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Geometric similarity invariants of geometric operators
Kui JiJoint with Chunlan Jiang
Hebei Normal University
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 1 / 36
outline
1 Cowen-Douglas Operators
2 New class of operators
3 Results
4 Reference
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 2 / 36
outline
1 Cowen-Douglas Operators
2 New class of operators
3 Results
4 Reference
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 2 / 36
outline
1 Cowen-Douglas Operators
2 New class of operators
3 Results
4 Reference
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 2 / 36
outline
1 Cowen-Douglas Operators
2 New class of operators
3 Results
4 Reference
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 2 / 36
Cowen-Douglas Operators
Let H be a complex separable Hilbert space and let L(H) be the algebra ofbounded linear operators on H.For an open connected subset Ω of the complex plane C, and n ∈ N, Cowenand Douglas introduced the class of operators Bn(Ω) in their Acta paper [1].
Definition (Bn(Ω))
An operator T ∈ Bn(Ω) if for each w ∈ Ω, is an eigenvalue of the operatorT of constant multiplicity n, these eigenvectors span the Hilbert space Hand the operator T − w , w ∈ Ω, is surjective.
It was showed that the map w → ker(T − w) is holomorphic andπ : ET → Ω, where
ET (w) = ker(T − w) : w ∈ Ω, π( ker(T − w) ) = w
defines a Hermitian holomorphic vector bundle on Ω.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 3 / 36
Cowen-Douglas Operators
First of all, we need introduce some complex geometry notations:Let ξ(Ω) be the algebra consist of the C∞ functions and ξp(Ω) denote thep-differential form of C∞ functions. Thus we have
ξ0(Ω) = ξ(Ω), ξ1(Ω) = fdz + gdz : f , g ∈ ξ(Ω),
ξ2(Ω) = fdzdz , f ∈ ξ(Ω)
For any vector bundle E which has C∞ differential structure, let ξp(Ω,E )denotes p-differential forms with the coefficients in E . Then each elementin ξ0(Ω,E ) is one of sections of E .
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 4 / 36
Cowen-Douglas Operators
Definition (Connection and Curvature)
The connection D can be regarded as a differential operator which mapsξ0(Ω,E ) to ξ1(Ω,E ). Let σ ∈ E (w), and h = ((〈σj , σi 〉))n×n. Then thecanonical connection D which keeping the metric and satisfying thefollowing equality:
D(n∑
i=1
fiσi ) =n∑
i=1
dfi ⊗ σi +n∑
i=1
n∑j=1
fiθj ,iσj
where θ = h−1∂h. And
D2 = dθ + θ ∧ θ = ∂(h−1∂h)
then −∂(h−1∂h) is called as the curvature of E denoted by KE
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 5 / 36
Cowen-Douglas Operators
Definition (Second fundamental form)
Let T ∈ B2(Ω), and σ1(w), σ2(w) ∈ Ker(T − w). Applying the Schmidtorthogonal progress to σ1, σ2, then we have e1, e2. Suppose
De1 = D1,0e1 + D0,1e2 = θ11e1 + θ21e2
and De2 = θ12e1 + θ22e2, then θ12 = 〈De2, e1〉 is called as the secondfundamental form of ET
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 6 / 36
Questions
Cowen-Douglas’ Unitary Classification Theorem
For any T ∈ Bn(Ω), when n = 1, the curvature is the completely unitaryinvariant. When n > 1, then the curvature and it’s covariant partialderivatives are the completely unitary invariants
Question 1(Similarity of Cowen-Douglas Operators)
For the similarity of Cowen-Douglas operators A,B ∈ B1(D), whetherA ∼s B if and only if
limw→∂D
KA(w)
KB(w)= 1.
Or can we use some geometric invariants involving curvature to describe thesimilarity of Cowen-Douglas operators?
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 7 / 36
Questions
Cowen-Douglas’ Unitary Classification Theorem
For any T ∈ Bn(Ω), when n = 1, the curvature is the completely unitaryinvariant. When n > 1, then the curvature and it’s covariant partialderivatives are the completely unitary invariants
Question 1(Similarity of Cowen-Douglas Operators)
For the similarity of Cowen-Douglas operators A,B ∈ B1(D), whetherA ∼s B if and only if
limw→∂D
KA(w)
KB(w)= 1.
Or can we use some geometric invariants involving curvature to describe thesimilarity of Cowen-Douglas operators?
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 7 / 36
D.N. Clark and G. Misra’s result
D. N. Clark and G. Misra gave a counter example of this conjecture. Let S0
be the backward unilateral shift operator and T be a weighted (backward)
shift operator with sequence αn =(
n∑j=1
1/j)1/2
(n+1∑j=1
1/j)1/2
and
KTKS0
= 1 + [1− ln(1− |w |2)]−1
−|w |2[1− ln(1− |w |2)]−2
Then KTKS0→ 1, when |w | goes to 1. However, T and S0 are not similar.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 8 / 36
D.N. Clark and G. Misra’s result
Theorem [D.N.Clark and G.Mirsa]Michigan Math. J. 1983
Let S denote a backward weighted shift operator with weight sequenceαn = [ (n+1)
(n+2) ]α/2 and T is a backward weighted shift operator with ||T || ≤ 1.Set αw to be the ratio of the normalized sections of ES and ET . Then
(i) T is similar to S if and only if αw is bounded and bounded from 0.
(ii) T is similar to S with T = XSX−1, X = U + K where U is unitaryand K is compact if and only if αw tends to a non-zero limit when|w | → 1.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 9 / 36
D.N. Clark and G. Misra’s result
Theorem [D.N.Clark and G.Mirsa]Michigan Math. J. 1983
Let S denote a backward weighted shift operator with weight sequenceαn = [ (n+1)
(n+2) ]α/2 and T is a backward weighted shift operator with ||T || ≤ 1.Set αw to be the ratio of the normalized sections of ES and ET . Then
(i) T is similar to S if and only if αw is bounded and bounded from 0.
(ii) T is similar to S with T = XSX−1, X = U + K where U is unitaryand K is compact if and only if αw tends to a non-zero limit when|w | → 1.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 9 / 36
D.N. Clark and G. Misra’s result
Theorem [D.N.Clark and G.Mirsa]Michigan Math. J. 1983
Let S denote a backward weighted shift operator with weight sequenceαn = [ (n+1)
(n+2) ]α/2 and T is a backward weighted shift operator with ||T || ≤ 1.Set αw to be the ratio of the normalized sections of ES and ET . Then
(i) T is similar to S if and only if αw is bounded and bounded from 0.
