Paranormal Operators on a Hilbert Space Citation Luecke, Glenn Richard (1970) Paranormal Operators on a Hilbert Space. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/6me8-nc92. https://resolver.caltech.edu/CaltechTHESIS:08112015-093404966 Abstract In this thesis an extensive study is made of the set P of all paranormal operators in B( H ), the set of all bounded endomorphisms on the complex Hilbert space H . T ϵ B( H ) is paranormal if for each z contained in the resolvent set of T, d(z, σ(T))//(T-zI) -1 = 1 where d(z, σ(T)) is the distance from z to σ(T), the spectrum of T. P contains the set N of normal operators and P contains the set of hyponormal operators. However, P is contained in L , the set of all T ϵ B( H ) such that the convex hull of the spectrum of T is equal to the closure of the numerical range of T. Thus, N ≤ P ≤ L . If the uniform operator (norm) topology is placed on B( H ), then the relative topological properties of N , P , L can be discussed. In Section IV, it is shown that: 1) N P and L are arc-wise connected and closed, 2) N, P, and L are nowhere dense subsets of B( H ) when dim H ≥ 2, 3) N = P when dim H ˂ ∞ , 4) N is a nowhere dense subset of P when dim H ˂ ∞ , 5) P is not a nowhere dense subset of L when dim H ˂ ∞ , and 6) it is not known if P is a nowhere dense subset of L when dim H ˂ ∞. The spectral properties of paranormal operators are of current interest in the literature. Putnam [22, 23] has shown that certain points on the boundary of the spectrum of a paranormal operator are either normal eigenvalues or normal approximate eigenvalues. Stampfli [26] has shown that a hyponormal operator with countable spectrum is normal. However, in Theorem 3.3, it is shown that a paranormal operator T with countable spectrum can be written as the direct sum, N ⊕ A, of a normal operator N with σ(N) = σ(T) and of an operator A with σ(A) a subset of the derived set of σ(T). It is then shown that A need not be normal. If we restrict the countable spectrum of T ϵ P to lie on a C 2 -smooth rectifiable Jordan curve G o , then T must be normal [see Theorem 3.5 and its Corollary]. If T is a scalar paranormal operator with countable spectrum, then in order to conclude that T is normal the condition of σ(T) ≤ G o can be relaxed [see Theorem 3.6]. In Theorem 3.7 it is then shown that the above result is not true when T is not assumed to be scalar. It was then conjectured that if T ϵ P with σ(T) ≤ G o , then T is normal. The proof of Theorem 3.5 relies heavily on the assumption that T has countable spectrum and cannot be generalized. However, the corollary to Theorem 3.9 states that if T ϵ P with σ(T) ≤ G o , then T has a non-trivial lattice of invariant subspaces. After the completion of most of the work on this thesis, Stampfli [30, 31] published a proof that a paranormal operator T with σ(T) ≤ G o is normal. His proof uses some rather deep results concerning numerical ranges whereas the proof of Theorem 3.5 uses relatively elementary methods. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Mathematics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): De Prima, Charles R. Thesis Committee: Unknown, Unknown Defense Date: 30 January 1970 Record Number: CaltechTHESIS:08112015-093404966 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:08112015-093404966 DOI: 10.7907/6me8-nc92 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 9094 Collection: CaltechTHESIS Deposited By: INVALID USER Deposited On: 17 Aug 2015 21:28 Last Modified: 16 May 2024 23:42 Thesis Files Preview PDF - Final Version See Usage Policy. 8MB Repository Staff Only: item control page

PARANORMAL OPERATORS ON A HILBERT SPACE Thesis by Glenn Richard Luecke In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy Cal i fornia Institute of Technology Pasadena, California 1970 (Submitted January 30, 1970) ii ACKNOWLEDGMENTS I would like to thank the National Science Foundation for their financial support which has helped to make possible my graduate study. I have appreciated the friendly cooperation of the entire Mathematics Department of the California Institute of Technology . I would espe- cially like to thank my thesis advisor Professor Charles DePrima for his helpful assistance and guidance in the preparation of this thesis. iii ABSTRACT In this thesis an extensive study is made of the set f paranormal operators in B(~), the complex Hilbert space ?=t. the set of all bounded endomorphisms on T ~ B(?:1) is the distance from z is par anormal if for each d (z, cr(T)) II (T-zl)- 1// = 1 contained in the resolvent set of T, d(z,~(T ) ) to ~(T), contains the set 77 of normal operators and hyponormal operators. T € B('ft) where rP the spectrum of T. 6J contains the set of such that the convex hull of the spectrum of T is equal to the Thus, ll ~ fP ~ J. If the uniform operator (norm) topology is p laced on the relative topological properties of fl, ? , and In Section IV, it is shown that: nected and closed, dim Fl < oa, set of "f... when 2) 71 , IP , and 3) n = ~ when dim Fl ~ 2 , subset of IP when when z However, ~ is contained in ~. the set of all closure of the numerical range of T. when of all dim fl = oo and dim fl 1) fl, rP, and B('1'), J. can be discussed. J are arc-wise con- i are nowhere dense subsets of dim Fl ~ then B(U) 4) Jl is a nowhere dense 00 , 5) ~ is not a nowhere dense subset of I. 6) it is not known if fY is a nowhere dense sub- = o0 The spectral properties of paranormal operators are of current interest in the literature. Putnam [22, 23] has shown that certain points on the boundary of the spectrum of a paranormal operator are either normal eigenvalues or normal approximate eigenvalues. Stampfli (26] has shown that a hyponormal operator with countable spectrum i s normal. However, in Theorem 3.3, it is shown that a paranormal opera- tor T with countable spectrum can be written as the di rect sum , N ~A, iv of a normal operator N with ~(A) ~(N) = a(T) a subset of the derived set of need not be normal. and of an operator A with cr(T). It is then s hown that A If we restrict the countable spectrum of to lie on a c2 -smooth rectifiable Jordan curve normal [see Theorem 3.5 and its Corollary]. G0 , Te~ then T must be If Tis a scalar paranor- ma l operator with ~ountable spectrum, then in order to conclude that T is normal the condition ~(T) ~ G0 can be relaxed [see Theorem 3.6]. In Theorem 3.7 it is then shown that the above result is not true when T is not assumed to be scalar. with ~(T) ~ G0 , It was then conjectured that if then T is normal. TE~ The proof of Theorem 3 . 5 relies heavily on the assumption that T has countable spectrum and cannot be generalized. Tc~ with subspaces. However, the Corollary to Theorem 3.9 states that if ~ (T) ~ G0 , then T has a non-trivial lattice o f invariant Af ter the completion of most of the work on this thesis, Stampf li [30, 31] published a proof that a paranormal operator T with ~(T) f G0 is normal. His proof uses some rather deep results concern- ing numerical ranges whereas the proof of Theorem 3 . 5 uses relatively elementary methods . 