单叶算子的（U+K）-轨道闭包及Putnam-Fuglede定理
摘要
设H是可分的无穷维Hilbert空间，L(H)为H上所有的有界线性算子的集合，κ(H)为紧算子理想，(εN)(H)是所有的本性正规算子的集合，我们定义了这样一类算子。
定义 2．1 T∈L(H)∩(εN)(H)称为单叶的，如果存在连通的开集Ω(?)σ(T)∩ρ_s-F(T)，使得
(1)σ(T)=(?)；
(2)dim ker(T-λ)~*=1，(?)λ∈Ω；
(3)ker(T-λ)={0}，(?)λ∈Ω；
(4)(?)λ_0∈Ω，使得∩_(n=1)~∞ Ran(T-λ_0)~n={0}，且T|_(ran(T-λ_0))≌_(u+k)T，
根据定义2．1有
引理3．1设算子T是单叶的，由定义2．1知(?)_0∈Ω，使得T|_(Ran(T-λ_0))≌_((u+k))T，则对(?)λ∈Ω，都有T|_(Ran(T-λ))≌_((u+k))T，
再由文献[6]中的命题2．7以及单叶算子的定义有
引理3．2算子T是如定义2．1的，则对(?)λ∈Ω，都有λI′(?)T∈(?)，这里I′为一维空间上的单位算子。
由引理3．1及数学归纳法有
推论3．1设{λ_1，…，λ_n}(?)Ω，且λ-i≠λ_j，i≠j，则有
T|_((V_(i=1)~n{ker(T-λ_i)~*})~⊥)≌_(u+k)T，
考虑T在分解H=(V_(i=1)~n{ker(T-λ_i)~*})(?)(V_(i=1)~n{ker(T-λ_j)~*})~⊥上的表示，再利用有限维矩阵的性质可以得到
推论3．2设F_d是以{λ_1，…，λ_n}(?)Ω为对角线元的对角阵，则存在一个算子C，使得
T≌_(u+k)(?)
吉林大学硕士学位论文
我们得到
推论3.3算子T是如定义2.1中的单叶算子，F是有限维空间上
的算子，且a(F)〔.，则有
FOT任(U十尤)(T).
与
子
算
八习
.自口
、、、，...，Z
弓.理。.。设二是，*维又寸角阵，形女。。一f式了
、‘毛，
洲u+胡一相似当且仅当凡的对角线元林l，…，凡，}。贝，_互不相同，
伪，的第、列不在R。叹T一凡I)里，1三*兰，2.
Fti()
)
算子，做个小扰动，可使得姚:的第*列不
姚IT
/了矛!、、飞
一一
C
如
形
在几州T一入:)里，在根据引理3.;
推论3一形女口一(
凡()
姚IT
)
有
算子，这里Fti是有限维对角阵，
对角线元互不相同，且都在几里，则c钊u+门(T).
由BDF定理，谱的上半连续性，推论3.1，划等，我们可以得到
定理2.1设T是如定义2.1的单叶算子，则(u+门(T)={A。
￡(万):A是本性正规的，且满足(、)(**)(、、*)}，
(，).二(A)=‘2;
(乞乞).二。(A)=Jg之;
(乞z乞).乞，“l(A一入)=一1 .V入任百2.
由定理2.1立即可以得到
定理2.2T与s是近似相似的两个单叶算子，则有T与s是近似
(u+均一等价的.特别的，如果T与s是相似的，也有T与s是近
似(u+均一等价的.
类似引理3.1的证明，利用解析函数的性质可以得到定理2.3
定理2.3设T是如定义2.1的单叶算子，则必有一个单位圆盘D
内的单叶解析函数。，使得兑〔!l( D)c彭，这里彭表示的是豆的内
部.
吉林大学硕士学位论文
由定理2.3易得如下两个推论
推论2.1设T是如定义2.1的单叶算子，若有以任优，，灭祥二。(尹)，
则蛇一定是单连通的.
推论2.2设T是如定义2.1的单叶算子，若有几一彭，彭表示瓦
的内部，则几一定是单连通的.
利用推论2.2我们得到
命题3.1两个本性正规算子相似，不一定有他们是(u+门一等价
的.
所以定理2.2中，我们不能用(u+门一相似，而用近似(“+门一
相似.这也说明了(u一十们一等价分类是比相似分类更细的分类.
在文献[ls}中，定义一了类单边移位算子，并考虑了单边移位算子
与类单边移位算子的(u+均一轨道闭包，结论是他们的(l1+门一轨
道闭包相同.这里我们同样也给出类单叶算子的定义.