(ii) T is similar to S with T = XSX−1, X = U + K where U is unitaryand K is compact if and only if αw tends to a non-zero limit when|w | → 1.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 9 / 36
Kehe Zhu’s result
In [3], K. Zhu introduced the spanning holomorphic cross-section forCowen-Douglas operators. Let T ∈ Bn(Ω). A holomorphic section of vectorbundle ET is a holomorphic function γ : Ω→ H such that for each w ∈ Ω,the vector γ(w) belongs to the fibre of ET over w . We say γ is a spanningholomorphic section for ET if Span γ(w) : w ∈ Ω = H.
Theorem [K. Zhu] Illinois J. Math. 2000
For any Cowen-Douglas operator T ∈ Bn(Ω), ET has a spanningholomorphic cross-section. Suppose T and T belongs to Bn(Ω), then Tand T are unitarily equivalent ( or similarity equivalent) if and only if thereexist spanning holomorphic cross-sections γT and γ
Tfor ET and ES ,
respectively, such that γT ∼u γT (or γT ∼s γT ).
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 10 / 36
R.G.Douglas, S.Treil and H. Kwon’s result
Theorem [R.G. Douglas, H. Kwon and S.Treil] J. Lond. Math. Soc.2013
For T ∈ Bm(D) that is an n-hypercontraction, let P : D→ L(H) denotethe function whose values are orthogonal projections onto ker(T − w).
Then T is similar tom⊕
S∗n if and only if there exists a boundedsubharmonic function ψ defined on D such that
||∂P(w)||22 −mn
(1− |w |2)2= ∆ψ(w),
Remark[Y. Hou, K. Ji and H. Kwon] Studia Math. 2017
The Hilbert-Schmidt norm ||∂P(w)||22 is pointed out to be −traceKT .
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 11 / 36
R.G.Douglas, S.Treil and H. Kwon’s result
Theorem [R.G. Douglas, H. Kwon and S.Treil] J. Lond. Math. Soc.2013
For T ∈ Bm(D) that is an n-hypercontraction, let P : D→ L(H) denotethe function whose values are orthogonal projections onto ker(T − w).
Then T is similar tom⊕
S∗n if and only if there exists a boundedsubharmonic function ψ defined on D such that
||∂P(w)||22 −mn
(1− |w |2)2= ∆ψ(w),
Remark[Y. Hou, K. Ji and H. Kwon] Studia Math. 2017
The Hilbert-Schmidt norm ||∂P(w)||22 is pointed out to be −traceKT .
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 11 / 36
Homogeneous operators
In 1984, G. Misra defined a class of homogeneous Cowen-Douglas operatorsas the following: an operator T is said to be homogeneous if φ(T ) isunitarily equivalent to T for each Mobius transformation φ, and he provedthe following theorem:
Theorem[G. Misra]Proc. Amer. Math. Soc.1984
Let T ∈ B1(D) is a homogenous operator, then T is unitarily equivalent tothe adjoint of multiplication operator Mz on the analytic functional spaceHK, where K(z ,w) = 1
(1−zw)λ, for some λ > −1.
An operator T is said to be weakly homogeneous if φ(T ) is similarityequivalent to T for each Mobius transformation φ. A natural question iswhat is the set of all of the weakly homogenous operator at least forCowen-Douglas class?
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 12 / 36
New class of Cowen-Douglas operators
Inspire of the structure of homogenous Cowen-Douglas operators, weintroduced the following new class of operators:
FBn(Ω)
We let FBn(Ω) be the set of all bounded linear operators T defined onsome complex separable Hilbert space H = H0 ⊕ · · · ⊕ Hn−1, which are ofthe form
T =
T0 S0,1 S0,2 · · · S0,n−1
0 T1 S1,2 · · · S1,n−1...
. . .. . .
. . ....
0 · · · 0 Tn−2 Sn−2,n−1
0 · · · · · · 0 Tn−1
,
where the operator Ti ∈ B(Hi ) is assumed to be in B1(Ω) andTiSi ,i+1 = Si ,i+1Ti+1, 0 ≤ i ≤ n − 2.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 13 / 36
Second fundamental form in the case of FB2(Ω)
The 2× 2 block(
Ti Sii+1
0 Ti+1
)in the decomposition of the operator T in
FB2(D) because of the intertwining property . Hence the correspondingsecond fundamental form θi ,i+1(T ) is given by the formula
θi ,i+1(T )(z) =KTi
(z) dz(‖Si,i+1(ti+1(z))‖2
‖ti+1(z)‖2 −KTi(z))1/2
. (2.1)
Remark
For any T , T ∈ FBn(Ω), when KTi= K
Ti, then
θi ,i+1(T )(z) = θi ,i+1(T )(z)⇔‖Si ,i+1(ti+1(z))‖‖ti+1(z)‖
=‖Si ,i+1(ti+1(z))‖‖ti+1(z)‖
So we also use‖Si,i+1(ti+1(z))‖‖ti+1(z)‖ as the second fundamental form θi ,i+1(T ).
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 14 / 36
Unitarily equivalence of operators in FBn(Ω)
For the unitarily classification problem of Cowen-Douglas operators, we havethe following result:
Theorem 1[Jiang,Ji,Dinesh and Misra]JFA, 2017
Let T , T ∈ FBn(Ω).
T ∼u T ⇔
KTi
= KTi
θi ,i+1(T ) = θi ,i+1(T )〈Si,j (tj ),ti 〉‖ti‖2 =
〈Si,j (tj ),ti 〉‖ti‖2
Note that numbers of unitarily invariants of common case are n2. Buttogether with the curvature and the second fundamental form, we find a setof n(n − 1)/2 + 1 invariants, which are less and easy to compute.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 15 / 36
Similarity of operators in FBn(Ω)
Definition
Let T ∈ FBn(Ω). The operator T is called as quasi-homogeneous operator ,i.e. T ∈ QBn(Ω), if Ti is homogenous operator and
Si ,j(tj) ∈∨t(k)
i , k ≤ j − i − 1.
For the similarity classification of Cowen-Douglas operators, we have thefollowing result:
Theorem 2[Jiang, Ji and Misra] JFA,2017
Let T , S ∈ QBn(Ω), then we haveKTi,i
= KTi,i
θi ,i+1(T ) = θi ,i+1(T )=⇒ T ∼s T if and only if T = T
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 16 / 36
Similarity of operators in FBn(Ω)
Definition
Let T ∈ FBn(Ω). The operator T is called as quasi-homogeneous operator ,i.e. T ∈ QBn(Ω), if Ti is homogenous operator and
Si ,j(tj) ∈∨t(k)
i , k ≤ j − i − 1.
For the similarity classification of Cowen-Douglas operators, we have thefollowing result:
Theorem 2[Jiang, Ji and Misra] JFA,2017
Let T , S ∈ QBn(Ω), then we haveKTi,i
= KTi,i
θi ,i+1(T ) = θi ,i+1(T )=⇒ T ∼s T if and only if T = T
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 16 / 36
New progresses
We would like to introduce the following a class of geometric operatorsdenoted by CFBn(Ω).
