strongly compact algebras and composition operatorsccowen/iwota12/shapiroslides.pdf · 2012. 7....
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Strongly compact algebras and composition operators
Joel H. Shapiro
Portland State University
IWOTA, July 2012
Lomonosov, 1980:
Strongly compact for an algebra of operators means
All bounded subsets strongly precompact.
Which algebras?
alg(T ) = the algebra generated by operator T and the identity.
com(T ) = the commutant of T .
Examples:
alg(I ) = CI : strongly compact.
com(I ) = all operators: not strongly compact.
L (finite dim’l Hilbert space): strongly compact.
Lomonosov, 1980:
Strongly compact for an algebra of operators means
All bounded subsets strongly precompact.
Which algebras?
alg(T ) = the algebra generated by operator T and the identity.
com(T ) = the commutant of T .
Examples:
alg(I ) = CI : strongly compact.
com(I ) = all operators: not strongly compact.
L (finite dim’l Hilbert space): strongly compact.
Selected background
(1980) Lomonosov: Introduced ”SC”to study Invariant Subspace Problem.
(1990) Marsalli: Independently “discovered” SC.Spectral suff. conditions for SC.Characterized SC for self-adjoint algebras.
(2006) Lacruz, Lomonosov, Rodrıguez-Piazza:Examples, constructions, counterexamples,normal operators, weighted shifts.
(2011) Fernandez-Valles and Lacruz: Weighted shifts,Cesaro operator, composition operators.
Composition Operators
ϕ holomorphic on U, ϕ(U) ⊂ U
Cϕf = f ◦ ϕ (f ∈ H(U))
Cϕ : H(U)→ H(U) linear transf.
Littlewood Subord’n Thm. (1920’s)
Cϕ : H2 → H2 (bounded) linear operator
General Question
For which ϕ is alg(Cϕ), com(Cϕ) strongly compact?
For today: ϕ ∈ LFT(U)
Composition Operators
ϕ holomorphic on U, ϕ(U) ⊂ U
Cϕf = f ◦ ϕ (f ∈ H(U))
Cϕ : H(U)→ H(U) linear transf.
Littlewood Subord’n Thm. (1920’s)
Cϕ : H2 → H2 (bounded) linear operator
General Question
For which ϕ is alg(Cϕ), com(Cϕ) strongly compact?
For today: ϕ ∈ LFT(U)
ϕ ∈ LFT(U) with a fixed point on ∂U
(Shaded results due to Lacruz and Fernandez-Valles 2011)
Marsalli 1990: Sufficient conditions for SC
Equivalent Defn :
A1x is rel. compact in H ∀ x ∈ H (or a dense subset of H).
Consequence—Useful Sufficient Condition:
∃ family of finite dimensional A -invariant subspaces that havedense linear span.
Corollaries:
Suff. for alg(T ) SC:∃ densely spanning family of eigenvectors.
Suff. for com(T ) SC:∃ densely spanning family of finite dim’l eigenspaces.
Marsalli 1990: Sufficient conditions for SC
Equivalent Defn :
A1x is rel. compact in H ∀ x ∈ H (or a dense subset of H).
Consequence—Useful Sufficient Condition:
∃ family of finite dimensional A -invariant subspaces that havedense linear span.
Corollaries:
Suff. for alg(T ) SC:∃ densely spanning family of eigenvectors.
Suff. for com(T ) SC:∃ densely spanning family of finite dim’l eigenspaces.
Marsalli 1990: Sufficient conditions for SC
Equivalent Defn :
A1x is rel. compact in H ∀ x ∈ H (or a dense subset of H).
Consequence—Useful Sufficient Condition:
∃ family of finite dimensional A -invariant subspaces that havedense linear span.
Corollaries:
Suff. for alg(T ) SC:∃ densely spanning family of eigenvectors.
Suff. for com(T ) SC:∃ densely spanning family of finite dim’l eigenspaces.
Application: ϕ elliptic
WLOG: ϕ(z) = ωz , |ω| = 1
I ω arbitrary
Eigenvectors, Eigenvalues: zn, ωn (n = 0, 1, 2, . . .)
⇒ alg(Cϕ) strongly compact.
I ω not a root of unity:
All eigenvalues have multiplicity 1
⇒ com(Cϕ) strongly compact
I ω is a root of unity: ωN = 1
Cϕ = I ⊕ ωI ⊕ · · · ⊕ ωN−1I
& com(Cϕ) ⊃ L (H0)⊕L (H1)⊕ · · · ⊕L (HN−1)
⇒ com(Cϕ) not strongly compact.
Application: ϕ elliptic
WLOG: ϕ(z) = ωz , |ω| = 1
I ω arbitrary
Eigenvectors, Eigenvalues: zn, ωn (n = 0, 1, 2, . . .)
⇒ alg(Cϕ) strongly compact.
I ω not a root of unity:
All eigenvalues have multiplicity 1
⇒ com(Cϕ) strongly compact
I ω is a root of unity: ωN = 1
Cϕ = I ⊕ ωI ⊕ · · · ⊕ ωN−1I
& com(Cϕ) ⊃ L (H0)⊕L (H1)⊕ · · · ⊕L (HN−1)
⇒ com(Cϕ) not strongly compact.