算子s。侧川川:N)(II)，称为类单叶的，如果存在单连通的开集
‘ZC。(s)自户、一。(S)，使得
(1)。(S)=百2;
(2)d该，;‘k(:r(S一入)*=1，V入任人2;
(3)ker(S一入)={O}，V入任公2;
(4)己入。任‘老，5 .t·S}，、u，‘(、一入())望;，十、S·
问题:如果T是单叶算子，￡是相应的类单叶算子，那么T是
不是在i百不画画里呢?
Let H be an infinitely dimentionai Hilbert space,and denote by (/f)the set of all linear operators acting on H.K,(H) denotes the ideal of compact operators on // and (sN)(H) denotes the set of essentially normal operators. We defined this kind of operator.
Definition 2.1 T 6(H) f](sN)(H) is called nnivalent operator, if there exists connected open set,
(2)dimker(T - A) = 1. VA 2; (3)ker(T- A) = {()},VAe JJ;
(4)3A0 ,s.t.fXL, Rnn(T - A0)" = {0},and r|/?n,l(r_A()) =. T. From Definition 2.1, we have
Lemma 3.1 Let T be univalent operator as 2.1, then1 exists a AO e JLs.t. m(r-A) +A: T,tlien for all A we have T|, T. From proposition 2.7 of [6] and the definition of univalent operator Lemma 3.2 T is given by Definition 2.1, then for all A T e
(/C)(T) hold, here / is the identity operator acting on one-dimensional space.
From Lemma 3.1 and induction,we obtain
Corollary 3.1 Let {A,, ,Xn} C Q,and A, Ay j.When T is restricted to the invariant snbspaoe formed by (,s'pan{ker(T - A,7)}f=, J.then the resulting operator is (U + /C)-similar to T.
Considering the representation of T on H = (v;.'=1 {ker(T-A,)})(B(v; {ker(T-A,;)})x.using property of finite dimensional matrix, we have
Corollary 3.2 Let F,i be a diagonal matrix {A). , A, C il. then there exists
operator C,s.t.
I Fd 0
T I
C T We have Corollary 3.3 If T is a univalent operator as 2.1.F is an operator on a finite
dimensional space, whose spectrum subsets SJ,thon F T(U + K.)(T). Lemma 3.3 An operator of the form
Fd 0 C-2, T
where F,i is a diagonal matrix is (U + /C) -similar to T if and only if the diagonal entries { A] n } C J2. are distinct and the ith column of Ci is not in R(i:n,(TAj/),l i n.
( Fd 0 \ An arbitrarily small perturbation of C = will g(;t the ith column
\C,t T)
of C'2\ out of range (T ?A,;/). Then by the Lemma 3.3,we have
Corollary 3.4 If C is an operator of the form
c= F
T whore F,/ is a diagonal matrix with distinct diagonal entries in SJ , then C €
By BDF Theorem,the upper semi-continuity of the spectrum,and Corollary. 1, 3.4,we can obtain
Theorem 2.1 Let T be univalent operator as 2.1, then (U + K,)(T) = {A 6 L(//);A is essentially normal and satisfies (i)(ii)(ni)}. (i).rr(A) = JI;
('m).ind(A - A) = -1, VA H.
By Theorem 2.1 we can obtain
Theorem 2.2 T and 5" arc two nnivalent operators .T is approximately similar to S1, then T and S are approximately (ZY + /C) similar. Particularly T and 5 are two similar nnivalent operators, then T and S are approximately (U + /C)similar.
Similar to the proof of Lemma 3.1,using the properties of analytic function we can get
Theorem 2.3 Let T be univalent operator as 2.1,then there exists / which is
an nnivalent analytic function on unit disk D.s.t.li C )(D) C O , here il denotes the interior of J7.
From Theorem 2.3 we can conclude
Corollary 2.1 In Definition 2.1 if for all A 6 dil.X rrp(T),then II must be simply connected.
Corollary 2.2 In Definition 2.1 if il ~ il , then ii must be simply connected.
Proposition 3.1 Two essentially normal operator are similar, we can't conclude they are (U + K.) - similar.
So in Theorem 2.2 we can't use (14 + 1C)similar, but approximately (U + /C) similar instead. So the partition by (U + 1C)similar equivalence is finer than that of similar equivalence.
In [13].it defined a kind of operator called shift-like , and proved TheoremrSuppose
T is a shift-like operator,.? is the unilateral shift, then (U + K,)(T) = (U + K.)(S). Here we will give the definition of univalent-like operator.