1 INTRODUCTION In this thesis an extensive study i s made of the topological and (p spectral properties of a subset of endomorphisms on the Hilbert space 'Fl • the set of bounded B ('R') , An element T of rP , called a paranormal operator, is defined by the relationship d (z, <T(T)) II (T-zl)- 11/ = 1 for all z tp(T), where jJ(T) is the ! resolvent set of T', and the spectrum of T. //(T-zl) -l )/ is the distance from d(z,o-(T)) z to <T (T ), Operators with the above growth condition on arise naturally in spectral operator theory [ 8]. ~is a very large set in that it contains all normal operators , n. and a ll hyponormal operators, yet (? is smal l enough to be contained in the set f. of all T in B(?f) such that the convex hull of the spectrum of T is equal to the closure of the numerical range of T. Thus It £ (j> f. ~. If the uniform operator (norm) topology is placed on B&), then the relative topological propert i es of 71, fP, and As an illustration, it is shown [see Section IV] that Tl., discussed. ~, and B(';/) J. can be £ are arc-wise connected, closed, nowhere dense subsets of when dim 71 :? 2, 7{ = (]> when dense subset of rP when dim ff < o0 , and fl is a nowhere dim JI = oo The spectral properties of paranormal and hyponormal operators have received considerable attent i on i n the current literature [22, 23, 30, 31]. Stampfli [25, 26, 27] has shown that a hyponormal operator whose spectrum is a sufficiently "thin" subset of the complex plane, must indeed be normal. In particular, a hyponormal operator with countable spectrum is normal. On the other hand, a paranormal operator 2 T with countable spectrum can be decomposed as the direct sum, of a norma l operator N with o-(N) = 0-(T), o-(A) a subset of the derived set of need not be normal. and of an operator A with 0-(T) [Theorem 3.3]. is in fact normal. G0 , However, A Stampfli [26, 27] has also proved that a hypo- normal operator whose spectrum is a subset of a Jordan curve N ~ A, c2 -smooth rectifiable It was then conjectured that this result would hold for T paranormal. This conjecture with the added assumption that T has a countable spectrum is proved here [see Theorem 3.4 and its Corollary], but the conjecture without the countability assumption on the spectrum remained unproved. However, using the Dunford spectral operator theory, it was shown that a paranormal operator whose spectrum is not countable and is a subse t of G , 0 must indeed have a nontrivial lattice of invariant subspaces [see Theorem 3.9 and its Corollary] . After having completed most of the work for this thesis, Starnpfli [30] announced a solution to a problem that generalizes a result stated above. Namely, he proved that a paranormal operator whose spectrum is a subset of G0 , is normal. In his proof [31] several rather deep results about numerical ranges are crucial, whereas the result proved in this thesis [Theorem 3 . 4] uses relatively elementary metho4s which depend heavily on the assumption that the spectrum is countable. It would be interesting to see a relatively simple proof of Stampfli' s result. Orland [19 ] has characterized operators T for which the convex hull of ~(T), co ~(T), equals the closure of the numerical range of T by 3 the condition d(z , co tr(T))l/(T-zl) - 1// ~ 1 for all z<f co cr(T), d{z, co ~(T)) is the distance from co ~(T). Thus, an extremal condition on /j (T-zl)-l/J to has been characterized by an extremal geometri c condition, namely numerical range on T. z where co ~{T) is equal to the closure of the In general [7, p . 556] d(z,~(T))fi(T-zl)- 1 h ~ 1 d(z,CT (T)) /l(T- zl )- 1// = 1, -1 for paranormality is an extremal condition on V(T-zl) ~. Therefore, for each zE,t{T), so that the condition, it seems reasonable to expect that the set of paranormal operat ors can be characterized by some extremal property; however, t his remains an open question. 4 Section I: Definitions and Basic Hilbert Space Properties Let 1=I be a complex Hilbert space with the i nner product of two elements x, ye 11 denoted by An operator on "fl is a con- (x,y) . tinuous linear transformation mapping 11 on~ all ope rator s B(~). by B(~) operator topology.; IE B(?t) For I/Tl/= sup{l/TxJJ: Tc B{'/¥), let i nto /<I • wi ll a lways be given the uniform will always denote t he identity operator. T and T* denote the adjoint o f T. x€'/:I, // xii• l} T € B()I) such that ST = TS = I. unique and i s denoted by S -1 • j?(T) = { zc I[ : where ([ denotes the complex field. of T and ( - jJ {T) 0-(T) poi nt spectrum of T and [see 11] . In this event S is T-zl is invertible} f7{T) is called the resolvent set is called the spectrum toof T. ~ff(T) ocr{T) f. er7T(T) , <T p (T) d S. [ 11, problem 63]. It is an impor- z t=. er 7/T) is a normal approximate eigenvalue for T when: 1) //xn II= 1 and //{T-zl)xn// --'>- 0 n~ =, 2) /JYnlf = 1 as n~ oo . as and n-?oo imply //(T*-ZI)y //--) 0 n z € a-P (T) Tx = zx} = {xe'R': //(T*-zl)xn// -+ 0 as n~oa as and //(T-zl )y //~O n imply is a normal eigenvalue for T when T*x = z xf . is the is the approximate point spectrum of T, The boundry of a set S is denoted by tant fact that denote the norm of is invert ible i f and only S € B(U) i f there exists Denote the set of { x E i:/ : It is well known that the eigenspaces of distinct norma l eigenva l ues of an opera t or T are ortho gonal [14, p. 233]. If ily true t h at is a normal e igenvalue f or T, then it is not neces sar- z z is a normal approximate eigenvalue f or T. An operator T is quasi-nilpotent if tion if T2 = T. cr(T) = {o}. T is an orthogonal projection if T2 T is a projecT and 5 T* = T . It is well known that T is an orthogonal projection if and only i f T T ~ 0, if z E cr(T) } 2 = T II T/I and (Tx,x) > 0 ~ 1. T is a positive operator, denoted by for all xE 'J:i. Let Rsp(T) =sup {1z1: denote the spectral radius of an operator T. By [7, lelmlla 4, p. 567], R (T) = lim l/Tn//l/n sp n_,,.oo For R(T ,z) z E/(T), = (T-zI) -1 An operator T is normal if T*T - TT* ~ 0 or if TT* - T*T l/d(z,cr(T)) for all zt=j'(T) Let is called the resolvent of T at z. T*T - TT* :::: 0. J(T{/ = Rsp(T), i f T is normal then and I t is well known that [11, p. 115]. ~ 0, where T is hyponormal i f and T is paranormal i f // R(T ,z)// = d(z,o-(T)) = inf{1z-w/: rP denote the set of all paranormal operators on fl . For T an operator on 7::/ the numerical range of T, to wea-(T)}. {(Tx,x): xG.f/, 11x/I= complex plane and l}. cr(T) £ W(T), W(T), is equal W(T) is a convex, bounded subset of the [see 6]. In [ 6] Donoghue proves the following theorem. Proposition 1.1 'Ff, then W(T) If T is an operator on a two dimensional Hilbert space is the convex hull of an ellipse whose foci are the eigenvalues of T. If the eigenvalues are distinct, then the eccentric- ity of the ellipse is sin t / (x,y)/ y where x and where 0 ~ t ~ TT/2 such that cos t are unit vectors in 'fl each generating a 6 dist i nct eigenspace. W(T) is the closed c ircular disc about A. Wintner [32] z~W(T). then //T-zl II . for all [ 7, corollary 3, p. 566] that //R(T, z)// ~ Z€f(T). I f S is a set o f complex numbers, then i = {TE BOf'): = G. Orland [ 19] has proved the following c haracte rization of J. . denote the convex hull of S. Proposit ion 1.2 all of diameter z, co o-(T) co S W(T) }. for all z // R (T ,z)I/ ~ l/d (z,W(T)) has shown that I t is well known l/d(z,!T(T)) let I f the eigenvalues are the same, say T € ;J.... if and only i f Let /IR(T , z)I/ ~ l / d(z ,co u-(T)) for z ~ co cr(T). If T E B(;I), then T is the d i re ct sum o f Ac B(M) and wri tten T =A e B, i f Mis a closed subspace of /'::/ that reduces T, MI (0 ) and MI fl, T re stricted to M equals A, and T restricted to M~ equals B. .J. B € B(M ) , The fo l l owing theorem will be used several times [see 11]. Proposition 1. 