Composition Operators and Endomorphisms

Abstract

If $b$ is an inner function, then composition with $b$ induces an endomorphism, $\beta$, of $L^\infty(\mathbb{T})$ that leaves $H^\infty(\mathbb{T})$ invariant. We investigate the structure of the endomorphisms of $B(L^2(\mathbb{T}))$ and $B(H^2(\mathbb{T}))$ that implement $\beta$ through the representations of $L^\infty(\mathbb{T})$ and $H^\infty(\mathbb{T})$ in terms of multiplication operators on $L^2(\mathbb{T})$ and $H^2(\mathbb{T})$. Our analysis, which is based on work of R. Rochberg and J. McDonald, will wind its way through the theory of composition operators on spaces of analytic functions to recent work on Cuntz families of isometries and Hilbert $C^*$-modules

Quotient Algebras of Toeplitz-Composition $$C^{*}$$ C ∗ -Algebras for Finite Blaschke Products

“…Watatani and the author [10] showed that, if R is a finite Blaschke product of degree at least two with R(0) = 0, then the operator C * R T a C R is a Toeplitz operator T LR(a) . Courtney, Muhly and Schmidt [5] extend this to the case for a general finite Blaschke product. On the other hand, Jury [13] independently proved a covariant relation C * ϕ T a C ϕ = T Aϕ(a) for an inner function ϕ, where A ϕ is the Aleksandrov operator defined by Aleksandrov-Clark measures [1,4].…”

“…On the other hand, Courtney, Muhly and Schmidt [5] studied certain endomorphisms of B(L 2 (T)) and B(H 2 (T)), where B(L 2 (T)) and B(H 2 (T)) are the C * -algebra of all bounded operators on L 2 (T) and H 2 (T) respectively. Let ϕ be an inner function and let α be the induced endomorphism of L ∞ (T) defined by α(a) = a • ϕ for a ∈ L ∞ (T).…”

“…They considered a covariant relation S * M a S = M L(a) for a ∈ L ∞ (Ω, µ) under some conditions. We discuss the difference between the Hilbert bimodule X R and the Hilbert bimodule L ∞ (T) L considered in [5]. Two Hilbert bimodules are almost the same.…”

Extending Wavelet Filters: Infinite Dimensions, the Nonrational Case, and Indefinite Inner Product Spaces

“…Therefore g = S j f , and the operator S j is closed, and hence continuous. From (21) we have for u ∈ P(Θ )…”

“…(21) is in force.Let g(z) = ∑ N−1 k=0 z k g k (z N ) ∈ P(Θ N ) where the g k ∈ P(Θ ), and let u ∈ P(Θ ). Then,[S j u , g] P(Θ N ) = [z j u(z N ) , k (z N )] P(Θ N ) = [u , g j ] P(Θ ) = [u , S [ * ] j g] P(Θ ) ,where [ , ] P(Θ ) and [ , ] P(Θ N ) denote the indefinite inner products in the corresponding spaces.…”