Definition (CFBn(Ω))
A geometric operator T with index n is said to be in CFBn(Ω), if Tsatisfies the following properties:
(1) T can be written as an n × n upper-triangular matrix form ((Ti ,j))n×nunder a topological direct decomposition of H;
(2) diagT := T1,1 u T2,2 u · · ·u Tn,n ∈ T′, where T′ denotes thecommutant of T ;
(3) each entry Ti ,j = φi ,jTi ,i+1Ti+1,i+2 · · ·Tj−1,j , where φi ,j ∈ Ti ,i′;(4) T is a strongly irreducible operator, i.e. there are no nontrivial
idempotents in T′.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 17 / 36
New progresses
We would like to introduce the following a class of geometric operatorsdenoted by CFBn(Ω).
Definition (CFBn(Ω))
A geometric operator T with index n is said to be in CFBn(Ω), if Tsatisfies the following properties:
(1) T can be written as an n × n upper-triangular matrix form ((Ti ,j))n×nunder a topological direct decomposition of H;
(2) diagT := T1,1 u T2,2 u · · ·u Tn,n ∈ T′, where T′ denotes thecommutant of T ;
(3) each entry Ti ,j = φi ,jTi ,i+1Ti+1,i+2 · · ·Tj−1,j , where φi ,j ∈ Ti ,i′;(4) T is a strongly irreducible operator, i.e. there are no nontrivial
idempotents in T′.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 17 / 36
New progresses
We would like to introduce the following a class of geometric operatorsdenoted by CFBn(Ω).
Definition (CFBn(Ω))
A geometric operator T with index n is said to be in CFBn(Ω), if Tsatisfies the following properties:
(1) T can be written as an n × n upper-triangular matrix form ((Ti ,j))n×nunder a topological direct decomposition of H;
(2) diagT := T1,1 u T2,2 u · · ·u Tn,n ∈ T′, where T′ denotes thecommutant of T ;
(3) each entry Ti ,j = φi ,jTi ,i+1Ti+1,i+2 · · ·Tj−1,j , where φi ,j ∈ Ti ,i′;(4) T is a strongly irreducible operator, i.e. there are no nontrivial
idempotents in T′.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 17 / 36
New progresses
We would like to introduce the following a class of geometric operatorsdenoted by CFBn(Ω).
Definition (CFBn(Ω))
A geometric operator T with index n is said to be in CFBn(Ω), if Tsatisfies the following properties:
(1) T can be written as an n × n upper-triangular matrix form ((Ti ,j))n×nunder a topological direct decomposition of H;
(2) diagT := T1,1 u T2,2 u · · ·u Tn,n ∈ T′, where T′ denotes thecommutant of T ;
(3) each entry Ti ,j = φi ,jTi ,i+1Ti+1,i+2 · · ·Tj−1,j , where φi ,j ∈ Ti ,i′;
(4) T is a strongly irreducible operator, i.e. there are no nontrivialidempotents in T′.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 17 / 36
New progresses
We would like to introduce the following a class of geometric operatorsdenoted by CFBn(Ω).
Definition (CFBn(Ω))
A geometric operator T with index n is said to be in CFBn(Ω), if Tsatisfies the following properties:
(1) T can be written as an n × n upper-triangular matrix form ((Ti ,j))n×nunder a topological direct decomposition of H;
(2) diagT := T1,1 u T2,2 u · · ·u Tn,n ∈ T′, where T′ denotes thecommutant of T ;
(3) each entry Ti ,j = φi ,jTi ,i+1Ti+1,i+2 · · ·Tj−1,j , where φi ,j ∈ Ti ,i′;(4) T is a strongly irreducible operator, i.e. there are no nontrivial
idempotents in T′.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 17 / 36
New progresses
Definition (Similarity invariant set)
Let F = Aα ∈ B(H), α ∈ Λ. We call F is a similarity invariant set, if forany invertible operator X ∈ B(H),
XFX−1 = XAαX−1 : Aα ∈ F = F .
Proposition
CFBn(Ω) is a similarity invariant set
Remark
The set of homogenous operators in Cowen-Douglas class is not a similarityinvariant set.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 18 / 36
New progresses
Definition (Similarity invariant set)
Let F = Aα ∈ B(H), α ∈ Λ. We call F is a similarity invariant set, if forany invertible operator X ∈ B(H),
XFX−1 = XAαX−1 : Aα ∈ F = F .
Proposition
CFBn(Ω) is a similarity invariant set
Remark
The set of homogenous operators in Cowen-Douglas class is not a similarityinvariant set.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 18 / 36
New progresses
Definition (Similarity invariant set)
Let F = Aα ∈ B(H), α ∈ Λ. We call F is a similarity invariant set, if forany invertible operator X ∈ B(H),
XFX−1 = XAαX−1 : Aα ∈ F = F .
Proposition
CFBn(Ω) is a similarity invariant set
Remark
The set of homogenous operators in Cowen-Douglas class is not a similarityinvariant set.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 18 / 36
New progresses
Definition (Similarity invariant set)
Let F = Aα ∈ B(H), α ∈ Λ. We call F is a similarity invariant set, if forany invertible operator X ∈ B(H),
XFX−1 = XAαX−1 : Aα ∈ F = F .
Proposition
CFBn(Ω) is a similarity invariant set
Remark
The set of homogenous operators in Cowen-Douglas class is not a similarityinvariant set.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 18 / 36
New progresses
Similarity orbit Theorem(Special case, Apostol, Fialkow, Herrero, andVoiculescu)
Let T and S ∈ Bn(Ω), and spectral pictures of T and S be the same. Thenthere exist two sequences of invertible operators Xn∞n=1 and Yn∞n=1
such thatlimn→∞
XnAX−1n = B, lim
n→∞YnBY
−1n = A.
Notice that CFBn(Ω) is a similarity invariant set, by using the similarityorbit theorem, we can prove that
Theorem
CFBn(Ω) is norm dense in Bn(Ω).
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 19 / 36
New progresses
Similarity orbit Theorem(Special case, Apostol, Fialkow, Herrero, andVoiculescu)
Let T and S ∈ Bn(Ω), and spectral pictures of T and S be the same. Thenthere exist two sequences of invertible operators Xn∞n=1 and Yn∞n=1
such thatlimn→∞
XnAX−1n = B, lim
n→∞YnBY
−1n = A.
Notice that CFBn(Ω) is a similarity invariant set, by using the similarityorbit theorem, we can prove that
Theorem
CFBn(Ω) is norm dense in Bn(Ω).
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 19 / 36
New progresses
Definition
Let T1,T2 ∈ L(H). Define a Rosenblum operator σT1,T2 : L(H)→ L(H)as
σT1,T2(X ) = T1X − XT2, ∀X ∈ L(H),
and a Rosenblum operator σT1 : L(H)→ L(H) as
σT1(X ) = T1X − XT1, ∀X ∈ L(H).