3 If the operator T is the direct sum of the operat ors A and B, then 1. l/T// = Max{l/All, JIB!!] 2. cr(T) ll (A) V 0-(B) 3. W(T) co(W(A) U W(B)). If T is an operator on 14 and u(z) is a rational funct ion of z with no poles lying i n the spectrum o f T, t hen we can r e present u(T) by 7 2~i ~S u(z)R(T,z) dz u(T) where S is the union of a finite number of closed rectifiable curves that form the boundry of an open bounded set D of complex numbers, where o-(T) !; D and D contains no singularities of u(z) {see 24]. The mapping E from the Borel sets B in the complex plane r[ into ti is a resolution of the identity the set of projection operators on for TE B('}:/) 2. [see 8, p. 219] i f and only i f M>0 there exists such that for all S€ B II E(S)// ~ M, 3. if { Sn} a sequence of disjoint Borel sets, is x E fl then for each 4. for each TE(S) S.; B, E(S)T er (TIE (S) ) ~ S. and An operator T is a spectral operator if T has a resolution of the identity. If T~ B(~) is a spectral operator with resolution of the identity E, then we can form the following operator on 7::/ [8, p. 226Jl T = ( z dEZ . Jcr(T) 8 A spectral operator T is scalar if and only if scalar operator and if f A T ; T. If T is a is a rational function with no poles in t hen cr(T), f(T) = f(z) dEz . ( )ir(T) If T is a scalar operator, then f or each fixed unique complex Borel measure m is given by there exists a m such that (Tx,y) and x,y E fl 1 z dm(z) o-(T) m(S) = (E(S)x,y) for each Borel set S. We will need the following theorem [8 , p. 230]. Proposition 1 .4 T is a scalar spectral operator if and only if there exists a normal operator N and a self-adjoint operator Q such that T = QNQ-1. We shall always assume that the arc G0 is c2 -smoothly imbedded in a one parameter family of c losed rectifiable Jordan curves -1 ~ t $+1. Let T be Then Gt' R(T,z)x an operator on Fl and let x be a fixed element of fl . is a vector valued analytic function of z on f (T) and may possible be extended analytically to a larger open set. spectrum of T is a nowhere dense set of complex numbers, then If the R(T,z)x must have unique analytic extens ions since then the resolvent set is 9 dense so that any two analytic, or even continuous, extensions must agree on their conunon domain of definition. of Tis a subset of sions. G0 , then R(T,z)x Therefore, i f the spectrum has unique analytic exten- By taking the union of all open se t s each of which is the domain of an analytic extension of set on which R(T,z)x this open set and let R(T,z)x, we obtain a max imal open can be analytically extended. ~(T,x) Let j1(T,x) be the complement of ;O<T, x ) be in the complex plane [8]. An operator T on 'i=I is said to satisfy Dunford' s Boundedness Condition (B) [8, p. 226], if there exists a constant K depending only on T such that i f x,yE ")::/ and <T(T ,x)/l <T(T ,y) = 0, then // x /{ :5 K//x+y//. 10 Section II: Basic Properties of Paranormal Operators An operator T on t he Hilbert space Pl is paranormal if for each Z €f(T), l/d(z,cr(T)). //R(T,z)// paranormal operators on '):/ . f denote the cl ass of all Let Every normal operator is clearly hypoStampfli [26] has shown tha t normal [see Section I f or definition]. every hyponormal operator is paranormal: Proposition 2.1 If the operator T is hyponormal, then T i s paranormal . Proof. The proof follows from these fac ts about hyponormal operators [26 ]: (1) If aT + bl a and b is hy ponormal. are complex numbers and T is hyponor mal, the n (2) I f T i s hyponormal, then (3) If T is hyponormal and invertible, t hen Let z Ef(T) T- l // T // = R6 p (T). is hyponormal. then // R(T ,z)// 1 d(O,cr (T - zl)) 1 d(z ,a-('r)) Therefore T is paranorma l . Theorem 2.1 If T is a paranormal operator and if complex numbers, then aT +bl and T* a and are paranor mal. b are 11 Proof. If a= 0, z E f (aT + bl). the proof is trivial. Suppose a F0 and let Then I/ ((aT +bl) - zl)-l// = l /j(T la/ ~I)- 1 // a 1 /a/d((z - b)/a,O'"(T)) 1 d(z,cr(aT + bl )) Therefore If aT + bl Z"C:/(T*), is paranormal. then z E/(T) l/d(z,<T(T)) = l/d(Z,~(T*)). The fact that co ~(T) and l/(T* - zl)- 1 // = /)(T - zl)-1// Therefore T* is paranormal. = W(T) when T is paranormal f ollows immediat ely from Proposition 1.2. An elementary proof, i ndependent of Proposition 1.2 is given below. Theorem 2.2 Proof. If Tis a paranormal operator, then For any operator co ff(T) f W(T). co ff(T) = W(T). Suppose the the or em were false, then ,by a translation and rotation, if necessary, we may assume [see Theorem 2.1] z >0 1, and we have x sup Re co <T(T) ~ 0 be a unit vector in 'fl. and sup Re W(T) > 0. Then since Let d(z,CT(T))//R(T,z)// = 12 l/Txl/ 2 -:- z Therefore for al l z > 0, sup Re W(T) ~ 0 • Contradiction. 2 - 2zRe (Tx ,x). zRe(Tx ,x) ~ 0 so that Re(Tx,x) ~ 0. Hence, The following two theorems will be useful in constructing examples . Theorem 2. 3 If A is any operator on "'!:/ , then there exists a Hilbert and a normal operator N on I< with space 'k T = A f) N € B(?(e ~) Proof. Since · W(A) <T (N) = such t hat W(A) is paranormal. is a compact set of complex numbers, there exist s a Hilbert space ")< and a normal operator N on /< such that [7, p. 581]. If Let T =A ti N. Then by Proposit ion 1.3 z f JJ(T), then 1 1 //R(A, z)/I ~ d(z,W(A)) d ( z , CT (T) ) 1 d (z, o-(N)) d(z ,cr(T)) and //R(N ,z)// Therefore 1 ' u(N) = W(A) u(T) = ~(N). 13 II R(T ,z)I/ Max{//R(A,z)//, //R(N,z~Jj 1 d (z, a-(T)) so that T is paranormal. Later on we will need the fact that the normal ope ra tor N in Theorem 2.3 can be chosen so t ha t Theorem 2.4 CT(N) - ~(A) is a countable set . If A is any operator, then there exists a normal operator N such that 1. A~ N 2. CT(N) 2 3. ~(N) is paranormal, lr(A) , - u(A) and is a countable set whose points of accumulation are contained in Proof. and let Assume B(z,r) is compact and //A//= 1. For n = 1,2,3, ••• , u(A). let 2/(n+l):::; d(z,~(A)) ~ 2/n}, Sn {z: Mn Maxf 4, sup {I/ R(A,z)//: be the open disc of radius z £ snJJ , r about z. Since Sn 14 there exists Zni f Sn, 1 ~ i ~ nln I ffin I V such that B(zn.' l/Mu) . i=l l. Let N be a normal operator such that [see 10, p. 5811 a-(N) o-(A) U {zn· : l. Let T =A e N. and i If such that zt;f° (A) 1 ~ i and i f ~ Il\i• n = 1,2,3, • • • ] . lz/ ~ 2, zE:Sn/1B(zni' l/Mu). Hence d(z,cr(T)) }/R(A,z)// ~ /z - z If lz/ > 2, I ni then there exists n /z - Zni/ ~ l/Mu. Then l/M ~ 1. n then d(z,~(T)) d(z, ~ (N)) ~ d(z,W(A)). Therefore d(z,cr(T)) //R (A,z)// ~ d(z,W(A)) // R(A,z)J/ ~ 1. Since N