Quasi-parabolic Composition Operators on Weighted Bergman Spaces

The commutative C*-algebras of Toeplitz operators on the classical (weightless) Bergman space were classified in Chapter 9 by pencils of geodesics on the unit disk, considered as the hyperbolic plane. Theorem 10.4.1 shows that the same classes of defining symbols generate commutative C*-algebras of Toeplitz operators on each weighted Bergman space. At the same time the principal question, whether the above cases are the only possible sets of defining symbols which might generate the commutative C * -algebras of Toeplitz operators on each weighted Bergman space, has remained open.

Algebras of Toeplitz operators in spaces of smooth functions

Construction of some classes of commutative Banach and C⋆-algebras of Toeplitz operators

We prove that there exist a unitary operator between the super weighted super Bergman spaces of the super-disk and the super upper plane, and we find the form of the functions invariant under the action of super reals over the super upper plane.We prove that, generalizing the parabolic classical case, every super Toeplitz operator with super-reals-invariant symbol is diagonal. Finally we prove that the algebra of Toeplitz operators with symbols invariant under the action of the super reals is commutative.

<p>In this article, we considered a class of composition operators on Lebesgue spaces with variable exponents over metric measure spaces. Taking advantage of the compatibility between the metric-measurable structure and the regularity properties of the variable exponent, we provided necessary and sufficient conditions for this class of operators to be bounded and compact, respectively. In addition, we showed the usefulness of the variable change to study weak compactness properties in the framework of non-standard spaces.</p>

In this paper, we study composition operators on a Banach space of analytic functions, denoted byX, which includes the Bloch space. This space arises naturally as the dual space of analytic functions in the Bergman spaceL α 1 (D) which admit an atomic decomposition. We characterize the functions which induce compact composition operators and those which induce Fredholm operatorson this space. We also investigate when a composition operator has a closed range.

The topological structure of the set of (weighted) composition operators has been studied on various function spaces on the unit disc such as Hardy spaces, the space of bounded holomorphic functions, weighted Banach spaces of holomorphic functions with sup-norm, Hilbert Bergman spaces. In this paper we consider this problem for all Bergman spaces $$A_{\alpha }^p$$ with $$p \in (0, \infty )$$ and $$ \alpha \in (-1, \infty )$$ . In this setting we establish a criterion for two composition operators to be linearly connected in the space of composition operators; furthermore, for the space of weighted composition operators, we prove that the set of compact weighted composition operators is path connected, but it is not a component.

In this paper, we investigate the boundedness of composition operators defined on a quasi-Banach space continuously included in the space of smooth functions on a manifold. We prove that the boundedness of a composition operator strongly limits the behavior of the original map, and it provides an effective method to investigate the properties of composition operators using the theory of dynamical system. Consequently, we prove that only affine maps can induce bounded composition operators on any quasi-Banach space continuously included in the space of entire functions of one variable if the function space contains a nonconstant function. We also prove that any polynomial automorphisms except affine transforms cannot induce bounded composition operators on a quasi-Banach space composed of entire functions in the two-dimensional complex affine space under several mild conditions.

Chapter 39 Spaces of analytic functions with integral norm

LetVbe an arbitrary system of weights on an open connected subsetGofℂN(N≥1)and letB(E)be the Banach algebra of all bounded linear operators on a Banach spaceE. LetHVb(G,E)andHV0(G,E)be the weighted locally convex spaces of vector-valued analytic functions. In this survey, we present a development of the theory of multiplication operators and composition operators from classical spaces of analytic functionsH(G)to the weighted spaces of analytic functionsHVb(G,E)andHV0(G,E).

Aluthge transforms of weighted shifts on directed trees

We prove that the class 𝒞 ( H Λ α , H Λ β p ) = { ϕ : C ϕ ( H Λ α ) ⊂ H Λ β p } , where HΛα and H Λ β p are holomorphic Lipschitz spaces, has the f-property for p⩾1, 0<β<1: If ϕ I ∈ 𝒞 ( H Λ α , H Λ β p ) where I is an inner function, then ϕ ∈ 𝒞 ( H Λ α , H Λ β p ) Also we characterize inner functions that belong to 𝒞 ( H Λ α , H Λ β p ) for p>0, 0<α⩽1 and 0<β⩽1. The results are deduced from a new, derivative-free, characterization of 𝒞 ( H Λ α , H Λ β p ) that extends some results of Dyakonov [Acta Math. 178 (1997) 143–167.] and the author [Acta Math. 183 (1999) 141–143; J. Math. Anal. Appl., 326 (2007) 1–11].

The single equality [formula omitted] does not imply the quasinormality of weighted shifts on rootless directed trees

A commutative Banach algebra is called symmetric if, regarded as a function algebra on its maximal ideal space, it is closed under complex conjugation. Varopoulos, [5], proved the asymmetry of the tensor algebra , where T is the unit circle. It is the object of this paper to prove the asymmetry of the Schur multipliers of the space , where m is the Lebesgue measure. In the second part of the paper we study the Hankel multipliers of the space and give an application to it.1. The asymmetry of. Let C(T) denote the space of continuous functions on T and A(T) be the space of those functions in C(T) that have absolutely convergent Fourier series. Consider the mapping F: C(T) → C(T × T) defined by F(f)(x, y) = f (x + y).

A bounded linear operator C T {C_T} on L 2 ( X , Σ , m ) {L^2}(X,\Sigma ,m) is a composition operator if it is induced by a point mapping T : X → X T:X \to X via C T f = f ∘ T {C_T}f = f \circ T . Normal and quasinormal composition operators on a finite measure space are characterized: C T {C_T} is normal iff T is measure preserving and T − 1 ( Σ ) {T^{ - 1}}(\Sigma ) is (essentially) all of Σ ; C T \Sigma ;{C_T} is quasinormal iff T is measure preserving.