexoid, then T itself is normal; in fact, T is the direct sum of at most three rotated positive operators. Using these results, we prove: Let T ∈ β(ℋ), ℋ infinite dimensional and separable. If T n ∉ {Y ∈ β(ℋ) : Y = AX - XA, A,X ∈ β(ℋ), A positive}, n = 1, 2, 3, ..., then there is an odd integer m and a compact operator K o such that T m + K o is normal. Moreover, T is a normal plus a compact if and only if ∩ {Cl(W(T + K)) : K compact} is a closed polygon (possibly degenerate). Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Mathematics and Engineering Science) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Minor Option: Engineering Thesis Availability: Public (worldwide access) Research Advisor(s): De Prima, Charles R. Thesis Committee: Unknown, Unknown Defense Date: 12 May 1975 Funders: Funding Agency Grant Number Caltech UNSPECIFIED Ford Foundation UNSPECIFIED Record Number: CaltechTHESIS:08242021-184834279 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:08242021-184834279 DOI: 10.7907/344c-kv97 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 14337 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 25 Aug 2021 00:08 Last Modified: 07 Aug 2024 18:14 Thesis Files PDF - Final Version See Usage Policy. 15MB Repository Staff Only: item control page

NUMERICAL RANGES OF POWERS OF OPERATORS Thesis by Elias Sai Wan Shiu In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1975 (Submitted May 12, 1975) ii To my Parents and Grandmother iii ACKNOWLEDGMENTS I wish to give my sincerest thanks to my adviser Professor c. R. DePrima for his guidance and for his infinite patience. It is also a pleasure to express my gratitude to Professor w. A. J. Luxemburg for the numerous mathematical discuss~ons he had with me in the past few years. Equally, I am grateful to my various teachers at C~ltech, particularly Professors J. H. Anderson, H.F. Bohnenblust and O. Taussky-Todd, as well as to my friends R. G. Lautzenheiser, B. K. Richard and A. L. Rubin for their i.nnumerable contributions to my mathematical experience. The teaching experience I had as a Graduate Teaching Assistant under the supervision of Professors T. M. Apostol, R. A. Dean, c. R. DePrima, F. B. Fuller, w. A. J. Luxemburg and J. Todd has been very valuable to me and is deeply appreciated. I would also like to thank Professor N. S. Mendelsohn, of the University of Manitoba, for his help and guidance during my undergraduate years there. For financial support I am indebted to the California Institute of Technology and the Ford Foundation. Finally I must thank Mrs. Frances Williams for her excellent typing of the manuscript. iv ABSTRACT We study the relations between a Hilbert space operator and the numerical ranges of its powers in this thesis. Let B(W) denote the set of bounded linear operators on a complex Hilbert space. For TE 8('1/), let a(T) and W(T) denote its spectrum and numerical range, respectively. The following are proved using von Neu- mann' s theory of spectral sets: (i) ·If a(T) c (v,oo) with v > 0 and if Tis not self- adjoint, then there is an index N such that {z E ~ : lzl ~ '111 c W(~) whenever n > N. r1 is accretive, n = 1, 2, ••• , k, if and only if the closed sector (ii) {z E a: !Arg zl ~ n/2kt U {01 is spectral for T. In this case llrmTx\l ~ tan(n/2k) !IReTx!I for each x E U. (i) remains valid if we replace ,t1 by t1n, where Dis a surjective operator commuting with T. Various situations in which the commutativity assumption is relaxed a.re examined. A theorem for finite dimensional matrices by C. R. Johnson is generalized to the operator case: an odd power of Tis normal. If O ¢ Cl(W(t1)), n = 1, 2, 3, ... , then Furthermore, if Tis a convexoid, then T itself is normal; in fact, Tis the direct sum of at most three rotated positive operators. Using these results, we prove: Let TE B('ff), 'fl infinite dimensional and separable. If ef1 ¢ (Y E /3(U) : Y = AX - XA, A,X E B('II), A positive}, n = 1, 2, 3, ... , then there is an odd integer m and a compact operator K such that r/11 + K is normal. 0 0 Moreover, Tis a normal plus a compact if and only if n (Cl(W(T + K)) : K compact1 is a closed polygon (possibly degenerate). V TABLE OF CONTENTS Page Acknowledgments iii Abstract iv Chapter 1 1. 1 Notation 1 1.·2 Hilbert Space Operators •. 2 1.3 Numerical Range . . . . . 5 1.4 2 x 2 Operator Matrices •. 7 1. 5 Essential Numerical Range & Essential Spectra 10 1.6 n-th Root & Commutativity . . . . . . . . . . . 11 2. 1 Introduction 15 2.2 Mappings of Spectral Sets . . 15 2.3 The Ma.in Theorem 16 2.4 A 2.5 Perturbations of the Hypothesis of the Main Theorem 26 2.6 Other Related Results . . . . . . . . . . . . . . . 34 3.1 Introduction 56 3.2 A Theorem of J. H. Anderson. 36 3.3 A Theorem of Johnson&. Newman. 37 Chapter 2 Theorem of Johnson, DePrima ~Richard . . 22 Chapter 3 vi Page 3.4 0 ¢ Cl(W(Tn)) . 38 3.5 Tn f. R . . . 42 3.6 Sufficient Conditions for Normality 43 REFERENCES 47 CH.APrER 1 In this chapter we shall state certain basic results, techniques and terminology that will be required later. Many of these results have appeared in the literature and are well-known. 1. NOTATION We let~ denote the set of complex ntnnbers and R denote the set of real nmnbers. For O ca, co(O) denotes the convex hull, Cl(O) the clo- sure, Int(O) the interior and c(O) the boundary of 0. We write O > (~)r, r E IR, if O CR,and each number in O> (~)r. Let .6.(r) denote the closed disc centered at the origin with radius r, .6.