is normal, z ff (T) . Let z E //R(N,z)// = l/d(z,cr(N)) f1 (T) , l/d(z,~(T)), then //R(T,z)j/ =Max {//R(A,z)//, // R(N,z)//] d (z, O""(T)) for all 15 Therefore T is paranormal. As we shall see, the class of all hyponormal operators on 'fl is distinct, in general, from the class ~ of paranormal opera tors. know [ 25] that if T is hyponormal, then invertible, then T- 1 is hyponormal. If T // = R sp (T) We and i f T is also These properties do not gener- alize to paranormal operators. Theorem 2.5 Proof. There exists an invertible paranormal operator T such that 1. T is not hyponormal 2. T2 is not paranormal 3. /IT/I > Rsp (T), and 4~ T-1 is not paranormal. Let Let N be a normal operator such that Then by Theorem 2.3, T is paranormal. ~(N) = W(A), and let We know from [25] that a hypo- normal operator is hyponormal on invariant subspaces. since A is not hyponormal, T is not hyponormal . W(A) is the closed disc of radius ~ about closed disc of radius 1 about z = 1. T =A e N. By Proposition 1.1, z = 1, Therefore Therefore, and W(A 2 ) is the 16 But normal. I W(T 2 ) cou(T 2 ) Therefore, Let x = (j{J and so by Theorem 2.2 then I/xii 1 ~fiO > 3/2 /IT/I ) //Ax// Therefore }/ T l/) R5 p(T). 11 Tl/ · t · Con t ra d ic-ion. If T -1 = 11 R(T- 1 ,o)// Hence T- l and T2 is not para- II Ax If = ,lzflO. were paranormal , then 1 a (o, fl (r 1)) · no t paranorma 1 . is Then 17 Section III: Spectral Properties of Paranormal Operators In addition to discussing the relatively elementary spectral properties of paranormal operators, the goal of this sect i on is to find out what can be said about a paranormal operator whose spectrum is a "thin" subset of the complex plane. In particular, if T has countable spectrum, then T can be decomposed as the direct sum, of a = ~(T), and of an operator A with a subset of the derived set of ff(T). In general the operator normal operator N with T = N ©A ~(N) N ~ A, need not be normal, however if rectifiable Jordan curve G0 , ~(A) lies on a c2-smooth cr(T) then T is indeed normal. I f T is a scalar paranormal operator with countable spectrum, then i n order to show that T is normal, the condition that to the following condition: such that lw-zl For each = d(w,a-(T)). u(T) z E cr(T) ~ G0 can be weakened there exists w ~f(T) An example is then given to show that a (non-scalar) paranormal operator T with countable spectrum satisfying the above property, need not be normal. If the assumption that the paranormal operator T has countable spectrum is dropped, then T has a non-trivial lattice of c losed invariant subspaces when ~(T)f G • 0 I was unable to prove the more general statement that a paranormal operator T with er (T) ~ G0 is normal. recently Stampf li f31) published a proof of this statement (see the introduction for a more detailed discussion of this). In ge~eral the points on the boundary of the spectrum of an operator T are not normal eigenvalues nor normal approximate Just 18 (} 0-(T) f <r 7r(T) , [ 11]). eigenvalues (although on However, certain points when T is paranormal must be either normal eigenvalues or dv(T) The theorem i s due to Putnam [22, 23], normal approximate eigenvalues. but the proof given here utilizes a trick of von Neumann and is shorter. Theorem 3.1 If T is an operator with eigenvalue of T and i f there exists 1, then z z an eigenvalue or approximate w E j)(T) such that I w-zl l/R(T ,w)I/ = is a normal eigenvalue or normal approximate eigenvalue, respectively. Proof. Suppose z is an approximate eigenvalue of T. stated in the theorem. Then there exists a sequence vectors in "!=I such that I-S R(T,w)(T-zl) so that 2 = 1. - Let //(I-S)xn//~O so that l/S II = /z-w/ //R(T ,w)// //(1-S*)Xu// //(T-zl)Xu// -7 O. and Let {Xu] w be as of unit S = (z -w) R(T ,w). llS~ll~l. Then Moreover, Now i/(I-S)xn!/ l/(I - S*)Xn//--70. 2 = l/S*xdl Then since 2 - //Sx0 // 2 ~ l - /I SxJI I -S = R(T,w)(T-zl) 2 , = (T-zl) R(T,w), //(T*-wl) (I-S*)Xnfl Therefore 'Ft /I (T*-zI)xn//~ O. such that // (T*-zI)y0 // --'>- 0, //(T-zl )ynl/ ~O . proof for z If Therefore z { y0 1 is a sequence o f unit vectors in then by a similar argument is a normal approximate eigenvalue. an eigenvalue of T is similar. The 19 Corollary If T is a paranormal operator with approximate eigenvalue of T and i f there exists = d( z,~(T)), lw-z/ z then z an eigenvalue or w Ef (T) such that is a normal eigenvalue or nor mal approxi- mate eigenvalue , respectively. Proof . Let z and w J be as stated. Then since T is paranormal / w- z/ w- zl II R(T ,w)O d (w, O"(T)) 1. The corollary now follows from Theorem 3.1. If T is an operator, then in general isolated points of the spectrum of Tare not eigenvalues [see 11, problem 147]. If T is para- normal, then the following is true. Theorem 3.2 Isolated points of the spectrum of a paranormal operator are normal eigenvalues. Proof. Let T be paranormal and let Let C be a circle about then z be an isolated point of of small enough radius so that if II (w-z)R(T,w)/I = 1 Since /w-z/ = d(w,cr(T)) . z w w = z. (w-z)R(T,w) is an analytic function of (w- z)R(T,w) is analytic on an open disc containing C. P - - -f 1 27Ti c at as R(T ,w) dw. tr(T). wt C, w-"t- z , Therefore Let 20 Then TP - zp = - _1_ 27Ti TP = zP. so t h at Since corollary to Theorem 3.1, Corollary PF 0 z fc (w-z)R(.T w) dw 0, ' [see 24, p . 421], ze ~p(T). By the is a normal eigenvalue . If T is a paranormal operator on a fini t e dimensional Hilbert space, then T is normal. Proof . Since the dimension of the Hi l bert space is finite , finite set. value. ~(T) By Theorem 3.2, each element of u(T) is a is a normal eigen- Recall that the eigenspaces of distinct normal eigenva lues a r e orthogona l [14, p. 233]. If O"(T) = {z1,z 2 , ••. ,znl identit y operator on the eigenspace of zi, i and i f = 1,2 , .•. , n , Ii i s the then T can be wri t t en as Therefore T is norma l . Paranorma l Operat ors with ~(T ) Count able Only paranormal operators with countable spectra will be conside r ed in this part. It will be shown that a paranormal operator with a countable spectrum i s not necessarily a normal operator. general we can make the following statement: However , i n 21 If T is a paranormal operator on "Fl with countable Theorem 3 . 