∃ fixed point on ∂U
Typeof ϕ
Fixed pt.position
alg(Cϕ) com(Cϕ) Eigenvec,eigenval
Eigsp.dim’n Rmks
PA ∂U only SC not SC eλz+1z−1 , e−λa ∞ λ ≥ 0
Re a = 0
PNA ∂U only SC SC eλz+1z−1 , e−λa 1 λ ≥ 0
Re a > 0
HA ∂U only SC not SC ( 1+z1−z )λ, aλ ∞ |Reλ| < 1
20 < a < 1
HNA∂U & U not SC not SC ********** ** ********
∂U & Ue SC ??? (1− z)λ, ( 12 )λ ∞ Reλ > − 12
ϕ(z) = 1+z2
1
Multipliers in the commutantFor f ∈ H∞: f ◦ ϕ = f =⇒ CϕMf = Mf Cϕ
Theorem. alg(Mf ) strongly compact =⇒{ζ ∈ ∂U : |f (ζ)| = ‖f ‖∞} has measure zero.
Typeof ϕ
Fixed pt.position
alg(Cϕ) com(Cϕ) Eigvec f(z),Eigval
λ s.t.Cϕf = f
alg(Mf )
PA ∂U only SC not SC eλz+1z−1 , e−λa λ = 2π
αa = iαα > 0
not SC
HA ∂U only SC not SC ( 1+z1−z )λ, aλ λ = 2πi
log a not SC
1+z2 ∂U & Ue SC ??? (1− z)λ, ( 12 )
λ λ = 2πilog 2 is SC
1
Multipliers in the commutantFor f ∈ H∞: f ◦ ϕ = f =⇒ CϕMf = Mf Cϕ
Theorem. alg(Mf ) strongly compact =⇒{ζ ∈ ∂U : |f (ζ)| = ‖f ‖∞} has measure zero.
Typeof ϕ
Fixed pt.position
alg(Cϕ) com(Cϕ) Eigvec f(z),Eigval
λ s.t.Cϕf = f
alg(Mf )
PA ∂U only SC not SC eλz+1z−1 , e−λa λ = 2π
αa = iαα > 0
not SC
HA ∂U only SC not SC ( 1+z1−z )λ, aλ λ = 2πi
log a not SC
1+z2 ∂U & Ue SC ??? (1− z)λ, ( 12 )
λ λ = 2πilog 2 is SC
1
Multipliers in the commutantFor f ∈ H∞: f ◦ ϕ = f =⇒ CϕMf = Mf Cϕ
Theorem. alg(Mf ) strongly compact =⇒{ζ ∈ ∂U : |f (ζ)| = ‖f ‖∞} has measure zero.
Typeof ϕ
Fixed pt.position
alg(Cϕ) com(Cϕ) Eigvec f(z),Eigval
λ s.t.Cϕf = f
alg(Mf )
PA ∂U only SC not SC eλz+1z−1 , e−λa λ = 2π
αa = iαα > 0
not SC
HA ∂U only SC not SC ( 1+z1−z )λ, aλ λ = 2πi
log a not SC
1+z2 ∂U & Ue SC ??? (1− z)λ, ( 12 )
λ λ = 2πilog 2 is SC
1
Review
Suppose 0 and 1 fixed:
∴ ϕ(z) =s z
1− (1− s)z( = z
2−z if s = 12).
∴ Cϕ = I0 ⊕ (Cϕ|H20) ≈ I0 ⊕ s C ∗sz+(1−s)︸ ︷︷ ︸
Cowen’s Adjoint Th.
Thm. alg(C ∗ψ) SC ⇐⇒ ψ fixes a point of U.
“ ∴ ” alg(Cϕ) not SC.
Suppose 0 and 1 fixed:
∴ ϕ(z) =s z
1− (1− s)z( = z
2−z if s = 12).
∴ Cϕ = I0 ⊕ (Cϕ|H20)
≈ I0 ⊕ s C ∗sz+(1−s)︸ ︷︷ ︸Cowen’s Adjoint Th.
Thm. alg(C ∗ψ) SC ⇐⇒ ψ fixes a point of U.
“ ∴ ” alg(Cϕ) not SC.
Suppose 0 and 1 fixed:
∴ ϕ(z) =s z
1− (1− s)z( = z
2−z if s = 12).
∴ Cϕ = I0 ⊕ (Cϕ|H20) ≈ I0 ⊕ s C ∗sz+(1−s)︸ ︷︷ ︸
Cowen’s Adjoint Th.
Thm. alg(C ∗ψ) SC ⇐⇒ ψ fixes a point of U.
“ ∴ ” alg(Cϕ) not SC.
Suppose 0 and 1 fixed:
∴ ϕ(z) =s z
1− (1− s)z( = z
2−z if s = 12).
∴ Cϕ = I0 ⊕ (Cϕ|H20) ≈ I0 ⊕ s C ∗sz+(1−s)︸ ︷︷ ︸
Cowen’s Adjoint Th.
Thm. alg(C ∗ψ) SC ⇐⇒ ψ fixes a point of U.
“ ∴ ” alg(Cϕ) not SC.
Some References
* Aurora Fernandez-Valles & Miguel Lacruz, A spectral condition for strongcompactness, J. Adv. Res. Pure Math. (JARPM) 3 (4) 2011, 50–60.
* Miguel Lacruz, Victor Lomonosov, & Luis Rodrıguez-Piazza, Stronglycompact algebras, Rev. R. Acad. Cien. Ser. A Mat. 100 (2006) 191–207.
I Miguel Lacruz & Marıa Del Pilar Romero de la Rosa, A local spectralcondition for strong compactness with some applications to bilateralweighted shifts, Proc. Amer. Math. Soc., to appear.
I Miguel Lacruz & Luis Rodrıguez-Piazza, Strongly compact normal operators,Proc. Amer. Math. Soc. 137 (2009) 2623–2630.
I Victor Lomonosov, Construction of an intertwining operator, Funksional.Anal. i Prilozhen., 14 (1980), 67–78 (Russian). English translation:Functional Analysis and its Applications 14 (1980) 54–55.
I Michael Marsalli, A classification of operator algebras, J. Operator Theory24 (1990) 155-163.