Definition (Property H)
Let T ∈ CFBn(Ω). We call T satisfies the Property (H) if and only if thefollowing statements hold: If Y ∈ B(Hj ,Hi ) satisfies
(i) Ti ,iY = YTi+1,i+1,
(ii) Y = Ti ,iZ − ZTi+1,i+1, for some Z , i < j = 1, · · · , n.
Then Y = 0. That is equivalent to kerσTi,i ,Ti+1,i+1∩ ranσTi,i ,Ti+1,i+1
= 0.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 20 / 36
New progresses
Definition
Let T1,T2 ∈ L(H). Define a Rosenblum operator σT1,T2 : L(H)→ L(H)as
σT1,T2(X ) = T1X − XT2, ∀X ∈ L(H),
and a Rosenblum operator σT1 : L(H)→ L(H) as
σT1(X ) = T1X − XT1, ∀X ∈ L(H).
Definition (Property H)
Let T ∈ CFBn(Ω). We call T satisfies the Property (H) if and only if thefollowing statements hold: If Y ∈ B(Hj ,Hi ) satisfies
(i) Ti ,iY = YTi+1,i+1,
(ii) Y = Ti ,iZ − ZTi+1,i+1, for some Z , i < j = 1, · · · , n.
Then Y = 0. That is equivalent to kerσTi,i ,Ti+1,i+1∩ ranσTi,i ,Ti+1,i+1
= 0.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 20 / 36
New progresses
Definition
Let T1,T2 ∈ L(H). Define a Rosenblum operator σT1,T2 : L(H)→ L(H)as
σT1,T2(X ) = T1X − XT2, ∀X ∈ L(H),
and a Rosenblum operator σT1 : L(H)→ L(H) as
σT1(X ) = T1X − XT1, ∀X ∈ L(H).
Definition (Property H)
Let T ∈ CFBn(Ω). We call T satisfies the Property (H) if and only if thefollowing statements hold: If Y ∈ B(Hj ,Hi ) satisfies
(i) Ti ,iY = YTi+1,i+1,
(ii) Y = Ti ,iZ − ZTi+1,i+1, for some Z , i < j = 1, · · · , n.
Then Y = 0. That is equivalent to kerσTi,i ,Ti+1,i+1∩ ranσTi,i ,Ti+1,i+1
= 0.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 20 / 36
New progresses
Definition
Let T1,T2 ∈ L(H). Define a Rosenblum operator σT1,T2 : L(H)→ L(H)as
σT1,T2(X ) = T1X − XT2, ∀X ∈ L(H),
and a Rosenblum operator σT1 : L(H)→ L(H) as
σT1(X ) = T1X − XT1, ∀X ∈ L(H).
Definition (Property H)
Let T ∈ CFBn(Ω). We call T satisfies the Property (H) if and only if thefollowing statements hold: If Y ∈ B(Hj ,Hi ) satisfies
(i) Ti ,iY = YTi+1,i+1,
(ii) Y = Ti ,iZ − ZTi+1,i+1, for some Z , i < j = 1, · · · , n.Then Y = 0. That is equivalent to kerσTi,i ,Ti+1,i+1
∩ ranσTi,i ,Ti+1,i+1= 0.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 20 / 36
New progresses
Proposition
Let T1,T2 ∈ L(H) and S2 be the right inverse of T2. If limn→∞
‖T n1 ‖·‖Sn
2 ‖n = 0,
then the Property (H) holds i.e. If there exists X ∈ L(H) such thatT1X = XT2 and X = T1Y − YT2 for some Y , then X=0.
Example
Let A,B ∈ B1(D) be backward shift operators with weighted sequences
ai∞i=1 and bi∞i=1. If limn→∞
n
n∏k=1
bk
n∏k=1
ak
=∞, then the Property (H) holds.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 21 / 36
New progresses
Proposition
Let T1,T2 ∈ L(H) and S2 be the right inverse of T2. If limn→∞
‖T n1 ‖·‖Sn
2 ‖n = 0,
then the Property (H) holds i.e. If there exists X ∈ L(H) such thatT1X = XT2 and X = T1Y − YT2 for some Y , then X=0.
Example
Let A,B ∈ B1(D) be backward shift operators with weighted sequences
ai∞i=1 and bi∞i=1. If limn→∞
n
n∏k=1
bk
n∏k=1
ak
=∞, then the Property (H) holds.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 21 / 36
New progresses
Definition
We call T ∼U+K S , if there exists a unitary operator U and a compactoperator K such that U + K is invertible and (U + K )T = S(U + K ).
Lemma
Let T , S ∈ B1(D), where S ∼u (M∗z ,HKS,KS), then we have
T ∼U+K S ⇔ KS − KT = ∆lnφ,
where φ is a bounded function with
φ(w) = 1 +
m∑i=1
2Refi (w)gi (w) +m∑i=1|gi (w)|2
KS(w ,w),
where m is the rank of K and fimi=1, gimi=1 ∈ HKSare orthogonal sets,
||fi || = 1, ||gi || → 0. When KS ≥ KT , then lnφ is subharmonic.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 22 / 36
New progresses
Definition
We call T ∼U+K S , if there exists a unitary operator U and a compactoperator K such that U + K is invertible and (U + K )T = S(U + K ).