(r) = { z E C : Iz I 5 r}. Lett(~) denote the closed sector of the complex plane symmetric with respect to the real axis, with vertex at the origin and angular opening 2~, I:(cp) = {z E ~ : IArg zl s_ cp} U {o}. Note that ~(n/2) denotes the closed right half plane. For a,e E ~ and e E ~, 0 < € s_ 1, we let S(a,~;e) denote the closed elliptical disc with foci at a and a and eccentricity€, Note that two degenerate cases are included in the definition: 2 2. (i) R(a,$;1) is the line segment joining a and S, (ii) B(a,a,E) is the singleton {a}. HILBERT SPACE OPERATORS We let 'II denote a complex Hilbert space with inner product( ·,·) and 13('11) the algebra of all bounded endomorphisms of Jl. For TE B(U), T* denotes its adjoint and a(T) denotes its spectrum. a(T) is a nonempty compact set in~- The spectral radius r(T) of Tis de- fined -by A E a (T)}. It can be shown that r(T) = lim \\Tnl\ 1/n . n-+oo If T1 and T2 are two commuting operators in B('ii), then and These are simple consequences of the Gelfand representation for commutative Banach algebras. See Chapter 11 of [32). o(·) is an upper semicontinuous set function on B('N) with respect to the uniform operator topology, i.e., for T E/S('II) and E > O, there exists 6 > 0 such that sup [dist(A,o(T)) h E a(s)} < E 3 whenever \\s - T\\ < 6. sional. cr( ·) is not continuous unless',/ is finite-dimen- However, o(T) does change continuously with T if the perturba- tion commutes with T. See §IV.3 of (24]. For TE B(U), an isolated pointµ of cr(T) is either a pole or an essential singularity of the resolvent 1 (A - T)- , depending on whet her the Laurent development of the resolvent in powers of A - µ has a fi nite or an infinite number of nonvanishing terms in negative powers of A - µ,respective:cy. See §s.8 of [38]. We let .7(T) denote the family of functions analytic on some neighborhood of o(T). For f E .7(T), let O be an open set in~, containing o(T), whose boundary o(O) consists of a finite number of rectifiable Jordan curves, oriented in the positive sense. If Cl(O) is contained in the domain of analyticity off, then we define f(T) by the DunfordTaylor integral f(T) does not depend on Oas long as O satisfies the above conditions. We shall find the following facts useful: (i) For SE B(U), ST= TS, then Sf(T) = f(T)S. (ii) SPECTRAL MAPPING THEOREM. (iii) If g E 7(f(T)) and h(z) = g(f(z)), then h E 7(T) and h(T) = f(cr(T)) = o(f(T)). g(f(T)). For :further details of the operational calculus, refer to Chapter VII of (11 ]. A consequence of the above three properties is: 4 (1.1) PROPOSITION. Let TE B(U) and f E .7(T). Suppose f is one-to-one on cr(T) and that for each A E a(T ) such that A is not a simple pole of 1 1 (A - T)- , we have f (A) / 0. Then T and f(T) have identical commutants. PROOF. It is sufficient to construct a function g E .7(f(T)) such that T = g(f(T)). 1 Let 0 1 = {A E a(T) : f (A) IO}. o(T) \ cr 1 are nonempty. o(T) \ o 1 We assume both o 1 and consists of finitely ma.ey points, say, _Al, ••• , Ak. There exists an open neighborhood O, of o is one-to-one and f 1 is nonzero, and c,(O,) n a(T) = ¢. 1 on which f Let N ., j = 1, ••• , k, be disjoint open sets in «: \ Cl(f(O_)) and f(A.) E: N .. J ·1 J J We put z E f(o ) g(z) = 1 z EN. J , j = 1, •.. ,k. ■ Then g E .7(f(T)) and T = g(f(T)). Proposition 1.1 generalizes the theorem in [14). In Section 6 we shall show that its converse also holds if JI is finite dimensional . For TE B('II), we let n(T) denote the nullity of T, i.e., the dimension of its nullspace. We let d(T) denote the defect of T, i.e., the dimension of the (algebraic) quotient space "/rr,t· Tis cal.led semi-Fredholm if T has closed range and either n(T) < oo or d(T) < ~- T is called Fredholm if T has closed range and both n(T) and d(T) are finite. (1 .2) PROPOSITION. Let T E 8 (,t) with O E o(a(T)). statements are equivalent. Then the following 5 (i) Tis Fredholm. (ii) d(T) is finite. (iii) Tis semi-Fredholm. Furthermore, if a:ny one of these conditions hold, n(T) = d(T) and O is a pole of (A - T) - 1 with finite rank. For a proof, see Theorem 2.7 in (26). 3. NUMERICAL RANGE The numerical range of an operator TE B(';/) is the set W(T) = {(Tx,x) : x E W, \\x\\ = 1} . The numerical radius of T, w(T), is the number w ( T) = sup [ p, I : A E W( T) } • A detailed discussion on numerical ranges may be found in Chapter 1 7 of [15]. The following list contains same of the well- known properties of numerical ranges: (i) (Toeplitz-Hausdorff) (ii) a(T) c Cl(W(T)). (iii) If U is unitary, then W(T) = W(U*TU). (iv) Let P be a nonzero (orthogonal.) projection on W. W(T) is convex. T1 = PTjw, the compression of T to "£91, then W(T 1 ) c W(T). (v) If Tis normal, then Cl(W(T)) = co(o(T)). (vi) Tis Hermitian if and only if W(T) CIR. If 6 If W(T) is real. and nonnegative, we say Tis positive and write 1 ~ T2 , we mean (T 1 - T2 ) > O. Note that a positive operator is necessarily Hermitian by (vi) and the set of all positive operaT > 0. By T tors forms a cone under " >" . * ~ o, or equivalently, Tis called accretive if (T + T) W(T) C l:(11/2). The determination of the numerical range of an operator is o:t'ten difficult. However, the following theorem describes the numerical range·s of all 2 x 2 matrices ( [ 46], [ 4 3], [ 10], [ 42], [ 15, p. 109] ) • ( 1. 3) THEOREM • (~ ~). Then Let A be the 2 x 2 upper triangular matrix W(A) (1.4) COROLIARY. a = a. For each positive integer n, n W(An) = n 2 @(a , e ; (1 + h/(a - S) I ) { an+ 6(nl'Pn-l l/2) -½ ) a == S • PROOF. a I s 7 Remark: If A is a. matrix with distinct eigenvalues o: and ~, then the numerical ranges of powers of A a.re elliptical discs with a constant eccentricity. For our purpose line segments and singletons a.re also If o:n = Sn for some integer n, then W(An) = {an}. elliptical discs. 4• 2 X 2 OPERATOR MATRICES .Let V EB'( denote the direct sum of two Hilbert spaces ,t and '(. An operator on 'JI EB 'II is expressed as a 2 x 2 matrix whose entries are operators. (1. 5) See Chapter 7 of [15]. LEMMA. Let T E A("i/ EB'<), Then W(T) = U {W( ( PROOF. (Ax,x) (By,x)) ) (ex, y) (Dy, y) x E ';I, y E '(, \\x\\ = \\y\\ = 1} . Let X E ';I, y E ?(, a,~ E ~. (T(ax EB ~Y), o:x EB Sy) Then = <(~ ~) (~;), (ax\) ayJ = <((Ax,x) (By,x)) (o:) (a)) . (Cx,y) (Dy,y) S ' ~ ■ 8 (1.6) COROLIARY. Let TE B(W ffi '() and If O I Int(W(T)), then B = 0. PROOF. ■ Apply Theorem 1 . 3. Let TE A(~) with disconnected spectrum, i.e., there are two disjoint, nonempty and closed sets a 1 and ~2 whose union is ~(T). (Some authors, e.g. [11], [38], call a 1 and cr 2 spectral sets, but we shall reserve this tenn for another concept.) a 1 such that Cl(O) n u 2 Let O be an open set containing =¢and d(n) consists of a finite number of pos- tively oriented rectifiable Jordan curves. 1 E = - . 