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.