Lemma
Let T , S ∈ B1(D), where S ∼u (M∗z ,HKS,KS), then we have
T ∼U+K S ⇔ KS − KT = ∆lnφ,
where φ is a bounded function with
φ(w) = 1 +
m∑i=1
2Refi (w)gi (w) +m∑i=1|gi (w)|2
KS(w ,w),
where m is the rank of K and fimi=1, gimi=1 ∈ HKSare orthogonal sets,
||fi || = 1, ||gi || → 0. When KS ≥ KT , then lnφ is subharmonic.Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 22 / 36
New progresses
Proposition
Let A,B ∈ B1(D) be backward weighted shift operators with weightedsequences ak∞k=1 and bk∞k=1 respectively. Then the followingstatements are equivalent:
(i) A ∼s B implies A ∼U+K B,
(ii) limn→∞
n∏k=1
ak
n∏k=1
bk
exists and is not equal to zero.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 23 / 36
New progresses
Proposition
Let A,B ∈ B1(D) be backward weighted shift operators with weightedsequences ak∞k=1 and bk∞k=1 respectively. Then the followingstatements are equivalent:
(i) A ∼s B implies A ∼U+K B,
(ii) limn→∞
n∏k=1
ak
n∏k=1
bk
exists and is not equal to zero.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 23 / 36
New progresses
Proposition
Let A,B ∈ B1(D) be backward weighted shift operators with weightedsequences ak∞k=1 and bk∞k=1 respectively. Then the followingstatements are equivalent:
(i) A ∼s B implies A ∼U+K B,
(ii) limn→∞
n∏k=1
ak
n∏k=1
bk
exists and is not equal to zero.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 23 / 36
Similarity involving U + K
Main Theorem 1 [Jiang and Ji]
Let T , T ∈ CFBn(Ω). Suppose the following statements hold
(1) T and T satisfy the Property (H);
(2) Ti ,i ∼s Ti ,i implies Ti ,i ∼U+K Ti ,i
then we have
T ∼s T ⇔
KTi− K
Ti= ∆lnφi
φiφi+1
θi ,i+1(T ) = θi ,i+1(T )
where φi are the bounded subharmonic functions in the Lemma above.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 24 / 36
Similarity involving U + K
Main Theorem 1 [Jiang and Ji]
Let T , T ∈ CFBn(Ω). Suppose the following statements hold
(1) T and T satisfy the Property (H);
(2) Ti ,i ∼s Ti ,i implies Ti ,i ∼U+K Ti ,i
then we have
T ∼s T ⇔
KTi− K
Ti= ∆lnφi
φiφi+1
θi ,i+1(T ) = θi ,i+1(T )
where φi are the bounded subharmonic functions in the Lemma above.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 24 / 36
Similarity involving U + K
Main Theorem 1 [Jiang and Ji]
Let T , T ∈ CFBn(Ω). Suppose the following statements hold
(1) T and T satisfy the Property (H);
(2) Ti ,i ∼s Ti ,i implies Ti ,i ∼U+K Ti ,i
then we have
T ∼s T ⇔
KTi− K
Ti= ∆lnφi
φiφi+1
θi ,i+1(T ) = θi ,i+1(T )
where φi are the bounded subharmonic functions in the Lemma above.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 24 / 36
Strongly Property (H)
Definition (Strongly Property (H))
Let T ∈ CFBn(Ω). We call T satisfies the strongly property (H) if andonly if the following statements hold: If Y ∈ B(Hj ,Hi ) satisfies
(i) Ti ,iY = YTj ,j ,
(ii) Y = Ti ,iZ − ZTj ,j , for some Z , i < j = 1, · · · , n.
Then Y = 0. That is equivalent to kerσTi,i ,Tj,j∩ ranσTi,i ,Tj,j
= 0.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 25 / 36
Strongly Property (H)
Definition (Strongly Property (H))
Let T ∈ CFBn(Ω). We call T satisfies the strongly property (H) if andonly if the following statements hold: If Y ∈ B(Hj ,Hi ) satisfies
(i) Ti ,iY = YTj ,j ,
(ii) Y = Ti ,iZ − ZTj ,j , for some Z , i < j = 1, · · · , n.
Then Y = 0. That is equivalent to kerσTi,i ,Tj,j∩ ranσTi,i ,Tj,j
= 0.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 25 / 36
Strongly Property (H)
Definition (Strongly Property (H))
Let T ∈ CFBn(Ω). We call T satisfies the strongly property (H) if andonly if the following statements hold: If Y ∈ B(Hj ,Hi ) satisfies
(i) Ti ,iY = YTj ,j ,
(ii) Y = Ti ,iZ − ZTj ,j , for some Z , i < j = 1, · · · , n.Then Y = 0. That is equivalent to kerσTi,i ,Tj,j
∩ ranσTi,i ,Tj,j= 0.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 25 / 36
Main Theorem 2
Main Theorem 2 [Jiang and Ji]
Let T = ((Ti ,j))n×n and T = ((Ti ,j))n×n be any two operators inCFBn(Ω), where Ti ,j = Ti ,j = 0, i > j . Suppose that T satisfies thestrongly property (H). Then we have
T ∼s T ⇔
XiTi ,i = Ti ,iXi ,
XiTi ,j = Ti ,jXj , i = 1, 2, · · · , n
where Xi ∈ L(Hi , Hi ), i = 1, 2, · · · , n are invertible operators.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 26 / 36
Application
In the following theorem, Soumitra Ghara give a way to decide when anoperator T ∈ FB2(D) to be a weakly homogeneous operator.
Theorem (S. Ghara, Thesis,IISC, 2018)
Let 1 ≤ λ ≤ µ ≤ λ+ 2 and ψ be a non-zero function in C (D) ∩ Hol(D).
The operator T =(
M∗z M∗
ψ
0 M∗z
)on H(λ) ⊕H(µ) is weakly homogeneous if and
only if ψ is non-vanishing on D.
Although the description of the weakly homogeneous operators in FB2(D)is more or less clear. However, the computation will become very difficultwith the growth of the rank n.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 27 / 36
Application
Thus, in general case, we need the intertwining operator between T andφα(T ), α ∈ D could be diagonal. That means we need to consider theoperators in CFBn(D) which satisfy the strongly Property (H). In the endof this talk, we will show that there also exists a lot examples of non-weaklyhomogeneous operators in CFBn(D)
Theorem 3[Jiang and Ji]
Let T =( T1,1 T1,2 T1,3
0 T2,2 T2,3
0 0 T3,3
)∈ CFB3(D). If T satisfies the strongly Property
(H), then T is not weakly homogeneous.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 28 / 36
End
Thank you!!
Email: [email protected] [email protected] Normal University
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 29 / 36
Reference
M. J. Cowen and R. G. Douglas, Complex geometry and operatortheory, Acta Math. 141 (1978), 187–261.
D. N. Clark and G. Misra, On curvature and similarity, Michigan Math.J. 30(1983), no. 3, 361-367.
D. N. Clark and G. Misra, On weighted shifts, curvature and similarity,J. London Math. Soc.,(2) 31(1985), 357 - 368.
R. Curto and N. Salinas, Generalized Bergman kernels and theCowen-Douglas theory. Amer. J. Math. 106 (1984), no. 2, 447–488.
R. G. Douglas, Operator theory and complex geometry, Extracta Math.24 (2009), no. 2, 135-165.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 30 / 36
Reference
M. J. Cowen and R. G. Douglas, Complex geometry and operatortheory, Acta Math. 141 (1978), 187–261.
D. N. Clark and G. Misra, On curvature and similarity, Michigan Math.J. 30(1983), no. 3, 361-367.
D. N. Clark and G. Misra, On weighted shifts, curvature and similarity,J. London Math. Soc.,(2) 31(1985), 357 - 368.
R. Curto and N. Salinas, Generalized Bergman kernels and theCowen-Douglas theory. Amer. J. Math. 106 (1984), no. 2, 447–488.