5 6 (H) D (eN)(H).is called univalent-like operator if there exists simply connected domain il C a(S) n/t//(5), s.t.
(l)a(S) =Tl:
(2)dijikn-(S - A) = LVA 6 H:
(3)kor(S- A) = {()},VA eH;
(4)3A0
Question: If T is an univalent operator,5 is a univalent-like operator corre-
-
spondingly. then whether or not T is in (U + K,)(S)
定义 2．1 T∈L(H)∩(εN)(H)称为单叶的，如果存在连通的开集Ω(?)σ(T)∩ρ_s-F(T)，使得
(1)σ(T)=(?)；
(2)dim ker(T-λ)~*=1，(?)λ∈Ω；
(3)ker(T-λ)={0}，(?)λ∈Ω；
(4)(?)λ_0∈Ω，使得∩_(n=1)~∞ Ran(T-λ_0)~n={0}，且T|_(ran(T-λ_0))≌_(u+k)T，
根据定义2．1有
引理3．1设算子T是单叶的，由定义2．1知(?)_0∈Ω，使得T|_(Ran(T-λ_0))≌_((u+k))T，则对(?)λ∈Ω，都有T|_(Ran(T-λ))≌_((u+k))T，
再由文献[6]中的命题2．7以及单叶算子的定义有
引理3．2算子T是如定义2．1的，则对(?)λ∈Ω，都有λI′(?)T∈(?)，这里I′为一维空间上的单位算子。
由引理3．1及数学归纳法有
推论3．1设{λ_1，…，λ_n}(?)Ω，且λ-i≠λ_j，i≠j，则有
T|_((V_(i=1)~n{ker(T-λ_i)~*})~⊥)≌_(u+k)T，
考虑T在分解H=(V_(i=1)~n{ker(T-λ_i)~*})(?)(V_(i=1)~n{ker(T-λ_j)~*})~⊥上的表示，再利用有限维矩阵的性质可以得到
推论3．2设F_d是以{λ_1，…，λ_n}(?)Ω为对角线元的对角阵，则存在一个算子C，使得
T≌_(u+k)(?)
吉林大学硕士学位论文
我们得到
推论3.3算子T是如定义2.1中的单叶算子，F是有限维空间上
的算子，且a(F)〔.，则有
FOT任(U十尤)(T).
与
子
算
八习
.自口
、、、，...，Z
弓.理。.。设二是，*维又寸角阵，形女。。一f式了
、‘毛，
洲u+胡一相似当且仅当凡的对角线元林l，…，凡，}。贝，_互不相同，
伪，的第、列不在R。叹T一凡I)里，1三*兰，2.
Fti()
)
算子，做个小扰动，可使得姚:的第*列不
姚IT
/了矛!、、飞
一一
C
如
形
在几州T一入:)里，在根据引理3.;
推论3一形女口一(
凡()
姚IT
)
有
算子，这里Fti是有限维对角阵，
对角线元互不相同，且都在几里，则c钊u+门(T).
由BDF定理，谱的上半连续性，推论3.1，划等，我们可以得到
定理2.1设T是如定义2.1的单叶算子，则(u+门(T)={A。
￡(万):A是本性正规的，且满足(、)(**)(、、*)}，
(，).二(A)=‘2;
(乞乞).二。(A)=Jg之;
(乞z乞).乞，“l(A一入)=一1 .V入任百2.
由定理2.1立即可以得到
定理2.2T与s是近似相似的两个单叶算子，则有T与s是近似
(u+均一等价的.特别的，如果T与s是相似的，也有T与s是近
似(u+均一等价的.
类似引理3.1的证明，利用解析函数的性质可以得到定理2.3
定理2.3设T是如定义2.1的单叶算子，则必有一个单位圆盘D
内的单叶解析函数。，使得兑〔!l( D)c彭，这里彭表示的是豆的内
部.
吉林大学硕士学位论文
由定理2.3易得如下两个推论
推论2.1设T是如定义2.1的单叶算子，若有以任优，，灭祥二。(尹)，
则蛇一定是单连通的.
推论2.2设T是如定义2.1的单叶算子，若有几一彭，彭表示瓦
的内部，则几一定是单连通的.
利用推论2.2我们得到
命题3.1两个本性正规算子相似，不一定有他们是(u+门一等价
的.
所以定理2.2中，我们不能用(u+门一相似，而用近似(“+门一
相似.这也说明了(u一十们一等价分类是比相似分类更细的分类.