2 1(J. Jd(O) (A - Put 1 T)- cU • Then Eis an idempotent, ET= TE and Usually, the operator Eis called a spectral projection. In order to emphasize that Eis not necessarily Hermitian, we call it a spectral idempotent in this paper. 9 (1 .7) PROPOSITION. Let T and Ebe as above and let P be the (ortho- gonal) projection on EW. E U'EB (E ~)'1, Then, with respect to the decomposition the operator matrix corresponding to T has the fonn and a(T.)::: a. l. Furthermore, T A - AT 1 PROOF. 2 1 i = 1,2. = O if and only if A = 0. From the relations EP = P and PE = E, we have Write Since TE= ET, we get C = 0 and B = T A - AT 2 . 1 The facts that ~(T.) = ~-, i = 1,2, follow from the following two equations. l. and (OO -AT 2) . (00 0T2) (0I A)I == ( OI A) I T 2 1 10 Hence or 5. ■ 0 = ~~ A = A. ESSENI'IAL NUMERICAL RANGE & ESSENTIAL SPECTRA Let~ be a complex Banach algebra with unit 1. dual space. Let~* denote its For a E il, u(a) denotes the spectrum of a and V(~J,a) de - notes the algebra numerical range of a, V ( ~, a) = { f ( a) : f E ij* , f ( 1 ) = 1 = \If\ I}. V(~,a) is a compact, convex set containing o(a). A detailed discussion of the numerical ranges of Banach algebras appears in [4] and [5]. Let ~l be a C* -algebra with unit, then by the Gelfand- Naimark theo rem there exists an isometric *-isomorphism T of~ onto a closed selfadjoint subalgebra of B(-lj), fa suitably chosen Hilbert space. Theorem 12.41 of (32). See Furthermore, we have V(~,a) = Cl(W(T(a))). ([3, Theorem 12], (2, Theorem 3]) For the rest of this section, we assume" is infinite dimensional and separable. We let '((U) denote the set of all compact operators and let ~(U) = B('ll)j'((U). [8). ij(U") is a c*-algebra called the Calkin algebra Let IT denote the canonical homomorphism from /3(',/) onto fil(.U) . The essential numerical range W (T) of an operator TE B(U) is by defie nition the algebra numerical range V(~(W), TI(T)). that we (T) = n {cl(W(T + K)) : K E K(W) L It is shown in [35) 11 ~(TI(T)) is called the Wolf (or Fredholm or Calkin) essential spectrum [39]. We define the Weyl essential spectrum aw(T) to be the largest subset of a(T) which is invariant under compact perturbations of T, Stampfli [36] has shown that there exists K0 E '<(W}, K0 depending on T, such that aw(T) = a(T + K0 ). It is proved in [13) that aW(T) consists of a(n(T)) together with some of the bounded components of the complement of a(TT(T)). Conse - quently, if o(TI(T)) lies on a simple arc, a(TI(T)) = aW(T). Most of the operators to be discussed in the rest of this paper will have essential spectra lying on finitely many disjoint line segments . (1 .8) THEOREM. [44], (7, p.62] Let T E 8(,t). Suppose TI(T) is normal and a(TI(T)) lies on a simple a.re. Then, there exists a compact operator K 0 such that T + K0 is normal. and a(T + K) = a(TI(T)). 0 NOTE: If the simple arc is a subset of the real axis, then Theorem 1.8 is obvious. Suppose TI(T) = (TI(T)) * . Then T - T* E '<(Al') and consequent- ly, T - Re(T) E '<(N'). 6. N-TH ROOTS & COMMUTATIVITY Given two n-th roots of an operator, we want to know when they are identical. (1 .9) The following theorem gives a sufficient condition. PROPOSITION. Let A,B E B(J/) such that 12 1 ~ j ~ n - 1, rn = exp(2rri/n). If An = Bn, th en A = B• Proposition (1.9) may be proved with the Dunford-Taylor integral, but it is a special case of (1 .10) THEOREM. Let A, BE /3('il) such that w = exp(2rri/n). If A~ = DBn for some D E B(JI), then AD = DB . With An= Bn, Theorem 1.10 is due to (12], and it is a special case of Proposition 1 .1 with f(z) c zn. We sketch two proofs for Theo- rem 1. 1O, the first one was suggested by De Prima and the second, Hille (18,I]. PROOF. For C E /3(W), define linear maps LC and RC on B(V) by LC(T) = CT and RC(T) = TC, respectively. n-1 I) Write J = \L L n-1-j R j , then O = A~ - DBn = J(AD - DB). . O A B J= Since LA¾=~ LA' we have n-1 ~(J) c { l an-l-j bj a E o(A), b E a(B)}. j=O (The above inclusion is actually an equality, see [27].) By hypothesis,O t a(A) n a(B) a.nd for a E a(A), b E o(B), a I b, then 13 n-1 \ an-l-j bj = (an - bn)/(a - b) ~ 0. L j=O Consequently, O I a(J) and AD - DB = O. II) Assume O ~ ~(A), then n-1 0 = D - A-nDB = TT 0 j:::O _, ■ Therefore D - A DB= 0. The following theorem gives the promised finite-dimensional converse of Proposition 1.1. ( 1 . 11 ) THEOREM. Hilbert space. Let A and B be two operators on a finite dimensional Then the following statements are equivalent. (i) A and B have identical commutants. (ii) A and B a.re polynomials of ea.ch other. (iii) A is a polynomial of Band they have identical eigenvectors. (iv) A is a polynomial of B, Ac p(B), pis one-to-one on the eigenvalues of Band p' is non-zero on those eigenvalues of B corresponding to nonlinear elementary divisors. (v) A and B commute and have identical invariant subspaces. 14 PROOF. (i) ⇒ (ii) The double commutant of A is the polynomial ring generated by A ([19, p.113, Corollary 1 ]). (ii) ⇒ (iii) and (ii) ⇒ (v) are obvious. (iii) ⇒ (iv) is not hard to see if we first assume that Bis in Jordan canonical form. (iv) ⇒ (i) is Proposition 1.1. ■ (v) ~ (ii) follows from Theorem 10 of [6]. The equivalence (iii)~ (iv) is the theorem in [~9). The following example shows that Proposition 1.1 does not have a converse in an infinite dimensional case: Let v = exp(2nia), where a is an irrational real number. Consider T € ,e(,½), T < So, 2 ~,' ~2' ••• > = < ~ ()' \)~,, \) ~2' •• > • e Then the commutant of every power of Tis the set of all diagonal operators. 15 CHAPl'ER 2 1• INI'RODOCTION The work in this chapter is motivated by the paper of De Prima and Richard [ 9). here. Many of the results in [ 9] are extended and generalized Using von Neumann's theory of spectral sets, we show that if T is a non-Hermitian operator with positive spectrum, then for large inn teger n, 0 lies in the numerical range of T. Hence, any semigroup of accretive operators is necessarily a connnutative semigroup of positive operators. Furthennore, the above theorem remains valid if we replace i e operat or commut a t·1ng wi·th T . Tn by TnD, where D 1.