R. G. Douglas, Operator theory and complex geometry, Extracta Math.24 (2009), no. 2, 135-165.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 30 / 36
Reference
M. J. Cowen and R. G. Douglas, Complex geometry and operatortheory, Acta Math. 141 (1978), 187–261.
D. N. Clark and G. Misra, On curvature and similarity, Michigan Math.J. 30(1983), no. 3, 361-367.
D. N. Clark and G. Misra, On weighted shifts, curvature and similarity,J. London Math. Soc.,(2) 31(1985), 357 - 368.
R. Curto and N. Salinas, Generalized Bergman kernels and theCowen-Douglas theory. Amer. J. Math. 106 (1984), no. 2, 447–488.
R. G. Douglas, Operator theory and complex geometry, Extracta Math.24 (2009), no. 2, 135-165.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 30 / 36
Reference
M. J. Cowen and R. G. Douglas, Complex geometry and operatortheory, Acta Math. 141 (1978), 187–261.
D. N. Clark and G. Misra, On curvature and similarity, Michigan Math.J. 30(1983), no. 3, 361-367.
D. N. Clark and G. Misra, On weighted shifts, curvature and similarity,J. London Math. Soc.,(2) 31(1985), 357 - 368.
R. Curto and N. Salinas, Generalized Bergman kernels and theCowen-Douglas theory. Amer. J. Math. 106 (1984), no. 2, 447–488.
R. G. Douglas, Operator theory and complex geometry, Extracta Math.24 (2009), no. 2, 135-165.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 30 / 36
Reference
M. J. Cowen and R. G. Douglas, Complex geometry and operatortheory, Acta Math. 141 (1978), 187–261.
D. N. Clark and G. Misra, On curvature and similarity, Michigan Math.J. 30(1983), no. 3, 361-367.
D. N. Clark and G. Misra, On weighted shifts, curvature and similarity,J. London Math. Soc.,(2) 31(1985), 357 - 368.
R. Curto and N. Salinas, Generalized Bergman kernels and theCowen-Douglas theory. Amer. J. Math. 106 (1984), no. 2, 447–488.
R. G. Douglas, Operator theory and complex geometry, Extracta Math.24 (2009), no. 2, 135-165.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 30 / 36
Reference
R. G. Douglas, H. Kwon, and S. Treil, Similarity of Operators in theBergman Space Setting, J. Lond. Math. Soc., 88 (2013), no. 3,637-648.
R. G. Douglas, G. Misra and C. Varughese, On quotient modules thecase of arbitrary multiplicity, J. Funct. Anal. 174 (2000), no. 2,364-398.
R. G. Douglas and G. Misra, Equivalence of quotient Hilbert modules.II., Trans. Amer. Math. Soc. 360 (2008), no. 4, 2229-2264.
P. R. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc.,76 (1970) 887-933.
P. R. Halmos, A Hilbert space problem book, (Second edition) GraduateTexts in Mathematics, Springer-Verlag, New York-Berlin, 1982.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 31 / 36
Reference
R. G. Douglas, H. Kwon, and S. Treil, Similarity of Operators in theBergman Space Setting, J. Lond. Math. Soc., 88 (2013), no. 3,637-648.
R. G. Douglas, G. Misra and C. Varughese, On quotient modules thecase of arbitrary multiplicity, J. Funct. Anal. 174 (2000), no. 2,364-398.
R. G. Douglas and G. Misra, Equivalence of quotient Hilbert modules.II., Trans. Amer. Math. Soc. 360 (2008), no. 4, 2229-2264.
P. R. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc.,76 (1970) 887-933.
P. R. Halmos, A Hilbert space problem book, (Second edition) GraduateTexts in Mathematics, Springer-Verlag, New York-Berlin, 1982.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 31 / 36
Reference
R. G. Douglas, H. Kwon, and S. Treil, Similarity of Operators in theBergman Space Setting, J. Lond. Math. Soc., 88 (2013), no. 3,637-648.
R. G. Douglas, G. Misra and C. Varughese, On quotient modules thecase of arbitrary multiplicity, J. Funct. Anal. 174 (2000), no. 2,364-398.
R. G. Douglas and G. Misra, Equivalence of quotient Hilbert modules.II., Trans. Amer. Math. Soc. 360 (2008), no. 4, 2229-2264.
P. R. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc.,76 (1970) 887-933.
P. R. Halmos, A Hilbert space problem book, (Second edition) GraduateTexts in Mathematics, Springer-Verlag, New York-Berlin, 1982.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 31 / 36
Reference
R. G. Douglas, H. Kwon, and S. Treil, Similarity of Operators in theBergman Space Setting, J. Lond. Math. Soc., 88 (2013), no. 3,637-648.
R. G. Douglas, G. Misra and C. Varughese, On quotient modules thecase of arbitrary multiplicity, J. Funct. Anal. 174 (2000), no. 2,364-398.
R. G. Douglas and G. Misra, Equivalence of quotient Hilbert modules.II., Trans. Amer. Math. Soc. 360 (2008), no. 4, 2229-2264.
P. R. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc.,76 (1970) 887-933.
P. R. Halmos, A Hilbert space problem book, (Second edition) GraduateTexts in Mathematics, Springer-Verlag, New York-Berlin, 1982.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 31 / 36
Reference
R. G. Douglas, H. Kwon, and S. Treil, Similarity of Operators in theBergman Space Setting, J. Lond. Math. Soc., 88 (2013), no. 3,637-648.
R. G. Douglas, G. Misra and C. Varughese, On quotient modules thecase of arbitrary multiplicity, J. Funct. Anal. 174 (2000), no. 2,364-398.
R. G. Douglas and G. Misra, Equivalence of quotient Hilbert modules.II., Trans. Amer. Math. Soc. 360 (2008), no. 4, 2229-2264.
P. R. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc.,76 (1970) 887-933.
P. R. Halmos, A Hilbert space problem book, (Second edition) GraduateTexts in Mathematics, Springer-Verlag, New York-Berlin, 1982.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 31 / 36
Reference
G. H. Hardy and J. E. Littlewood, Tauberian Theorems ConcerningPower Series and Dirichlet’s Series whose Coefficients are Positive,Proc. London Math. Soc., 13 (1914), 174 – 191
C. Jiang, Similarity, reducibility and approximation of theCowen-Douglas operators. J. Operator Theory 32 (1994), no. 1, 77–89.
C. Jiang, Similarity classification of Cowen-Douglas operators. Canad. J.Math. 56 (4) (2004), 742–775.
K. Ji, C. Jiang, D. K. Keshari and G. Misra Flag structure for operatorsin the Cowen-Douglas class, Comptes rendus - Mathematique, 352(2014), 511–514.
K. Ji, C. Jiang, D. K. Keshari and G. Misra Rigidity of the flagstructure for a class of Cowen-Douglas operators, preprint.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 32 / 36
Reference
G. H. Hardy and J. E. Littlewood, Tauberian Theorems ConcerningPower Series and Dirichlet’s Series whose Coefficients are Positive,Proc. London Math. Soc., 13 (1914), 174 – 191
C. Jiang, Similarity, reducibility and approximation of theCowen-Douglas operators. J. Operator Theory 32 (1994), no. 1, 77–89.