在文献[ls}中，定义一了类单边移位算子，并考虑了单边移位算子
与类单边移位算子的(u+均一轨道闭包，结论是他们的(l1+门一轨
道闭包相同.这里我们同样也给出类单叶算子的定义.
算子s。侧川川:N)(II)，称为类单叶的，如果存在单连通的开集
‘ZC。(s)自户、一。(S)，使得
(1)。(S)=百2;
(2)d该，;‘k(:r(S一入)*=1，V入任人2;
(3)ker(S一入)={O}，V入任公2;
(4)己入。任‘老，5 .t·S}，、u，‘(、一入())望;，十、S·
问题:如果T是单叶算子，￡是相应的类单叶算子，那么T是
不是在i百不画画里呢?
Let H be an infinitely dimentionai Hilbert space,and denote by (/f)the set of all linear operators acting on H.K,(H) denotes the ideal of compact operators on // and (sN)(H) denotes the set of essentially normal operators. We defined this kind of operator.
Definition 2.1 T 6(H) f](sN)(H) is called nnivalent operator, if there exists connected open set,
(2)dimker(T - A) = 1. VA 2; (3)ker(T- A) = {()},VAe JJ;
(4)3A0 ,s.t.fXL, Rnn(T - A0)" = {0},and r|/?n,l(r_A()) =. T. From Definition 2.1, we have
Lemma 3.1 Let T be univalent operator as 2.1, then1 exists a AO e JLs.t. m(r-A) +A: T,tlien for all A we have T|, T. From proposition 2.7 of [6] and the definition of univalent operator Lemma 3.2 T is given by Definition 2.1, then for all A T e
(/C)(T) hold, here / is the identity operator acting on one-dimensional space.
From Lemma 3.1 and induction,we obtain
Corollary 3.1 Let {A,, ,Xn} C Q,and A, Ay j.When T is restricted to the invariant snbspaoe formed by (,s'pan{ker(T - A,7)}f=, J.then the resulting operator is (U + /C)-similar to T.
Considering the representation of T on H = (v;.'=1 {ker(T-A,)})(B(v; {ker(T-A,;)})x.using property of finite dimensional matrix, we have
Corollary 3.2 Let F,i be a diagonal matrix {A). , A, C il. then there exists
operator C,s.t.
I Fd 0
T I
C T We have Corollary 3.3 If T is a univalent operator as 2.1.F is an operator on a finite
dimensional space, whose spectrum subsets SJ,thon F T(U + K.)(T). Lemma 3.3 An operator of the form
Fd 0 C-2, T
where F,i is a diagonal matrix is (U + /C) -similar to T if and only if the diagonal entries { A] n } C J2. are distinct and the ith column of Ci is not in R(i:n,(TAj/),l i n.
( Fd 0 \ An arbitrarily small perturbation of C = will g(;t the ith column
\C,t T)
of C'2\ out of range (T ?A,;/). Then by the Lemma 3.3,we have
Corollary 3.4 If C is an operator of the form
c= F
T whore F,/ is a diagonal matrix with distinct diagonal entries in SJ , then C €
By BDF Theorem,the upper semi-continuity of the spectrum,and Corollary. 1, 3.4,we can obtain
Theorem 2.1 Let T be univalent operator as 2.1, then (U + K,)(T) = {A 6 L(//);A is essentially normal and satisfies (i)(ii)(ni)}. (i).rr(A) = JI;
('m).ind(A - A) = -1, VA H.
By Theorem 2.1 we can obtain
Theorem 2.2 T and 5" arc two nnivalent operators .T is approximately similar to S1, then T and S are approximately (ZY + /C) similar. Particularly T and 5 are two similar nnivalent operators, then T and S are approximately (U + /C)similar.
Similar to the proof of Lemma 3.1,using the properties of analytic function we can get
Theorem 2.3 Let T be univalent operator as 2.1,then there exists / which is
an nnivalent analytic function on unit disk D.s.t.li C )(D) C O , here il denotes the interior of J7.
From Theorem 2.3 we can conclude
Corollary 2.1 In Definition 2.1 if for all A 6 dil.X rrp(T),then II must be simply connected.
Corollary 2.2 In Definition 2.1 if il ~ il , then ii must be simply connected.
Proposition 3.1 Two essentially normal operator are similar, we can't conclude they are (U + K.) - similar.
So in Theorem 2.2 we can't use (14 + 1C)similar, but approximately (U + /C) similar instead. So the partition by (U + 1C)similar equivalence is finer than that of similar equivalence.