·s an.inver t"bl Var - ious situations in which the commutativity assumption of T and Dis relaxed are examined. In the last section sane variants of these theo- rems, which are derived with Calkin algebra techniques , are given. 2. MAPPINGS OF SPECTRAL SETS Let TE B('H) and let A be a closed subset of~ containing cr(T). A is said to be spectral for T if, whenever q is a rational complexvalued function with poles outside A, Spectral sets were introduced by von Newna.nn. detailed discussion on spectral sets. Chapter XI of [31] has a We list some properties of spec- tral sets: (i) If A is spectral for T, then any closed set containing~ is 16 spectral for T. (ii) If A is spectral for T, then co A::, W(T). (iii) t(~/2) is spectral for T if and only if Tis accretive. (iv) is spectral for T if and only if T is Hermitian. [R If A, A, n = 1, 2, 3, .•. n are closed convex subsets of~ such that /\ ::, A, we say /\_ tends to /\, A n n n ➔ /\, whenever for each E > O and each compact set r, there exists a positive integer n (E,r) such that 0 n > n . ( E, r) - o implies A n r c (A n r) + f\( €). n The following two theorems about spectral sets are proved in [ 9]. They are the principal tools in the proof of the main theorem . (2 . 1 ) I\ n THEOREM. Let A, A be closed convex subsets of~ such that n ::, A and A ➔ A. n Let T E B(V) with A spectral for T. n n n If T n -+ T (in the uniform operator topology), then A is spectral for T. (2.2) THEOREM. Let f be an analytic function in Int(t(n/2)). Suppose Tis accretive and Re a(T) > O, then Cl(co(f(Int E(~/2)))) is spectral for f(T). 3. THE MAIN THEOREM In this section we investigate some of the relations between an operator and the numerical ranges of its powers. n C. A. Berger proved the n power inequality for ntnnerical radii w(T )!S(w{T)). (41] and [15) contain proofs, discussions and generalizations of the theorem. An important ap- 17 plication appears in (45]. The power inequality indicates the maximum rate of the growth of the numerical ranges of the powers of an operator. The following theorem shows that the numerical ranges of large powers of a non-Hennitian operator with positive spectrum must have a certain minimum rate of growth. (2.3) For TE B(U) with a(T) > V > O, then either (i) THEOREM. or '(ii) there is a positive integer n o such that l\(yn) c W(Tn) when - ever n > n . - 0 Recall that we write =(T) > (~)y if a(T) is real and for each A E a(T), A> (~)y. (2.4) COROLLA.RY. (ii) For TE B('!/) with a(T) > 1, then either (t) T ~ I or for each bounded subset O c i, there is a positive integer n (0) such that O c W(Tn) whenever n > n. 0 - 0 The following lemma is needed to prove Theorem 2.3. (2.5) LEMMA . Let T, T , n = 1, 2, 3, ••• E B(W') and T converge to T n n in the uniform operator topo1ogy. Let A be a neighborhood of o(T) and f, n = 1, 2, 3, •.. , a sequence of functions analytic on A. n is an integer N such that f f n n E 3(T) whenever n > N. n Then there Furthennore, if converges uniformly to a function f on A, then f (T) converges to f(T) in the unifonn operator topology. n n 18 PROOF. Let O be an open set containing a(T) such that Cl(O) c A and o(O) consists of a finite number of positively oriented rectifiable Jordan curves. T n Since o(•) is an upper semi-continuous set function and converges to Tin the unifo:nn topology, there is an integer N such that o(T ) c O, n > N. n - n E .7(T ) whenever n > N. n - Since fn ➔ f uniformly on ao, there is an integer N1 ~ N Pick€> 0. such that r Consequently, f 2 ~ €/2 if n ~ N 1 ([11, Lemma VIIo 3.13]). Put M = sup f j fn(),) j : A E au, n ~ 1} and t(orl) = length of ao. Applying Lemma VII. 6.3 of [11 ], we get another integer N2 ~ N such that ■ For A E B(U) with (-~,o] n a(A) =¢and for~ E ffi, define A~ E f?J...W) by A~ = - 1- 2ni Jr ~ LogA ( e A - A) - l dA (1) , where Log A is the principal logarithm of A and r is the positively oriented curve containing a(A) in its interior domain as shown in the 19 following figure. If~ E (-1,1), then a(A~) c Int(t(l$ln)) by the spectral mapping theorem. It follows :from Proposition 1.9 that for a positive integer m and BE B('iil), if Bm = A and a(B) c Int(E(n/m)), then B = Al/m_ Now we are ready to give the proof of Theorem 2.3. PROOF. Assume there is an infinite subset M of the natural number s and for each integer m E M, there is a complex number k , m k m t Cl(W(rfl)). Ikm I <- ym and We sha.ll show that Tis positive. For each m EM, O t Cl(W(T111 - ~)); hence there is a real number a E (-n, n:) such that A m m Cl(W(A )) c Int(E(~/2)). m == e iCXm ....m (·1· - km) is strictly accreti ve, i.e., By Theorem 2.2, the closed sector r.(rr/2m) l/m spec t r al f or Am . Consequently, lim \\km T-ml\ == 0. mEM is 20 Put By Lennna 2.5 Bl/mis m defined for large m 1 lim \\B /m - rl\ = o. mEM m and For m E M, set C = e m llXm B m therefore o(c) c Int(t(1t/2)). m Hence Cl/mis well-defined. m Since 1 m o(c /m) c Int(l:(1t/2m)), we have 1 We want to show C /m ➔ I in the uniform topology a.nd since r( n:/2m) m 1/m 1/m . is spectral for A = T C , the nonnegative real numbers form a m m spectral set for T according to Theorem 2.1. lim mEM \ k T-mll = 0 implies that there is a positive integer m such m o that for each m E M, m > m , - o Since a n Re(a(B )) > O. m E (-n:,n:} and Re(a(c )) > o, we have m i.a /m 1/ C m = e m m 1 Therefore lim lie /m - mEM m m EM, r\l = O. m>m. - 0 ■ 21 The following is an immediate consequence of Theorem 2.3. (2.6) COROLLARY. If O ¢ Int(W(Tn)) for Let TE B('J/) with a(T) > O. infinitezy many n's, then T ~ O. We note that if Tis singular with a(T) ~ o, then Corollary 2.6 is not applicable. CONJECTURE. We give the following If O ¢ Int(W(Tn)), Let TE 8(11) with a(T) > 0. n = 1) 2, 3, ... , then T ~ O. We shall prove this conjecture under the additional assumption that O is an isolated point of a(T). First let us state a simplified version of a theorem of Sinclair and Crabb [34]. (2.7) THEOREM. If TE B('J/) and O ¢ Int(w(rr2n)), n = O, 1, 2, ... , then \\Tl\ ~ 8 r(T). An elementary proof of Theorem 2.7 appears in (5, p.27]. One immediate corollary of Theorem 2.7 is: 2n a(T) = [o},and O ¢ Int(W(T (2.8) THEOREM. TE B(¾), T quasinilpotent, i.e., )), n = o, 1, 2, . •. , then T = O. Let TE B('J/) with o(T) ~ 0. If O ¢ Int(W(Tn)), n = 1, 2, 3, ... ,and if O is an isolated point of a(T), then T > 0. PROOF. Let Ebe the spectral idempotent associated with 0. sition 1 .7. With respect to E 1/EB (E YI) J. , See Propo- 22 and where a(Q) = {o1 and a(T 1 ) > O. Since n n n (Q Q A - AT1 ) T ' n _ Tn 0 1 W(Tn) _ contains both W(Qn) and W(T~). we get Q = O and T1 ?:_ O, respectively. 