3 spectrum, then either T is normal or ~(N) operator with of = ~ (T) T ~(A) and =A e N where N is a normal is a subset of the derived set cr(T). Proof. is an isolated point of z If is a normal eigenvalue of T; let E(z) 0-(T), then by Theorem 3.2 be the eigenspace of ~(T) . cr0 (T) denote the set of all isolated points of M lJ closed span z Let z. Let E(z). z e: U-0 (T) By the definition of a normal eigenvalue [see Section I J E(z) T and T is normal on reduces T and T is normal on M. isolated point, Suppose M f (0). M f ~ . z € <r (T). for each E (z), Since M =ti If 0 Then write ~ , ~(T) Consequently, M must have at least one then Tis normal. K e M and T restricted to Kand N is T restricted to M. N is normal. Suppose to the contrary that T = A e N where A is Clearly <:T(A) ~(N) = 0-(T) CT (T), then there exists such that w ~(T) . Then point of CT(A), Zf C, then /z-w/ so there exists a circle C about = d(z,O-(T)) = d(z,<T(A)). w 1 1 d(z,o-(T)) d (z ,O'(A)) wt: 0-(A) is an isolated w such that if Then for II R(A,z)IJ 6 Max {llR(A,z)I/, II R{N ,z)IJ and is not a subset of the set of all accumulation points of is an isolated point of reduces z€C l = II R(T ,z)// 22 Therefore we may use the method of proof of Theorem 3.2 to conclude that wt: up (A). of T. But 0-p (A) f; ifp (T), so that is a normal eigenvalue w Contradiction. With Theorem 3.3 we can easily classify all compact paranormal operators. Corollary If T is a compact paranormal operator, then either T is normal or T = A $ N where N is compact and normal and A is compact and quasi-nilpotent. Proof. The spectrum of a compact operator is countable with zero the only possible point of accumulation. To show that there exists a non-normal paranormal operator with countable spectrum, simply let N with o-(N) Then A = (g ~) and choose a normal operator counta ble as in Theorem 2.4 so that A ~ N is paranormal. A e N has countable spectrum and is not normal since A is not normal. However, i f the countable set restricted by assuming that Jordan curve G0 , ~(T) iT(T) for lies on the TE rP c2 -smooth rectifiable then T must indeed be normal. theorem will be proved. is further First, a more general 23 Theorem 3 .4 If A is an operator on ?=I with i solated point of G0 such that for ~(A) <r(A) f G0 and with w an and if there exists an open set U containing d(z,G0 )11R(A,z)/I ~ 1, z e U- G0 , then w i s a normal eigenvalue of A. Proof. Without loss of generality assume that an isolated point of (}(A), w = 0. Since zero is R(A,z) is analytic in a deleted open neighborhood D of zero with D f;. U. Let C be a circle about zero con- tained in D. z ED Then for each R(A,z) where A n Recall that G0 = - ~1- {z-n-l R(A,z) dz. 27Ti) C is c2 -smoothly embedded in a one parameter f amily of c l osed rect ifiable Jordan curves For a,b E G0 define Gt, -1 ~ t ca,b = c 1 + c 2 + C3 + C4 shown above and such that ~ 1. where the Ci's are as 24 1. for z € c2 , /z-al = d (z ,G0 ) , 2. for z € c4 , /z-b/ d ( z, G0 ) , 3. Cl f; Gt and ca,b ~ D and t h a t zero is contained in the interior of the re gion determined by 27Ti j - 2} 1 f i s independent of t I(a ,b) Then I(a ,b) I(a,b) - A._ 3 = _1_ 0 <: t < 1. for some C3f. G_t We shall always assume that and Now define (z-a) (z-b)R(A, z) dz ca , b C (z-a) (z-b)R(A,z ) dz. and we have ~ (c (-z 2 + (a+b)z - ab + z 2 )R(A ,z ) dz 27rl. ) (a+b) 2 ~ 1 {ab) 2 ~ 1 i•R(A,z) dz - £ R(A, z) d z. The ref ore I (a,b) lim a,b --? 0 a,b € G0 Therefore, to show A_ 3 = 0, it suffi ces to show I(a, b) Let G > 0, and fix 0 < t < 1 lengths less than E for every o. so that the arcs a ,b E G0 • Let c2 and c4 h ave 25 p~) = sup{i/R(A,z)//: Then there exists q(€) > 0 z€GVG t -t }<. 00 . such that whenever Jai,lb/ < q(€), we have If 1. the length o f c1 and C 3 2. if then izl zE C2 , zc:Cab• , /IR(A,z)// ~ 1/ I z -b I. t hen /al, lb/< q(t). A_3 = 0. Hence R(A,z) l/Jz-a}. Similarly, i f ztC4 (5 /rr) (2€) 2 , Therefore Therefore Thus p. 573). = Making the obvious est imates we obtain JI I(a,b)I/ whenever < 2€. //R(A ,z)// ~ l /d(z ,G0 ) then l /p (t), and is les s than A_0 = 0 l(a ,b) ~O for n as a,b~O; a,b € G0 • = 3 ,4 ,5 , .•. [7, Theorem 18, has a pole at z ero and henc e OE u (A) /j,ee 7, p p. 573]. To s ee that 0 E 0- P (A) is a normal eigenvalue , we choose such that d(u,G0 ) = lu/ = d (u,~(A)). l/d(u,G0 ) = l/ lu/, and since obtain /u/l/R(A,u)// = 1. Then since 1/R(A,u)// j l/R(A,u)# l /d(u , O-(A)) = ut D ~ 1/ Jul, we Thus by Theorem 3.1, zero is a n ormal eigen- value of A. Theorem 3.5 cr(T) f G , 0 for If T is an operator with countable spectrum and with and if there exists an o pen set U containing z6U-G0 , Proof . d(z,G0 )1/R (T ,z)I/ ~ 1, such t hat then Tis a normal operator. Let M be the largest clo sed subspace of T and T is normal on M. G0 fl such that M reduces By Theorem 3.4 each isolated po i nt z of 26 a-(T) is a normal eigenvalue. E (z) .f M. Hence If M = 'rt and T = A~ N, M. Since If E(z) is the eigenspace of M I (0). M :f ")::/ . then we are done, so assume [see Proposition 1.3], we know countable and hence has an isolated point maximality of M, tl M z. E(z) By Theorem 3.5, reduces T and Tis normal on E (z) £ M. is 0-(A) normal eigenvalue lof A and consequently the eigenspace, Thus ..L =M Wri te 'Jf where A is T restricted to ~ and N is T restricted to o-(A) (; o-(T) reduces A. then z' Contradiction, since E(z), E(z). z is a of z By the ...L E(z)~ M . Corol l ary If T is a paranormal operator with countable spectrum and if u(T) .s; G0 , then T is normal. Proof. For zEJ°(T), llR(T,z)/I = l/d(z,a-(T)) !( l/d(z,G0 ) . The corol- lary now follows from Theorem 3.5. In the Corollary to Theorem 3.5 it was shown that a paranormal operator T with countable spectrum and with u(T)~ G , 0 must be normal. If it is assumed that T is a scalar paranormal operator with countable, then to show that Tis normal, the condition o(T) ~(T)f G0 can be weakened. Theorem 3.6 If T is a scalar paranormal operator with countable spec- trum and if for each d(w,a-(T)), u E o(T) then Tis normal. there exists such that !w-u / = 27 Proof. that Let Un----7-U u E cr(T), /un-u/:::: d(~,<r(T)). and { ~l then there exists a sequence T :::: t;;/ (T) such Since Tis scalar ( z dEz. )cr(T) Therefore = (u0 -u)R(T , u ) n 1 u-un - - dEz . z-~ <r(T) Let x,y E fl be fixed and define m(S) = (E(S)x,y) to be the complex Borel measure for each Bo rel set S in fn(z) Then m Ifn (z) ) ~ 1 = U-\Jn and z-\ln and f f (z) n (z) -~f(z). 0-(T). For each {~ if z if z I u =u Therefore we may apply the Lebe s gue dominated convergence theorem: /1 jm({u})/ f (z) dm(z) / a-(T) = lim n~oo = lim n~= } ( fn (z) dm(z) ) )u(T) /((u-u0 )R(T,un)x,y)/ ~ / u-uJ //R(T , u0 )// //x//// y// = //x// //y//. Since m( { u} ) (E({uJ)x,y), zE O-(T) let we have that / (E((ul)x,y)) ~ l/x// )}yl/ . 28 Letting y = E({u} )x, II E({uJ)}/ ~ 1. we obtain i/E(fu])xJ/ ~ lixl/, Therefore E( f u}) and hence is an orthogonal projection f or each u E o-(T) [see Section I] . Let S~ ~(T) be a Borel set, then S is a countable set so write :z= 00 (E(S)x ,y) = 00 2: (x,E({zrJ )y) (E({znJ )x,y) n=l n=l 00 = conj L (E({z nf)y,x) n=l Therefore E(S) = E(S)* and hence (E (S)y ,x) = (x,E(S)y). E(S) is an orthogonal projection. Consequently, T i s a scalar operator with a resolution of the identity of orthogonal projec t i ons; and hen ce t is normal. To show that the condition o(T) ~ G0 is stronger than the condi- tion stated i n Theorem 3.6, let C be the f ol lowing countable compact set of complex numbers: c {O} V{l!n + n = 1,2,3, ..• }. Then C does not lie on a c 2 -smooth c losed arc, but C doe s satisfy the condition in Theorem 3. 6 , i.e. for each that /w-z/ = d(w,C ) . zE C there exists w~ C such To see this, observe that this condition is equivalent to the following: For each D not intersecting C such that z f oD. z E. C there exists an open dis c 29 We will now see that the following condition on omitted from Theorem 3.6: such that Jw-z/ For each = d(w,rr(T)). z E 0-(T) 0-(T) there exists cannot be wC:/(T) Stampfli [25] has shown that a scalar hyponormal operator whose spectrum has zer o area in the complex plane, must be normal. Stampfli's result does not generalize to paranormal operators. Theorem 3. 