In [13].it defined a kind of operator called shift-like , and proved TheoremrSuppose
T is a shift-like operator,.? is the unilateral shift, then (U + K,)(T) = (U + K.)(S). Here we will give the definition of univalent-like operator.
5 6 (H) D (eN)(H).is called univalent-like operator if there exists simply connected domain il C a(S) n/t//(5), s.t.
(l)a(S) =Tl:
(2)dijikn-(S - A) = LVA 6 H:
(3)kor(S- A) = {()},VA eH;
(4)3A0
Question: If T is an univalent operator,5 is a univalent-like operator corre-
-
spondingly. then whether or not T is in (U + K,)(S)
引文
[1] D. Voiculescu,A non-commutative Weyl-von Neumann, theorem, Rev. Romnaine Math. Pures Appl.21 (1976),97-113.MR54:3427
[2] C. Apostol, L. Fialkow, D. A. Herrero, D. Voiculescu, Approximation of Hilbert space operators, Ⅱ. Research Notes in Math.vol.102.Pitman Books, Ltd., London-Boston-Melbourne,1984.MR85m:47002
[3] F.Gilfeather,Stron9 irreducibility of operator,Indaina Univ Math. 22(1972)393-397.
[4] C.L.Jiang Strongly irreducible operators and Cowen-Douglas operators, Northeastern Math.1(1991),1-3.
[5] 蒋春澜，算子相似、约化和逼近博士论文(1992)吉林大学．
[6] D.A.Herrero and C.L.limits of strongly irreducible operators and the Riesz decomposition zheorem, Mich.Math.137 (1990) 283-297.
[7] C.K.Fong and C.J.Jiang.Approximation by Jordan type operators, Houston J.Math. 19(1980), 169-184.
[8] P.S.Guinand, L. Marcoux, Between the unitary and similarity orbits or normal operators, Pacific J. Math.159(1993),299-334.
[9] You Qing Ji and Jue Xian Li, The Quasiapproximate (U+K)-invariants of Essentially Normal Operator, Integr.equ.oper.theory 99.
[10] I.D.Berg and K.R. Davidson, Almost commuting matrices and quantitative version of the Brown-Douglas-Fillmore theorem, Acta Math 166(1991)121-161.
[11] R.G.Douglas Banach Algebra Techniques in Operator Theory, Aeademic press. New York,1972.
[12] 岳华，任何一个算子都可以分解成两个强不可约算子的直和，数学的实践与认识，(2003)，第六期．
[13] L. Marcoux The Closure of the (U+K)-Orbit of Shift-like OperatorsIndiana University Mathematies Journal, Vol.41,No.4(1992),1211-1223.
[2] C. Apostol, L. Fialkow, D. A. Herrero, D. Voiculescu, Approximation of Hilbert space operators, Ⅱ. Research Notes in Math.vol.102.Pitman Books, Ltd., London-Boston-Melbourne,1984.MR85m:47002
[3] F.Gilfeather,Stron9 irreducibility of operator,Indaina Univ Math. 22(1972)393-397.
[4] C.L.Jiang Strongly irreducible operators and Cowen-Douglas operators, Northeastern Math.1(1991),1-3.
[5] 蒋春澜，算子相似、约化和逼近博士论文(1992)吉林大学．
[6] D.A.Herrero and C.L.limits of strongly irreducible operators and the Riesz decomposition zheorem, Mich.Math.137 (1990) 283-297.
[7] C.K.Fong and C.J.Jiang.Approximation by Jordan type operators, Houston J.Math. 19(1980), 169-184.
[8] P.S.Guinand, L. Marcoux, Between the unitary and similarity orbits or normal operators, Pacific J. Math.159(1993),299-334.
[9] You Qing Ji and Jue Xian Li, The Quasiapproximate (U+K)-invariants of Essentially Normal Operator, Integr.equ.oper.theory 99.
[10] I.D.Berg and K.R. Davidson, Almost commuting matrices and quantitative version of the Brown-Douglas-Fillmore theorem, Acta Math 166(1991)121-161.
[11] R.G.Douglas Banach Algebra Techniques in Operator Theory, Aeademic press. New York,1972.
[12] 岳华，任何一个算子都可以分解成两个强不可约算子的直和，数学的实践与认识，(2003)，第六期．
[13] L. Marcoux The Closure of the (U+K)-Orbit of Shift-like OperatorsIndiana University Mathematies Journal, Vol.41,No.4(1992),1211-1223.