0 T = (O 4. By Theorem 2. 7 and Corollary 2. 6, However, 0 f Int(W (T)) and -AT 1 \ Tl ) , by Corollary 1.6, - AT 1 = 0 . ■ Consequently T ~ 0. A THEOREM OF JOHNSON, DEPRIMA AND RICHARD The following theorem was first stated and proved by C. R. Johnson ((20, Chapter 2], (21]) for finite dimensional matrices and itwas gen- eralized by DePrima and Richard [ 9] for arbitrary bounded operators. (2. 9) THEOREM. Let T E B(W). Then T > 0 if a.nd only if Tn is accre- ti ve, n = 1 , 2, 3, .•• PROOF. The necessity is clear . For the sufficiency, note tha.t a(T) ?:_ 0 by the spectral mapping theorem. also accretive, n = 1, 2, 3, ... For any y > O, (T + y) n is Applying Theorem 2.3, we get (T + V) ?:_ yI. At the end of this section we shall give a.n elementary proof of ■ 23 Theorem 2.9. (2.10) The proper way of viewing Theorem 2.9 appears to be: THEOREM. Let TE B('il). Then Tn is accretive, n = 1, 2, ... ,k, if a.nd only if E(rr/2k) is spectral for T. PROOF. The sufficiency follows from the definition of a spectral set. If Tn is accretive, n = 1, ... , k, then t(rr/2k) :::> ~(T}; consequently, for each y > O, ((T + y)k) spectral for T + y. 1/k = T + y. We let y tend By Theorem 2 .2, ~(rr/2k) is ■ to O and apply Theorem 2. 1 We are going to give some results related to Theorem 2.9 and Theorem 2.10 . Given an accretive operator A E B('il) and a E (0,1), we de- fine the fractional power Aaby (2) for each x E "· The integral in (2) is convergent in the Bochner or absolute sense; and if Of a(A), then the fractional power defined by (2) is the same as the one defined by (1) on page 18. Furthermore, lim \\(A+ y)a ~+ l 1 1\ = o. (2.11) For an accretive operator A E B(U) and a positive in- THEOREM. See (24, §v. 3.11 ]. k teger k, there exists a unique operator B such that A= B and ~(rr/2k) is spectral for B. Theorem 2.11 generalizes a theorem of Macaev a.nd Palant (28]. so see [25) and [37, Proposition 5.5). fied version of (23, Theorem 1.1 ]. Al- The next theorem is a simpli- 24 (2.12) Let A E B('/1) be accretive and let a E (0,1/2]. THEOREM. ~ = (Aa + AiE0)/2 and Ka = (Aa - A-Ma)/2i. (2.14) COROLLARY. Let TE e(,:). Put Then for each x E 'II If Tn is accretive for n = 1, ... , k, then \\rm T xii S tan(1t/2k) IIReTxl\, x E 'JI. We conclude this section by giving the promised elementary proof of Theorem 2.9. (2 • , 3) T~,. ~•ll'Jt1. • 2 If A = A*, B > 0 and B2 :::_ A , then L et A 1 B E /.;/ r:J('II) • B > A. PROOF. Pick A < 0 and x E 'II, \\x\l = 1 • Hence ll(B - A - ">--)xii:::_ \\(B - A)x\l - llrucl\ > 0. Since B - A is Hermitian, we have a(B - A) ~ 0. ( 2. 14) LEMMA. Let T E /3 (JI) . If T and 1f are accreti ve, then W(T) c l:(1t/4). PROOF. ( ( (Re T) 2 Consequently, B > A. ■ 2 - ( Im T) ) x , x) 25 2 = Re(T x,x). 2 2 Hence T is accretive if and only if (ReT,2 ~ (Im T) . By Lemma 2.13 and the fact that T is accretive, we get Re(T) :::, Im T and ReT ~ - Im T. ■ Therefore, W(T) c t(n/4). (2.15) Let TE 8('11). COROLIARY. I:(an/2), 0 ~ a 5 1, PROOF. By Lemma 2.14, If Tis accretive and W(T 2 ) c then W(T) c t(an/4). exp(±in/4)T are accretive. exp(± i(1-a)n/4)T are accretive. Hence 2 The hypothesis W(T ) c t(alf/2 ) im- plies that ex:p(±i(1 - a)n/2)ef are accretive. Applying Lemma 2.14 again, we get both exp(in/4)(exp(i(1 - a)1t/4)T) and exp(-in/4)(exp(-i(1 - a)n/4)T) are accretive. Therefore, exp(±i(1 - a/2)n/2)T are accretive and W(T) c t(an/4). (2. 9 ') Let T E /3(,t) . THEOREM. ■ 2n If T is accretive, n = o, 1, 2, ... , then T > 0. PROOF. Corollary 2. 15 shows that if 2n T is accretive, k+1 then W(T) C t(n / 2 ). n = O, 1, •.• , k, ■ 26 5. PERTURBATIONS OF THE HYPOTHESIS OF THE MAIN THEOREM n The sequence TD, n = 1, 2, 3, ••• is studied in this section. seek conditions which imply T is positive. This structure a.rises natu- ra.lly in the study of multiplicative commutators. T -- ADA-1D-1. ~ Let A, D E B(YI), put If Kr = TA, then Tn D = AIL.-n ilA. , n E ~. See [ 9, §4] a.nd [ 30]. The following theorem generalizes Theorem 2.3. (2.16) THEOREM. Let T,D E /3(~) with a(T) > y > 0 and TD == DT. If there are infinitely many n's such that ~(yn) ¢ W(TnD), then i8 a(D) c e O t(n:/2) for some e E ~. Furthermore, if Dis invertible, 0 then T ~ y I. The proof of this theorem follows lines similar to that of Theorem 2.3. PROOF. Assume there is an infinite subset M of the natural numbers and for each integer m EM there exists a ccmplex number k, lk I< vm and m km %. Cl(W(rfln)). Am= e IDm(,tnn - m - For each m E M, there is crm E (-n:,rr) such that \i) is strictfy accretive. 11m IIB mEM m - nil = o. Since TD= ur, a(B) c Int(e m there is a real number 9 0 Put Bm = (D - kmT-m), then -ia m t(~/2)) such that a(D) c e 1eo and o(B) ➔ o(D). m ~(~/2). If O I a(D), we may assume a(D) n (-~,o] = ¢. By Lemma 2.5, is a positive integer m such that Bl/mis defined form> m and o m Hence - o there 27 1 lim !IB /m mEM r!l = o. m For m E M, m > m , we have - 0 Al/m = exp(i(a + 2~€(m))/m) m m where €(m) = 1, 0 or -1. Applying Theorem 2.1 and Theorem 2.2, we get [o,~) as a spectral set for ■ T. As in ~2.3 we do not know what conclusions can be drawn if we re- place the eypothesis a(T) > 0 by a(T) ~ 0 in Theorem 2.6. However, if 1 we assume O is a pole of the resolvent (A - T) - , then we have the following (2.17) THEOREM. Let T,D E B('I:/) with ~(T) ~ 0 and D invertible. pose O f. Int(W(TnD)), n = 1, 2, 3, ... If O is a pole of (X - T) - Sup- 1 and if' TD = IJ.r, then T ~ o. Since O is a boundary point of ~(T), 0 is a pole of (A - T)- REMARK. 1 if either (i) Tis semi-Fredholm (Proposition 1.2), or (ii) the ascent of T, a(T), is finite and Ta(T)+k(J/) is closed for some positive integer k((26], Theorem 2.7). Recall that a(T) is the sma.llest nonnegative integer p such that~ and p+1 T have identical nullspaces. PROOF. Let E be the spectral idempotent associated with {o} . osition 1.7. With respect to E 11 ffi (E .,_,)~, we have See Prop- 28 and D = (; 3 ;2)4 . Let m be an integer such that if1 = 0. Since rf1n = DT111, we get D = 0. 