7 such that There exists a non-normal, scalar paranormal operator T <T(T) is countable. Moreover, Cl(T) has exactly two points of accumulation. Proof. Let N 0 be the normal operator (~ g), let Q be the self- (i 6), and let A be the non-normal operator QN Q -1 adjoint operator 0 (8 f). By Theorem 2.4 there exists a normal operator N such that: 1) ~(N) is countable with zero and one the only points of accumulation, and 2) T Let A e N is paranormal. B = Q e I. Then B = B* Since A is not normal, T is not normal. and B(N e N)B-l = QN Q-l ~ N 0 0 AeN T. Therefore, by Proposition 1.4, T is scalar. In light of Theorem 3.6, it seems reasonab le to conjecture the following theorem: If T is a paranormal opera tor with countable spec- trwn such that for each · 1w-z/ = d(w,~(T)), ZE 11(T) there exists then Tis normal. wcj?(T) such that This conjecture is false. 30 Stampfli [25] has shown that a hyponormal opera t or with countable spectrum is normal, however this result does not generalize to the paranorma l case. Theorem 3 . 8 1. There exists a paranormal operator T such that ~(T) is countable with zero the only point of accumulation, lz-21 ~ 2, and be the closed disc of radius n 2. · i f 3. Proof. n = 1,2. Let z E IJ(T), then T is not normal. Dn about n, for Let V be the Volterra integration operator, i .e . for 2 fE L (0,1 ) , t sof (x) dx. (Vf) (t) Let B = (I + V)- l, and let { 1J and Hence /IB// = 1. closed d i sc of radius A <r(A) II B// = 1. I - [OJ B. and Therefore By f 11, problem 150], W(B) is cont ained in the W(A) f n1 • For n = 1, 2, . •• , let Then 1. Fn = { z f o2 : 2. ~ sup{/IR(A,z)//: z E Fnt, 3. dn inf{ d (z ,W(A)): Z6 (.)D 2 )fl Fn} > 0, 4. Pn Max[Mu, l/dn 5. B(z,r) 4/(n+l) ~ /z/ ~ 4/n}. f, and be the open disc of radius <T (B) r about z. 31 Since Fn is compact, there exists o-(N) =foll) {z Let N be a normal operator with n = 1,2,3, . . . }, then ff(N) point of a ccumulation. Let such that n.). 1 ~ i ~ mn, is a countable set with zero the on l y T =A~ N, then ~(T) = u(N). and i We now verify that T is paranormal. If z €Dz , z 'I 0, then there exis t s d (z, <r(N )) l/R(A,z )// ~ n such tha t /z-zn./ //R(A,z)// 1 ~ If z (l/Mn)// R(A,z)// ~ 1. is real and negative, t hen 1 d(z,u(N))/IR(A,z)// ~ lzld(z,W (A)) Suppose z ~Dz and that z the point of intersection of 0 Dz necting z d(x,W(A)). and W(A). is not real and negative. n a nd Let i x 'I 0. suchthat Then /x-znJ ~ 1/Pn, and so be d( z,W(A) ) = / z-x / + x EFnflB(zn.,l/Pn) . 1 Then x with the shortest line segment con- Observe that Thereexists = 1. 32 ~ lz-x/ + d(x,W(A)) = d(z,W(A)). Therefore, d(z,0-(N))//R(A,z)// ~I z-z I I/ R(A,z)// ni ~ d(z,W{A))d(z,~(A)) Therefore, for each complex number 1. z ; 0, d{z,CT(N))// R(.A, z)// 6 1. Since N is normal, for each l / d (z, a-(T)). Hence, for z z t=j{T) =j7(N), l/d(z ,<T(N)) // R(N,z)// </' (T) /! R(T,z)// = Max{J/R(A,z)ij, //R(N,z)//} 1 d (z ,<T(T)) Therefore T is paranormal. For the rest of this section, the assumption that T has countable spectrum will be dropped. If T is an operator on Fl , then T is said to have a lattice of closed subspaces on the closed subsets of for each closed such that: S .S 0-(T) 1) M(~) = 0, closed subsets of o-(T), there exists a closed subspace, 2) M(~(T)) then =fl, and M(S1/J 82) is non- trivial if there exists a closed set M(S) I 0 and M(S) f fl. CT(T) M(S), if of 'fl 3) if S1 and S2 are M(S1)fl M(S2). S £ 0-(T) The lattice such that 33 An operator T is said t o satisfy Dunford's Boundedness condition (B) [8 , p . 226], if there exis t s a constant such that if x ,y E Ff and a-(T ,x) 17 o-(T ,y ) k depe nding only on T = 0 [see Section I] , then /l x//5; kl/x+y//. Theorem 3. 9 If set U containing TE B('H) G0 wit h (J (T) .f G0 such t hat f or each and i f there exists an open d (z ,G 0 )1! R(T ,z )I/ ~ 1, z f U-G0 , then either there exists a complex nwnber b s u ch that T = bl, or T has a non-tr ivial lat t i ce of invariant subspace s on the closed subse ts of <T(T). If T is a paranormal operator with z~G0 , cr(T) f G0 , d(z,G0 )//R(T,z)/I ~ d(z,<T(T))J/R(T,z)// = 1. then f or each Thus , as an immediate consequence of Theorem 3.9 , we have the following: Corollary If T is a paranormal operator with there exists a complex number b such that <T(T) f G0 , T = bl, then either or T has a non- trivia l lattice of invariant subspaces on t he closed subsets of 0-(T) • .. The following lerruna comes from [7, Lenuna 2, p. 240; and Theorem 18, p. 264]. Lemma 1 (a) 0-(T, x) = 0 if and only if x = 0. (b) If wit h and i f there exists an open T €. B('W) set U containing a-(T) .f G0 G0 such t h at f or e ach z E. U-G0 , 34 d (z ,G0 ) /IR(T ,z)/I ~ 1, ~atisfies then T is scalar if and only if T conditiori (B). If T is an operator on fl with Lemma 2 o-(T) q' (T) f U = closure G0 , then CT(T ,x). x~'JI Proof. ~(T,x) ~ ~(T) Clearly be false, then there exists N(a,f)/l/(T,x) Let f x(z) xE/:/. for all a f o-(T) and X€ '}/, where €: Suppose the lemma to >0 such that N(a,E) be the maximal analytic extension of Then define =0 XE?i . for each fx is analytic in Bz:fl~"):/ as N(a,E)l//(T). Bzx = fx(z). Bz(x + y) Fix = [z: J z-a/< €} R(T,z) x For x,yEJ/, for each z e(J (T)UN(a,t) , then for z E/(T) = fx+y(z) = R(T,z)(x + y) = R(T,z)x + R(T,z)y Since Bis a continuous function of that Bz(x + y) = Bzx + Bzy shows Bz is homogeneous and zEf7(T)VN(a,E'). Therefore, is one-to-one and onto. and a fj7 (T). for all z and since ~(T)~G 0 , Z€,P(T)VN(z,E). we have S i mi larly on e = (T-zl)Bz = I for al l B8 (T-al) = (T-al)Ba =I , so tha t Bz(T-zl) Thus, by [11, problem 41] , T-al T- al is invertible Contradiction. We now state the follow ing lemma from [7, Lemma 4, p. 254]. 35 Lemma 3 If TE BO¥) U containing G0 cr(T) S G0 with z e U-G0 , so that for each if Sis a closed subset of and i f there exists an open set M(S) = {xt:'f/: then lf(T), d(z,G0 )//R(T,z)// ~ 1, and s} is 0-(T,x) f a closed subspace invariant under T. Proof of Theorem 3.9. If by Theorem 3.4 T = bl. cr(T) = {b} for some complex number b, For the rest of the proof assume that then ff(T) has at least two points. Let M(S) and only i f x tf/, be defined as in Lemma 3. M(~) = 0. x 0, so that M(a-(T)) fl. It is obvious that whenever S1 and s2 By Lemma 1, Since = 0 ~ (T,x) O" (T ,x) f;- IT(T) if for each M(S1 fl Sz) = M(s 1 ) II M(Sz) are closed subsets of ~(T). Therefore, these sets form a lattice of closed, invariant subspaces on the closed subsets of O-(T). If this lattice is non-trivial, then the proof is com- plete. Suppose that this lattice is trivial, i.e. M(S) for every closed subset S of cr(T). xE1t'. M(o-(T,x)) ='fl for every nonzero cr(T,x) = <l(T) ~, either so that, by for all nonzero x = 0 or y = 0. Then, since xE/I. = 0 or M(S) 'jc/ xf M(cr (T,x)), Consequently, by Lemma 2, Hence whene ver cr(T,x)()O-(T,y) Thus T satisfies condition {B) trivially Lenuna 1, T is scalar. Therefore, since ~(T) has at least two elements, T has a non- trivial lattice {formed from the resolution of the identity of T [see Sect ion I]) of closed, invariant subspaces on the c losed subsets of ~(T). 