C. Jiang, Similarity classification of Cowen-Douglas operators. Canad. J.Math. 56 (4) (2004), 742–775.
K. Ji, C. Jiang, D. K. Keshari and G. Misra Flag structure for operatorsin the Cowen-Douglas class, Comptes rendus - Mathematique, 352(2014), 511–514.
K. Ji, C. Jiang, D. K. Keshari and G. Misra Rigidity of the flagstructure for a class of Cowen-Douglas operators, preprint.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 32 / 36
Reference
G. H. Hardy and J. E. Littlewood, Tauberian Theorems ConcerningPower Series and Dirichlet’s Series whose Coefficients are Positive,Proc. London Math. Soc., 13 (1914), 174 – 191
C. Jiang, Similarity, reducibility and approximation of theCowen-Douglas operators. J. Operator Theory 32 (1994), no. 1, 77–89.
C. Jiang, Similarity classification of Cowen-Douglas operators. Canad. J.Math. 56 (4) (2004), 742–775.
K. Ji, C. Jiang, D. K. Keshari and G. Misra Flag structure for operatorsin the Cowen-Douglas class, Comptes rendus - Mathematique, 352(2014), 511–514.
K. Ji, C. Jiang, D. K. Keshari and G. Misra Rigidity of the flagstructure for a class of Cowen-Douglas operators, preprint.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 32 / 36
Reference
G. H. Hardy and J. E. Littlewood, Tauberian Theorems ConcerningPower Series and Dirichlet’s Series whose Coefficients are Positive,Proc. London Math. Soc., 13 (1914), 174 – 191
C. Jiang, Similarity, reducibility and approximation of theCowen-Douglas operators. J. Operator Theory 32 (1994), no. 1, 77–89.
C. Jiang, Similarity classification of Cowen-Douglas operators. Canad. J.Math. 56 (4) (2004), 742–775.
K. Ji, C. Jiang, D. K. Keshari and G. Misra Flag structure for operatorsin the Cowen-Douglas class, Comptes rendus - Mathematique, 352(2014), 511–514.
K. Ji, C. Jiang, D. K. Keshari and G. Misra Rigidity of the flagstructure for a class of Cowen-Douglas operators, preprint.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 32 / 36
Reference
G. H. Hardy and J. E. Littlewood, Tauberian Theorems ConcerningPower Series and Dirichlet’s Series whose Coefficients are Positive,Proc. London Math. Soc., 13 (1914), 174 – 191
C. Jiang, Similarity, reducibility and approximation of theCowen-Douglas operators. J. Operator Theory 32 (1994), no. 1, 77–89.
C. Jiang, Similarity classification of Cowen-Douglas operators. Canad. J.Math. 56 (4) (2004), 742–775.
K. Ji, C. Jiang, D. K. Keshari and G. Misra Flag structure for operatorsin the Cowen-Douglas class, Comptes rendus - Mathematique, 352(2014), 511–514.
K. Ji, C. Jiang, D. K. Keshari and G. Misra Rigidity of the flagstructure for a class of Cowen-Douglas operators, preprint.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 32 / 36
Reference
C. Jiang and K. Ji, Similarity classification of holomorphic curves. Adv.Math. 215 (2) (2007), 446-468.
C. Jiang, X. Guo and K. Ji, K -group and similarity classification ofoperators, J. Funct. Anal., 225 (2005), 167-192.
C. Jiang and Z. Wang, Strongly irreducible operators on Hilbert space.Pitman Research Notes in Mathematics Series, 389. Longman, Harlow,1998. x+243 pp. ISBN: 0-582-30594-2.
C. Jiang and Z. Wang, The spectral picture and the closure of thesimilarity orbit of strongly irreducible operators, Integral EquationsOperator Theory 24 (1996), no. 1, 81-105.
S. Kobayashi, Differential geometry of complex vector bundles,Publications of the Mathematical Society of Japan, 15. Kano MemorialLectures, 5. Princeton University Press, Princeton, NJ; Iwanami Shoten,Tokyo, 1987.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 33 / 36
Reference
C. Jiang and K. Ji, Similarity classification of holomorphic curves. Adv.Math. 215 (2) (2007), 446-468.
C. Jiang, X. Guo and K. Ji, K -group and similarity classification ofoperators, J. Funct. Anal., 225 (2005), 167-192.
C. Jiang and Z. Wang, Strongly irreducible operators on Hilbert space.Pitman Research Notes in Mathematics Series, 389. Longman, Harlow,1998. x+243 pp. ISBN: 0-582-30594-2.
C. Jiang and Z. Wang, The spectral picture and the closure of thesimilarity orbit of strongly irreducible operators, Integral EquationsOperator Theory 24 (1996), no. 1, 81-105.
S. Kobayashi, Differential geometry of complex vector bundles,Publications of the Mathematical Society of Japan, 15. Kano MemorialLectures, 5. Princeton University Press, Princeton, NJ; Iwanami Shoten,Tokyo, 1987.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 33 / 36
Reference
C. Jiang and K. Ji, Similarity classification of holomorphic curves. Adv.Math. 215 (2) (2007), 446-468.
C. Jiang, X. Guo and K. Ji, K -group and similarity classification ofoperators, J. Funct. Anal., 225 (2005), 167-192.
C. Jiang and Z. Wang, Strongly irreducible operators on Hilbert space.Pitman Research Notes in Mathematics Series, 389. Longman, Harlow,1998. x+243 pp. ISBN: 0-582-30594-2.
C. Jiang and Z. Wang, The spectral picture and the closure of thesimilarity orbit of strongly irreducible operators, Integral EquationsOperator Theory 24 (1996), no. 1, 81-105.
S. Kobayashi, Differential geometry of complex vector bundles,Publications of the Mathematical Society of Japan, 15. Kano MemorialLectures, 5. Princeton University Press, Princeton, NJ; Iwanami Shoten,Tokyo, 1987.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 33 / 36
Reference
C. Jiang and K. Ji, Similarity classification of holomorphic curves. Adv.Math. 215 (2) (2007), 446-468.
C. Jiang, X. Guo and K. Ji, K -group and similarity classification ofoperators, J. Funct. Anal., 225 (2005), 167-192.
C. Jiang and Z. Wang, Strongly irreducible operators on Hilbert space.Pitman Research Notes in Mathematics Series, 389. Longman, Harlow,1998. x+243 pp. ISBN: 0-582-30594-2.
C. Jiang and Z. Wang, The spectral picture and the closure of thesimilarity orbit of strongly irreducible operators, Integral EquationsOperator Theory 24 (1996), no. 1, 81-105.