3 Thus TD = DT implies ND1 = D1 N and T1J\ = I\.T1 . Now the first half of Theorem 2.16 is applicable to the sequence T~ J\, n = 1, 2, 3, ... , and we get O I Int(a(J\)). By Proposition 1.2, Furthermore, D is onto since Dis invertible. 4 I\ is invertible. Therefore T1 >Oby the second half of Theorem 2.16 and D is also invertible [15, pp . 220-221 ]. 1 The conditions that the operator N is nilpotent and commutes with n imply 1 ND1 is also nilpotent. N = 0. Since O, Int W(ND 1 ) and D1 invertible, we have Hence TD= (0 -AT1 D4). o T1n4 By Corollary 1. 6 and O , and T = Int W(TD), - AT 1l\ = 0. Consequently A = 0 (g ~ ) with T1 :::_ 0. ■ 1 One may ask what happens to Theorem 2. 16 if the commutativity assumption on T and D is dropped. Since a(T) lies on a.n open half re,y originating from the origin, the condition TD a DT is equivalent to 29 that~= r:,r1Il for same nonzero integer m by Theorem 1.10 . In general, we cannot drop the commutativity condition as demonstrated in the fol- lowing (2.18) PROPOSITION. Let T,D E B('i/) with a(T) > O, D > 0. Suppose O ¢ Int (W(TnD)) for infinitely many n's, then T > O D invertible and if and only if TD = DT. PROOF. The sufficiency follows from Theorem 2.16. For the necessity note that By Corollary 2. 6 we have D-½TD½ ~ 0. imply that D commutes with D-½TD½. ,1 1 .!. Hence D2 TD 2 = D- 2 TD 3/2 , and we get Dr = TD. ■ Proposition 2.18 remains valid if we merely assume Dis normal and restrict" to be finite dimensional. (2.19) LEMMA: Let Pi, i = 1, ... , k, be k pairwise orthogonal prok jections on,: and k l Pi = I. i=1 l AiPi i=l with A1 > A2 > ··· > Ak:::: 0. infinitely many n's. Let T = Suppose DE 8('1/) and O, Int (W(T°n)) for Then P.D P. = 0 l. J < i < j 5 k. 30 PROOF. It is sufficient to consider the case T-_ (a0 0) a>S>O ~ and D = (~ :). We shall show that b = O. and O ¢ Int (W(T~)), there is a real nurnber 8 n such that i8 O < det (Re(e = a~n Re(e 1 I - 4 a¾e n(T~))) 1e i8 n i8 na) Re(e nd) + ~ n - -i8 , c e n 2 . Thus Therefore b = 0 because O 5 f3/a < 1 and ther e are infinitely many n's for which the above inequality holds. • (2.20) PROPOSITION. Let,: be finite-dimensional. T > O and D normal. If O ,- Int (W(T°:n)) for infinitely many n's, then TD= IJr. k PROOF. Let T = \L A.]. P.]. i=l Let T,D E R(W) with 31 where 11. 1 > A2 > • • • > Ak ~ 0 and [Pi} is a set of k pairwise orthogonal projections, t P. = I. l < i < j ~ k. Since Wis finite di - mensionaJ. and Dis normal = 0 i Yi,j ' I j. ■ There fore TD = ur. Proposition 2.20 is not true if J/ is infinite dimensional. The following counterexample is suggested by J. H. Anderson. Let U denote the unilateral shi:f't on t , 2 Let B denote the projection on t 2 given by B < ~O' ~,, ~2' •.• > = < ~O' O, O, .•. >. Put then VV* = V*V = I. * 21 then TnD = ( +U 0 TD - 0 ) 0 Put D = 2I + V; D is normal. Let T = (IQ\ O o) , whi ch i s accre t·ive, n = 1 , 2 , 3 , .•• Dr= (_~ g). We conclude this section with the following But 32 (2.21) THEOREM. Let T,D E B(JI) with the descent of T finite a.nd D strictly accretive. Suppose T~ is accretive for n = 1, 2, 3, ... If i11n = rJt11 for some positive integer m, then T::: O. Recall that T has finite descent if there exists a nonnegative in1 teger q such that T<\/ = Tq+ v. orem 2.21 if m = 1. (2.22) This hypothesis is not necessary in The- In fact we have PROPOSITION. (cf. [9], Theorem 3) a(D) c Int (E(~/2)). Let T,D E B(W) with Suppose T~ is accretive for n = 1, 2, 3, If TD = DT, then T ~ 0. PROOF. First we show ~(T) > 0. There is a real number O ~ ~ < 1 such that o(D) c Int(r(~~/2)). TD = DT implies that We note that the hypotheses of the theorem are not changed if T is replaced by T + r, r > 0. Hence ( r + a (T) ) n = ( a ( T + r) ) n c I ( ( 1 + S) rc/2) for a.11 r::: o, n = 1, 2, 3, •.. This is possible only if rr(T) ~ 0. any V > O, (T + y)~ is a.ccretive, n = 1, 2, 3, .•. For Now Theorem 2.16 is ■ applicable. Before we can give the proof of Theorem 2.21, we need one more lemma. (2.23) vector x LEMMA. 0 [30, Theorem 4.18] Let S,C E B(W). Suppose there is a E Ker S*2 \ Ker S* such that (CS * x ,s* x) 0 0 I 0. Then 33 0 E Int (W(SC)). PROOF. For a E ~, (sc(x0 + as * x) , 0 = x0 + as * x) 0 (C X' s * X) + a(cs*X' s * X) 0 0 0 0 Hence ti = t J ( ( SC (x + as 0 * x0 ) , x0 + as *x0 ) : a E ct} . ■ Consequently, 0 f Int (W(SC)). PROOF OF THEOREM 2.21. Since if1n = IJ.rm and a(D) c Int (t(~/2)), we have rffD' ~ 0 by Proposition 2.22. Hence the ascent of T, a(T) ~ m. By hypothesis the descent of Tis also finite, and applying Theorem 2.1 of (26), we have that O is a pole of (A - T)- 1 . Let Ebe the spectral idempotent associated with fo }. With respect .1. to E W © (E Ji'), put E = (~ ~), then ~ ~ O implies -rf1 = O, ~ ~ 0 and -A~= 0. Thus A== O and T1 > 0 . Hence T = N ffi T • 1 0 f W(D), 0 I Int (W(TD)) and by Lemma 2.23, we conclude that the * a(N)* < 2. ascent of N, Hence T > 0 . The operator N is nilpotent, therefore, N c 0. ■ 34 6. OTHER REIATED RESULTS In this section 'II is infinite dimensional and separable. For T E B('II) , let we (T) = n (Cl(W(T + K)) ~w(T) = n fa(T + K) as in §1 .5. K canpact} and : K compact} Corresponding to Theorem 2.3, Theorem 2.9 and Theorem 2.16, we have the following theorems: THEOREM. (2.25) Let TE B('i/) with ~W(T) > V > O, then either (i) there is a compact operator K such that T + K ~ VI, or there is a positive integer n (ii) o such that ~(yn) cw (Tn) whene ever n > n . - (2.26) 0 THEOREM. (cf. (9, Theorem 4]) For TE B(W), then W (Tn) c ~(~/2), n = 1, 2, 3, .•. , if and only if there is a compact e operator K such that T + K > O. (2.27) THEOREM. Let T,D E /3('11) with aw(T) > y > 0 and (T111n - mm) com- If there are infinitely many n's such n n... iGo that l:)_(y) ¢. W(T v), then ~W(D) c e t(~/2) for sane 8 ~ ~- Further- pact for some positive integer m. 0 more, if Dis semi-Fredholm, then there is a compact operator K such that T + K ~ YI. We sketch the proof of Theorem 2.27. As in §I.5, we let TT : 8(1 1 ) ➔ al{,t), the quotient map, and 35 a faithful* representation. ,- ,. 