36 Section IV: (j> denote the set of all paranormal operators on Hilbert Let fl . space Topological Properties of Paranormal Operators Let ;[ = {T eB('R'): col.f(T) 71 be the set of all normal operators on 'JI. and let B(~) be assumed that B('?f) 7/, fP and connected subsets of B(fl). When the dimension of 1/ is finite, dim 'ii = o0 , When nowhere dense subset of fP. 5 ~ dim Pl< DO dim ~ In rP relative to I/, '!.. , will be discussed. It will be shown that f? = (fJ • It will always has the uniform operator (norm) topology. this section, the topological properties of and W(T) } , = oo, , then then (}J ;J are closed, arc-wise In Section II, it was shown that then by the corollary to Theorem 3 . 2 , then i t will be shown that When dim 11 ~ 4, t he n <P has a nonempty interior i n I / /f <;_ (f> [:_ / . /l_ is a <P = f J. . and when When and it is not known i f (fJ is a nowhere Finally, it will be shown that J.. is a nowhere dense subset of ;j. dense subset of B("t{J, when dim i::/ ~ 2. The following notation will be used in this section: c o mpact set in the complex plane s + (6) ([ and if E 7 0, d(z,S) <: 6 ]. If S is a then let 37 If Sand Sn, {Sn n = 1,2,3, ... are compact sets in { , then the sequence I approaches S, written Sn ~S, if for every exists a positive integer N such that for n ~ N, ,;. /' 0 Sn f there S + (~) and S .S Sn + (E ) . In general (}(T) is not a continuous function of T i n ~(T) [see 11, problem 85], but Theorem 4. 1 If {Tn} B(~) is continuous if we restrict T to rP. is a sequence of paranormal operators approach- ing the operator T in norm, then Cf (Tu)~ 0-(T) as n~=. To prove this theorem we need the following lenuna f r om [11, problem 86] . Lenuna If Sf B('11') T f B(P/) and and II T-S}/< S' , Proof of Theorem 4.1 > 0, then there exists then o-(S ) f: cr(T) + (€). E t- > 0 (}(Tn)___,.CT(T), n ~ N, n ¢ N. o-(T 0 ) f; O-(T) + {€). u(T) ~ 0-(Tn) + (t::) If this does not hold, then without loss of generality we may assume that there exists such that there it suffices to show that for each there exists a positive integer N such that for all such that if E > 0 We know by the lemma that for each exists a positive integer N such that for Therefore, to show 6 > 0 d(zu,<f(Tn))~ E E > 0 for all n. and a sequenc.e Since o-(T) {Zn 1 ~ (}(T) is compact, we 38 may assume Zn-~ z E cr(T) . If I Zn-Z / < € /2, ~ E then - E/2 -:;::::. <; /2. Hence Now choose m so that J/{Tm-T)R(Tm,z)I/ is invertible [11, problem 173]. < 1, then I - (Tm- T)R(Tm,z) Let A Then A (T- z I) Theorem 4 .2 Proof. Since (T -zl)A ~is so that z E:f(T). Contradiction. an arc-wise connected, closed subset of T f (J> see that the ray in CP I implies B(~) is arc-wise connected. B('P{). aT E <P for every complex number through T is contained in cP . a, we Therefore 39 Suppose T~ B('W) . Tn---+ T, { Tn f a sequence of operators in rP, and By the lennna to Theorem 4.1, Let li m n --7 00 Therefore, since 1 :; 1 sup d ( z, cr(Tn)) d (z, (}(T ) ) II R(Tn,z)/I = l/d(z,O-(Tn)) whenever there exists a positive integer N such that the sequence { /IR(T0 ,z)//: is bounded. Then, since //R(T ,z )}j---71/R(T , z)// n as = R(T,z)(T-Tn) R(Tn,z), R(T,z) - R(Tn,z) n~ oo . // R(T ,z)J/ = N} n .:f Consequently, lim n--?C>O fl R (Tn ,z)// 1 lim n-C)oo d (z , a-(Tn)) ~ S ince in general J //R(T,z)// ~ l/d(z,(J(T)), T is paranormal. I. is an arc- wise conn ec ted , closed subset of B(i=f). Theorem 4.3 Proof. 1 d (z , cr(T)) Since T E: J:. implies that aTE/ for every complex number a, is arc- wise connected. Let l/Tn-T // Tn -7T, for (2/(T-Tn//). {TnJ s;.£ //xi/= 1, and TEB(y). W(Tn) ~ W{T)+(2/IT-Tnl/) Consequently, W{Tn)~W{T). /(Tnx ,x) -(Tx,x) ) ~ Since Let and E- W(T) f W{Tn)+ > 0, then by the 40 lemma to Theorem 4.1 there exists a positive integer N such that lf(Tn) ~ <r(T) + (E) n ~ N. for all co o-(T ) ~ co <J(T) + (E) n n ~ N, Therefore, for and hence W(T) lim --W(Tn) n~= lim n~oo ~ Since 6 / 0 is arbitrary, W(T) ~co ~(T). Since in ge ner al 7l be the set of all normal operators on fl // Tn -T // ~ 0 implies a, /{, 7l. I/ Tn*-T* If ~O, operator topology on complex co 0- (T) + (E ) • T G ;A_. co o-(T) S W(T), Let co !T(T ) n B(f{). Since Since is closed in t he uniform T 6 /{ implies aT GIJ. f or any is arc-wise connec ted. We know that 7( c (p c t_ f_ B ('fl) • Much more can be said about how the above four sets are related. It has already been shown [see the corollary to Theorem 3.2] that n = rP when dim 11 .::: "thin" subset of rP . 00 When dim 11 = o<:> , then I( is a very The f ollowing theorem makes this more precise. 41 Theorem 4 .4 Proof. ft. is a nowhere dense subset of (P when co Since 77. is closed, to show that 77 is a nowhere dense subset of fP , it suffices to show that Tf dim 'fl 77. and let €: ?? has empty interior in r? . Let > 0. First suppose that T has an eigenvalue of infinite multiplicity. We may assume that the eigenvalue is zero. zero . Then dim M = oo Let M be the eigenspace of M reduces T, ~nd we can write where Z is the zero operator on M. T = B~ Z Let be a non-normal paranormal operator [see Theorem 2.3] on M with N a normal operator such that ~(N) = closure w(~ ~)· Then B ~ S is a non-normal paranormal operator such that II T - B ~ S II = II B e Z - B -e SI/ /IS // ~. The last equality holds since l/N// f/2 and /} (g ~)JJ 6 . = 42 Therefore, since interior of If E > 0 is arbitrary, T is not contained in the 77 in rP o IT (T) Tc 17 , then is finite and eigenvalue of infinite multiplicity . () (T) = 0-P (T) We therefore assume that is infinite and that zero is an accumulation point of €/2. the open disc about zero of radius and T has an ~(T). (J(T) Let D be Let Ebe the resolution of I the identity for T [see Section I] so that T = ( zdEz. ) tr(T) Let M = E(D), ~(T) P - D, and let A = Then M reduces T, dim M = oo the zero operator on M. Then fp zdEz• and A is a normal operator. Ae Z Let Z be is a normal operator with zero an eigenvalue of infinite multiplicity, and I/ T - A $ Z II By the first part of this proof, there exists a non-normal paranormal operator s such that fl A e z - SI} < €I 2. (j T - S // ~ /IT - A e Z// Then + }J A e Z - S JI< € • 43 >0 ~ Therefore, since fl in tP • interior of is arbitrary, Tis not con t ained in the Hence the interior of 7( in (J> is empty. rP and 1. will be discussed. Next, the relationsh ip between c2 this define to be the set of al l opera tors T €J closed line segment or a point. set of all operators Te J: poly gon with k sides . For W(T) such that W(T) must be in the spectrum of T . T f ;1. , z E a-p (T) (\ () W(T) for T. Thus for the verticies of of T, when dim °JI < =n < dim f:I normal operators when 7? -/: '£ for i s a normal eigenvalue of W(T) are normal eigenvalues 00 5 ~ dim Pl < o0 Recall that [P = 7/_ Theorem 4. 5 If 5 ~ dim °):/ Proof. There exists SE. Cu U Cu-l • ~ are 1( • = fP = f when dim 'N ~ 4, The fo llowing theorem says 5 ~ dim Pl ..=: = dim ?