S. Kobayashi, Differential geometry of complex vector bundles,Publications of the Mathematical Society of Japan, 15. Kano MemorialLectures, 5. Princeton University Press, Princeton, NJ; Iwanami Shoten,Tokyo, 1987.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 33 / 36
Reference
C. Jiang and K. Ji, Similarity classification of holomorphic curves. Adv.Math. 215 (2) (2007), 446-468.
C. Jiang, X. Guo and K. Ji, K -group and similarity classification ofoperators, J. Funct. Anal., 225 (2005), 167-192.
C. Jiang and Z. Wang, Strongly irreducible operators on Hilbert space.Pitman Research Notes in Mathematics Series, 389. Longman, Harlow,1998. x+243 pp. ISBN: 0-582-30594-2.
C. Jiang and Z. Wang, The spectral picture and the closure of thesimilarity orbit of strongly irreducible operators, Integral EquationsOperator Theory 24 (1996), no. 1, 81-105.
S. Kobayashi, Differential geometry of complex vector bundles,Publications of the Mathematical Society of Japan, 15. Kano MemorialLectures, 5. Princeton University Press, Princeton, NJ; Iwanami Shoten,Tokyo, 1987.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 33 / 36
Reference
H. Kwon and S. Treil, Similarity of operators and geometry ofeigenvector bundles, Publ. Mat. 53. (2009), 417-438.
A. Koranyi and G. Misra, A classification of homogeneous operators inthe Cowen-Douglas class, Adv. Math., 226 (2011) 5338-5360.
A. Koranyi and G. Misra, Multiplicity-free homogeneous operators in theCowen-Douglas class. Perspectives in mathematical sciences. II, 83-101,Stat. Sci. Interdiscip. Res., 8, World Sci. Publ., Hackensack, NJ, 2009.
G. Misra, , Curvature and the backward shift operators, Proc. Amer.Math. Soc. 91 (1984), no. 1, 105-107.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 34 / 36
Reference
H. Kwon and S. Treil, Similarity of operators and geometry ofeigenvector bundles, Publ. Mat. 53. (2009), 417-438.
A. Koranyi and G. Misra, A classification of homogeneous operators inthe Cowen-Douglas class, Adv. Math., 226 (2011) 5338-5360.
A. Koranyi and G. Misra, Multiplicity-free homogeneous operators in theCowen-Douglas class. Perspectives in mathematical sciences. II, 83-101,Stat. Sci. Interdiscip. Res., 8, World Sci. Publ., Hackensack, NJ, 2009.
G. Misra, , Curvature and the backward shift operators, Proc. Amer.Math. Soc. 91 (1984), no. 1, 105-107.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 34 / 36
Reference
H. Kwon and S. Treil, Similarity of operators and geometry ofeigenvector bundles, Publ. Mat. 53. (2009), 417-438.
A. Koranyi and G. Misra, A classification of homogeneous operators inthe Cowen-Douglas class, Adv. Math., 226 (2011) 5338-5360.
A. Koranyi and G. Misra, Multiplicity-free homogeneous operators in theCowen-Douglas class. Perspectives in mathematical sciences. II, 83-101,Stat. Sci. Interdiscip. Res., 8, World Sci. Publ., Hackensack, NJ, 2009.
G. Misra, , Curvature and the backward shift operators, Proc. Amer.Math. Soc. 91 (1984), no. 1, 105-107.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 34 / 36
Reference
H. Kwon and S. Treil, Similarity of operators and geometry ofeigenvector bundles, Publ. Mat. 53. (2009), 417-438.
A. Koranyi and G. Misra, A classification of homogeneous operators inthe Cowen-Douglas class, Adv. Math., 226 (2011) 5338-5360.
A. Koranyi and G. Misra, Multiplicity-free homogeneous operators in theCowen-Douglas class. Perspectives in mathematical sciences. II, 83-101,Stat. Sci. Interdiscip. Res., 8, World Sci. Publ., Hackensack, NJ, 2009.
G. Misra, , Curvature and the backward shift operators, Proc. Amer.Math. Soc. 91 (1984), no. 1, 105-107.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 34 / 36
Reference
G. Pisier, A polynomially bounded operator on Hilbert space which isnot similar to a contraction, J. Amer. Math. Soc., 10 (1997), 351-369.
M. Rordam, F. Larsen and N. J. Laustsen An Introduction to K-Theoryfor C*-Algebras. Cambridge University press (2000).
Pit-Mann Wong and W. Stoll, On Holomorphic Jet Bundles,arXiv:math/0003226v1, 2000.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 35 / 36
Reference
G. Pisier, A polynomially bounded operator on Hilbert space which isnot similar to a contraction, J. Amer. Math. Soc., 10 (1997), 351-369.
M. Rordam, F. Larsen and N. J. Laustsen An Introduction to K-Theoryfor C*-Algebras. Cambridge University press (2000).
Pit-Mann Wong and W. Stoll, On Holomorphic Jet Bundles,arXiv:math/0003226v1, 2000.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 35 / 36
Reference
G. Pisier, A polynomially bounded operator on Hilbert space which isnot similar to a contraction, J. Amer. Math. Soc., 10 (1997), 351-369.
M. Rordam, F. Larsen and N. J. Laustsen An Introduction to K-Theoryfor C*-Algebras. Cambridge University press (2000).
Pit-Mann Wong and W. Stoll, On Holomorphic Jet Bundles,arXiv:math/0003226v1, 2000.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 35 / 36
Reference
A. L. Shields, Weighted Shift Operators And Analytic Function Theory,Mathematical Survey, 13(1974) 49-128.
M. Spivak, A Comprehensive introduction to differential geometry(Volume 3). Publish or Perish. ISBN 0-914098-72-1.
K. Zhu,Operators in Cowen-Douglas classes. Illinois J. Math. 44 (2000),no. 4, 767-783.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 36 / 36
Reference
A. L. Shields, Weighted Shift Operators And Analytic Function Theory,Mathematical Survey, 13(1974) 49-128.
M. Spivak, A Comprehensive introduction to differential geometry(Volume 3). Publish or Perish. ISBN 0-914098-72-1.
K. Zhu,Operators in Cowen-Douglas classes. Illinois J. Math. 44 (2000),no. 4, 767-783.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 36 / 36
Reference
A. L. Shields, Weighted Shift Operators And Analytic Function Theory,Mathematical Survey, 13(1974) 49-128.
M. Spivak, A Comprehensive introduction to differential geometry(Volume 3). Publish or Perish. ISBN 0-914098-72-1.
K. Zhu,Operators in Cowen-Douglas classes. Illinois J. Math. 44 (2000),no. 4, 767-783.
Kui Ji (Hebei Normal University) Geometric similarity invariants Oct 2018 36 / 36