0 TI(D) commute by Theorem 1. 10. Cl(W(ri1n)). 0 TI(T) and Furthermore, w(,- o TI(TnD)) c Applying the first half of Theorem 2 .1 6, we get i8 o(T • TI(D)) c e O I:(1t/2) for some 80 E [R. Consequently, 0 ¢ Int(o(D)). by Proposition 1.2. By a If Dis semi-Fredholm, then Dis Fredholm theorem of Atkinson, Dis Fredholm if and only if TI(D) is invertible ([40}, _ [15, Problem 142)). Thus it follows from the second half of Theorem 2.16 that T O TI(T) :::_ V I-j . Applying Theorem 1.8, we conclude that there is a compact operator K such that T + K~YI. ■ 36 CHAPTER 3 1• INTRO~TION In this chapter we give some applications of the theory developed ear lier. The main problem we study is the following: for T E B(J/) , if each of the powers of Tis not a commutator with a positive factor, then what conclusions can we draw about T? When JI is a.n infinite di- mensional separable Hilbert space, we show that there is a positive integer m and a compact operator K such that T11 + K is nonnal. The main tools used to prove this fact are (i) a characterization of commutators with self-adjoint factor due to J. H. Anderson [1 ], (ii) a number-theoretic result of C.R. Johnson and M. New-man (22], (iii) Theorem 2.3. and The author wishes to thank C. R. Johnson for informing him of the result in (22). 2. A THEOREM OF J. H. ANDERSON A derivation on the algebra 8(,t) is a linear map 6 from B(W) into itself with the property 6(XY) = o(X)Y + X 6 (Y) ators X, Y in B(W) . for every pair of oper- It is known that all derivations a.re inner, i.e., for each derivation o, there is an operator A in B(J/) such that 6(X) = 6A(X) = AX - XA. Let R denote the set U f 6A (,e (,t) ) A ?:_ 0) 37 Thus the problem we are interested in is the following: what are the operators T with the property that Tn ~ R, n = 1, 2, 3, ... ? For the rest of this section 1:1 will denote an infinite-dimensional separable Hilbert space. J. H. Anderson in his thesis [1, Theorem 7.2) proved the following deep result: (3.1) THEOREM. R =[TE 8(11) : 0 E we(T)}. As in §1 .5 we let n denote the quotient map of/S(U) onto the Calkin algebra a.nd let T denote a faithful* representation of the Cal.ki n algebra onto a self-adjoint subalgebra of S(F, sen Hilbert space. For TE S(V), We (T) = Cl(W(T•Il(T))). Hence the cypothesis that Tn 0/Cl(W(-r 0 f some suitably cho- t R, n = 1, 2, 3, ... is equivalent to Il(T)n)), n= 1, 2, 3, ••• In §4 of this chapter we shall show: if Of Cl(W(T°)) n = 1, 2, 3, •.• , then a power of Tis normal. 3. A THEOREM OF JOHNSON AND NEWMAN The following question is raised in [22]: how many distinct points a 1, ..• , at on the unit circle of~ are in general required to insure that for some positive integer m, 0 E cofcf,1, •.• , ~}? A complete solu- tion is given by (3.2) THEOREM [22]. Let a, ~, y be distinct complex numbers with !al= le!= Iv!= 1. Then there exists a positive integer m such that 0 E co{cl1, '3m, ymJ if a.nd only if fa, $, y} cannot be obtained ftom 38 f, , e2~i/7 , e tnr.i/7} or lr 1 , e2~i/7 , e 1(}J(i/7} via any combination of permutation, reflection and simultaneous rotation. . Let IR+ denote the set of strictly positive numbers, NOTATION. IR+ = (O,oo). For(!, c ~ \ {o}, let I Arg C, denote the cardinality of the set [Arg A : A Ee}, and let C,m denote the set {'111 : y E e}, m an integer. COROLIARY. (3.3) C. n R+ / ¢. Let e be a set of nonzero complex numbers such that If O, co(en), n = ,, 2, 3, ... , and if# Arg (!,~ 3, then . 7 # Ar g C, = 3 and C, + c R • For any three nonzero complex numbers a, 8, y, 0 E co[a, S, y} PROOF. ■ (3.4) 0 f THEOREM. Let TE B(J✓-) with a(T) n IR+/.¢. Cl(W(rt1)), n = 1, 2, 3, •.. Suppose We have the following cases: (i) # Arg c(T) = 1, then T > O. (ii) # Arg a(T) ~ 3, then# Arg a(T) = 3 and T7 > O. (iii) ~ Arg a(T) = 2, then either there is a positive odd number m such that rfil > O or there exists a closed subspace .u of JI and positive 1 operators T and T on Y. and 1 2 1 'r," respectively such that T == T1 EB e . 8T2 , e being irrational modulo 2~. PROOF. Case (i). Since c(T) > o, T ~ 0 by Corollary 2.6. 1. 39 0 ¢ Cl(W(Tn))::, co(a(T)n), n = 1, 2, 3, .• . Case (ii). lary 3.3, we have# Arg a(T) = 3 and a(T 7) > O. By Corol- Applying Corollary 2 . 6 again, we get T7 ~ 0. Case (iii). a(T) ClR + U e There exists a real number 8 E [0,21f) such that H~ + lR . If 9 is rational modulo 21t, there is a positive odd integer m such that a(T111) > o, thus rJ!ll ~ 0. Before we can treat the case where e is irrational modulo 21f, we need the following (3. 5) . LEMMA. Let € E (O, 1), a, ~ E (t, a·~ I 0. If IArg (a/~) I > arccos (-E 2 ), then OE~ (a,~;€). PROOF. By definition of an ellipse, Put += IArg (a/e) I. 0 E@ (a, e; E) if and only if Then we have to show or equivalently, The above inequalities hold if 2 .. I. 2 0 ~ (1 +cos"€) - (1/€2 - 1) 2 , l - 1 > I1 + cos ,Vl I , or l/ or 1/E 2 2 - 1 ~ -(1 + cos ,fe ) 2 since , ~ arccos ( -E ) • But the last inequality is always true. ■ 40 We now come back to the proof of the last part of Theorem 3. 4 . Ebe the spectral idempotent associated with a(T) nm+. 1.7. Let See Proposition With respect to E JI EB (E JI)~, put E = (~ t), then o, T2 ~ O and & is irrational modulo 2n:. We note that To show that T = T1 EB e iA T2, we have to show A= 0. Assume A~ O. where· T 1 ~ a positive integer n and y E (E For ui·, with I\Y\l == 1 and Ay I o, let 8 [n,y] denote the numerical range of the 2 x 2 matrix 0 By Lemma 1 .5, 8 [n,y] C W(Tn). By Theorem 1 .3, S[n,y) = 8 (a,a; E[n,y]) 9 where a E IR+, $ E ein lR+ and e[n,y] = Let y , m = 1, 2, 3, •.• be a sequence in (E V)~ such that IIY I! = 1 and m m 41 lim IIAymfl = \\All. For each n, m--+oe = 1 + ➔ Hence lim E[n,y ] = (1 m m-too 2 + l!All ) 1 as m ➔ a, • .l. -2 (cf. Corollary 1.4). Thus for each integer n, there is an integer m(n) such that Since e is irrational modulo 2~, pick an integer N for which jArg eiNE91::: arccos 3.5. 2 (-1/(1 + \IA!l /2)). Then OE@ [N,ym(N)] by Lemma However, O ~ W(TN) by hypothesis; therefore, A= 0 and T = T ~ e 1 18 T . 