¥ < oo . for = n c: = T t. Cn V Cn-l • Suppose Cn V Cn-l • rY and t._ are related when much more about how b e the z then S. Hildebrandt [15] has shown that and that ~ then e a ch vertex of Hence, all the operators in OCJ let a S. Hildebrandt [15] has shown that if TE Ck W(T) with is the convex hull of a k = 2,3 , ... , TE Ck, If = 3,4,5, .•. , k To do Since then the interior of (p in Cn U Cn-l f 7J, :J. T is nor mal. i , fl T - SI/< e , then To show this, _s uppose the statement were false. Then >0 there would exis t such that whenever fsn} f f such that SE I/ T-Sn // --7 0 and 44 n-2 l_) Ci. i=2 Then, since W(Sn)-7-W(T), T ~ Contradiction. n-2 \_) Ci. i=2 i. = J7 in 'J. . Hence, T i s an interior point of tP fP Let T be contained in the interior of T ~ Cn V Cn-l" Let e > 0. Since k ~ n - 2. Since dim ?I ~ 5 co <1(T) = W(T), and s i nce TE Ck, in TE Ck, Suppose for some there exists a normal operator N such that 1. /IT - N// < €/2 2. W(N) is a polygon with at least three sides, and 3. N has at least two eigenvalues interior of N = A e B Write Let a >0 z, w contained in the W(N). where B can be written as B and let then IJB - CJ/= //(g ~)JI Choose a > 0 Then since Hence small enough so that W(A) = W(N) A e C E t. Since and a. W(C) ~ W(N) cr (A e C) = CT(N), A e C and so that a< c /2. co CT(A e C) = W(A ~ C). is not normal a n d since 45 /IT - A e Cl/~ //T - N// + l/N - A~ Cf} ~ E/2 + c/2 = t' Contradiction. Hence f. . T is not an interior point of IP in It is an open question as to what t he interior of <Pini is when dim °JI dim "JI = oo • Theorem 4. 6 Proof. Write However , it can be shown that rJ> :f J = oo /}> :f ti. "Ji when .L M~ M d im ?t = oo where dim M 5. and N A where a,b,c when Let are three complex numbers that form a tr iangle with contained in the interior of the triangle . Consider A e N W(A e N). W(A) as an operator on M and observe that co ~(A e N) = W(N) A e N is not normal and since dim M < oo, A e N is not paranormal. z E/AA e N) s uch that l/R (A e N,z)// > d(z.~(A1 e N)) • Hence there exists .L Let I be the identity operator on M and let T A e N e al. Since 46 = CT(A ~ N) Then, cr(T) and Since d (z, o-(T)) 5f I z - af , II R(T ,z)// W(T) W(A ~ N). T € J:. Therefore • Max [ VR(A e N ,z)Q, _ l _ } Jz-aJ ff R(A ~ N ,z)// > d(z,<T(T)) 1 Therefore T is not paranormal. The main result of this section is to show that if then 1, is a nowhere dens e subset of (norm) topology. 2, in the uniform operator Once this is .shown, it follows iwnediately that and (]> are nowhere dense subsets of Theorem 4. 7 B('N) dim 11 ~ 77.. B(10. £ is a nowhere dense subset of B(H') when dim 'ti d3- 2. To prove this theorem we need the following two techni cal lerrm1as. Lemma 1 of If T ~ B(?t), vectors in '>I and are distinct, normal then there exists sequences [ x 11 { such that: = 0 1. (xn,yn) for all 2. //(T - z I)xn//~O as n--;ao 3. //(T - z 2 I)yn//-7'0 as n~o0 1 n, and approximate eigenvalues and { Yn} of unit 47 Lenuna 2 If TE f such that there exists distinc t then T is not coatained in the interior of Proof of Theorem 4. 7 ~ a,bE oW(T)/)crp(T), I.. . i. is closed [Theorem 4. 3], to show that Since :f. has empty interior. is nowhere dense it suffices to show that We first remark that if T is in the interior of roust contain at least two points. ((T-al)x,x) = 0 Then 2, write A E B(?i) ti = M -e M...1.. XE 11 for all where Suppose Te I. so that dim M = 2. Let I. , then and T 0-(T) al. =[a ]. dim'fl~ Since > 0 b cr(T) and define as A (g ~) on M, and Then IJ(T+A) {al and since Since JI A// = b >0 is arbitrary, A j_ 0 on M . ~a] I W(T+A). b I 0, T ~ interior Therefore T+A/f J.. J. . With the above remark completed, we can now finish the proof of Theorem 4.7. Suppose the theorem were false and there exists interior J; . Then there exists and I/ T - VII<= E, then tain at least two points. VE cf.. > 0 such that whenever From the above remark cr(T) e VE B('f/) must con- There must be at least two extreme points of W(T), since extreme points of ~(T). € T W(T) for TE cf. are extreme points of Hence, after a rotation, if necessary, we may assume there exists 48 Jm. -vV(T) z, z, l __ - - - - - + - - _ _...___ _ _ _ _ _ _ _ Re z, --?> Re. Pie ld.. such that 1. Re z.1 inf Re W(T), 2. Re z sup Re W(T), and 2 and are normal approximate eigenvalues of T [14, Theorem 2, p. 233]. By Lemma 1 there exists unit vectors x,yE 'ii = 0, //(T-z 1 I)x// < 08, Since such that (x,y) Let M be the closed subspace spanned by Cx { x,y j. and (/(T-z 2 I)y// < Define C € B('h') 08 . as -(r=./4) x, Cy = +(€ /4)y, and Cz Since J/ CIJ::; E/2, T + C 6 ;[. 0 for all S ince ((T + C)x ,x) we obtain . 7T (Tx,x) - €/4 inf Re W(T + C) < Re z . 1 a f 0- (TtC) / l ()W(T+C) such that J_ z t: M • Since Re a = inf and T+C E ~ , Re W(T+C). there exists Since C is a compact operator, Weyl 's spectral inclusion Theorem [11, problem 143] yields 49 ~(T+C) Therefore, a E IJ (T+C), - ~ p (T+C) ~ ~(T). so that p a Eu (T+C) 11 oW (T+C). p Similarly one shows t here exists b t: (f (T+C) /1 p Re b =sup Re W(T+C) >Re z , 2 such that Lemma 2, there e xists But oW(T+C) SE B(FI) such that and hence //SI!< € / 2 // T- (T+C+S)/J ~ II CjJ + II Sii <::: ~ so by assumption f b. a and By T+C+S~ J. T+C+S e f. Contradiction . Proof of Lerruna 1. n~ o0. There e x ists sequence s /(z 1 -z 2 ) (wn , y 0 )/ - > 0 ( w ,y ) - > 0 as n { Yn t of unit Sin ce z f. z 2 , Then Therefore, n and f wn ] n -->- oo . as n ~ oa . 1 50 There exists complex numbers inc;/ such that (x0 ,yn) = 0. an wn = 8nYn + bnxn, and /anJ 2 bn and unit vectors + /bn/ 2 = 1, From the above paragraph we have that Proof of Lennna 2. Let €- S ince 0. / and an ->"0, a,bE-0-p(T)/ ) Cl W(T), T [18, Theorem 2, p. 233]. are normal eigenvalues of ~ so a and b Let u,v€:'t/ { u,v} reduces be unit vectors such t hat Tu = au Then T. (u,v) = 0 Define and bv Tv and the closed subspace N spanned by Se- B("fl') as Su = €. v Then we may write Sv 0 Sz 0 T + S =Ae B for a 11 J. z EN . -1 corresponding t o ?o/ = N ~ N Then the matrix representation of A relative to { u,v} is A= ( ~~)· Hence o-(A) {a,bf £ 0-(T). Clearly Cl(B) f. o- (T). co v-(T+S) c co CT(T) A is not a normal operator. Therefore, W(T). So by Proposition 1.1, W(A) is the 51 convex hull of a nondegenerate ellipse (i.e., not a straight line) with foci at a and b. Since W(A) ~ W(T+S), co ~(T+S) Therefore, T+S ~ / . T ~ interior J:. • Thus, since we must have F W(T+S). J/SI} 6 >0 is arbitrary, 52 BIBLIOGRAPHY 1. S. K. Berberian, The Numerica l Range of a Normal Operator, Duke Math . J. 31 (1964), 479-483. 2. S. K. Berberian, A Note on Operators Whose Spectrum is a Spectral Se t, Acta. Sci. Math. 27 (1966), 201-203. 3. S. K. Berberian and G. H. Orland, On the Closure of the Numerical Range of an Operator, Proc . Amer. Math. Soc. 18 (1967), 499- 503. 4. C. A. Berger . and J. G. Stampfli, Mapping Theorems for the Numerical Range, Amer . J. Math. 89 (1967) 1047-1055. 5. William F. Donoghue, On the Numerical Range of a Bounded Operator, Michigan Math. J. 4 (1957), 261-263. 6. William F. Donoghue, On a Problem of Nieminen, Inst. Hautes Etude s Sci. 16 (1963), 127-129. 7. N. Dunford and J. T. 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