2 The proof of Theorem 3.4 is now completed. The following is an immediate consequence of Theorem 3 . 4. (3.6) COROLLARY. Let TE B(JI). If O ¢ Cl(W(Tn)), n = 1, 2, 3, .•. , then an odd power of T is normal. With JI finite dimensional Corollary 3.6 is first proved in (21 ]. ■ 42 (3.7) THEOREM. Let 1/ be an infinite dimensional separable Hilbert space and let T E B(';/J • Suppose Tn f. R, n = 1, 2, 3, • . . Then we have the following cases: # Arg aw(T) = 1, then there exist 8 E [0,2~) a.nd compact op- (i) erator K such that (e ·e T + K) ~ 0. 1 # Arg aw(T) ~ 3, then I Arg aw(T) = 3 and there exist (ii) · ·e 7 8 E [0,2~) and compact operator K such that (e 1 T + K) ~ O. I Arg aW(T) = 2, then either there exist a positive odd inte- (iii) ·e germ, 8 E [0,2n) and canpact operator K such that (e 1 r'J!1 + K) ~ o, or there exist a closed subspace 11 of 11 and positive operators T and T 1 2 1 on ,: and 11 .1. respectively such that (T - e 1 1 where (8 PROOF. 1 - 1e 1 T1 ~ e u2 T ) is compact 2 e2 ) is a number irrational modulo 2n. By Theorem 3. 1, R = [ S E 8(J/) : 0 I. Cl(W(T O TI(S)))}. Now, most of the conclusions in the theorem follow directly from Theorem 3. 4. How- ever, a little more explanation is needed in the second half of case 1~, (iii). We know T • Il(T) = e v1 EB e i82 v2 on f 1 ffi f 1.1. = ?-, where v1 ~ O and v2 ~ O. Thus Il(T) is normal and a(Il(T)) lies on a simple arc. By Theorem 1 .8, there is a compact operator K such that T + K is normal and o-(T + K) = c(Il(T)). Consequently, there exist '11 closed subspace of 'fl and positive operators T andT 1 i82 i81 (T - e 2 on .u1 and 111.L , respectively such that T1 EB e T2 ) is compact. ■ 43 ( 3. 8) COROLIARY. Let T E B(',/), 'JI infini te-d.imensional and separable. If Tn ~ R, n = 1, 2, 3, ••. , then an odd power of Tis a normal plus a compact. REMARK. One may ask if a stronger conclusion can be drawn when the hypothesis that each of the powers of Tis not a commutator with a positive factor is replaced by the hypothesis that each of the powers of T is not a commutator with a normal factor. However, these two hypotheses are actually equivalent, i.e., R = U {~N(B('JI)): N normal}. It follows immediately from Theorem 3. 1 that ~, is a norm closed subset of B(?I) . The theorem in (16] states that for each normal operator N, there is a Hermitian operator A and a function~ continuous on o(A) such that ~(A)= N. By the Weierstrass approximation theorem, oN(B('JI)) is a sub- set of the norm closure of oA(B(¾)) ( 1, Corollary 13. 11 ] . Consequently, R = U {o N(B('JI)) : N normal}. 6. SUFFICIENT CONDITIONS FOR NORMALITY In this section we present some variants of the results in the last two sections. We give additional conditions which make the operator T itself normal or normal plus compact. (3: 9) THEOREM. Let T E ~('ft) and O ,. Cl(W(Tn)), n = l, 2, 3, . . . If T is a convexoid, i.e., co(cr(T)) = Cl(W(T)), then for 1 ~ j ~ k, where k is some positive integer~ 3, ' there exist positive operators T. E B(W.) J k and real numbers 8 j such that 'I/= l EB j=l k 'Jlj and T = i8. l EB e JT j=1 j. J 44 i78, Moreover, if k = 3, then e Put k = ~ Arg a(T) and k ~ 3 by Theorem 3.4. PROOF. First, we consider the case k = 2, i.e., there are two real numbers 8 1 and 8 o(T) c e i81 ~ with a(T) n e + U e i81 i82 + ~. + ~. 2 such that Let Ebe the spectral idempotent associated With respect to E 'JI EB (EN')~, put E = (~ ~) , then , T where T1 ~ 0 and T2 ~ 0. .Assume A I O; thus there is a two-dimensional compression of T whose numerical range consists of an elliptical disc 18. + with foci on each of the two half-raJ'S e J ~, j = 1, 2, and eccentricity strict]s- less than unity. However, Tis a convexoid by hypothesis and co(cr(T)) is a quadrilateral, a triangle or a line segment with all of its vertices ]s-ing on the two half-rays e A= 0 a.nd T = e i81 T1 ffi e i82 18. + J IR , J· = 1 , 2 • T he re :fore, T2 • The case that l. Arg rr(T) = 3 is treated in a similar fashion. Nev- ertheless, we note that the above geometric argument fails if # Arg cr(T) ~ 4. (3.10) Fortunately this case cannot arise. COROLLARY. ■ Let TE 8(9/) and suppose O ¢ Cl(W(Tn)), n = 1, 2, ... Then Tis normal if and only if Cl(W(T)) is a closed polygon (possibly degenerate). 45 PROOF. Cl(W(T)) is a closed polygon implies that Tis a convexoid ■ ([17], [33, Corollary 2.1 ]). We note that the polygon mentioned in Corollary 3.10 may have at most six sides. (3.11) THEOREM. Let TE 8(11), W infinite dimensional and separable. Suppose ,t1, R, n = 1, 2, 3, ... Then Tis a normal plus a compact if and only if We(T) is a closed polygon (possibly degenerate) . PROOF. ■ Apply Corollary 3.10 and Theorem 1 .8. We conclude this chapter with the following theorem on finite dimensional ma:brices (cf. (21 ]). (3.12) THEOREM. Let T be a finite dimensional square matrix. 0 ~ Int(W(Tn)), n = 1, 2, 3, •.. Suppose If for each positive integer n, T and Tn have identical eigenvectors or identical com.mutants, then Tis nor- ma.l. PROOF. By a theorem of Schur, U*TU is upper triangular. there is a unitary matrix U such that We have to show that U*TU is actually diago- nal. Assume U*TU is not diagonal; then we can find, if necessary after applying a suitable simultaneous row and column permutation (which of course will preserve the upper triangular structure), a 2 x 2 submatrix T 1 = (~ ;) along the main diagonal with y -/ O. n For each n, W(T) 46 contains W(T~) . Suppose a=~= o, then OE Int(W(T1)). If a=~ Io, W(T~) is the closed circular disc an+ Ll(nl-.,on-l l/2) by Corollary 1.4. Hence for n > 2" la/y I, o E Int(W(T~)). Suppose a/ S, then W(T~) is the closed elliptical disc (possibly n a singleton) e (an, ~ ; €), where E= (1+lv/(a - ~)I OE Int(T~) if Ian!+ IS 0 1 < Ian - A0 1/e. If !al I 2 J,_ -2 ) . Consequently, l$I, this inequality holds for large n since E < 1 • . For the case a i a and lal = lal, we apply the additional hypothe- sis and Proposition 1.11 to conclude that an I en, n = 1, 2, 3, . . . , or equivalently, a/9 = exp(2ni8) for sane irrational real number 8. W(T~) is the ellipse with foci at an and ~n and constant eccentricity€< 1 . 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