Fredholm multiplication and composition operators on the Hardy space

Complex symmetry of invertible composition operators

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 study 2-complex symmetric composition operators with the conjugation J, defined by Jf(z)=(f(z¯))¯, on the Hardy space H2. More precisely, we obtain the necessary and sufficient condition for the composition operator Cϕ to be 2-complex symmetric with J when ϕ is an automorphism of D. We also characterize 2-complex symmetric with J when ϕ is a linear fractional self-map of D.

Approximation numbers of weighted composition operators

Let S be the shift operator on the Hardy space H2 and let S* be its adjoint. A closed subspace \({\mathcal F}\) of H2 is said to be nearly S*-invariant if every element \({f\in\mathcal F}\) with f(0) = 0 satisfies \({S^*f\in\mathcal F}\). In particular, the kernels of Toeplitz operators are nearly S*-invariant subspaces. Hitt gave the description of these subspaces. They are of the form \({\mathcal F=g (H^2\ominus u H^2)}\) with \({g\in H^2}\) and u inner, u(0) = 0. A very particular fact is that the operator of multiplication by g acts as an isometry on \({H^2\ominus uH^2}\). Sarason obtained a characterization of the functions g which act isometrically on \({H^2\ominus uH^2}\). Hayashi obtained the link between the symbol \({\varphi}\) of a Toeplitz operator and the functions g and u to ensure that a given subspace \({\mathcal F=gK_u}\) is the kernel of \({T_\varphi}\). Chalendar, Chevrot and Partington studied the nearly S*-invariant subspaces for vector-valued functions. In this paper, we investigate the generalization of Sarason’s and Hayashi’s results in the vector-valued context.

For a compact subset K in the complex plane, let A(K) denote the algebra of all functions continuous on K and analytic on K° and let R(K) denote the uniform closure of the rational functions with poles off K. Let G is a bounded open subset whose complement in the plane has a finite number of components. Suppose that $$A(\overline{G} ) = R(\overline{G} )$$ and every function in H∞(G) is the pointwise limit of a bounded sequence of functions in $$R(\overline{G} )$$ . The purpose of this paper is to characterize all subnormal operators similar to M z , the operator of multiplication by the independent variable z on the Hardy space H2(G). We also characterize all bounded linear operators that are unitarily equivalent to M z in the case when each of the components of G is simply connected. In particular, our similarity result extends a well-known result of W. Clary on the unit disk to multiply connected domains.

We provide a precise description of the lattice of invariant subspaces of composition operators acting on the classical Hardy space, whose inducing symbol is a parabolic non-automorphism.This is achieved with an explicit isomorphism between the Hardy space and the Sobolev Banach algebra W 1,2 [0, ∞) that induces a bijection between the lattice of the composition operator and the closed ideals of W 1,2 [0, ∞).In particular, each invariant subspace of parabolic non-automorphism composition operator always consists of the closed span of a set of eigenfunctions.As a consequence, such composition operators have no non-trivial reducing subspaces.For the sake of completeness, we also include a characterization of the closed ideals of the Banach algebra W 1,2 [0, ∞).Although such a characterization is known, the proof we provide here is somehow different.

We prove that the norm of a weighted composition operator on the Hardy space H 2 of the disk is controlled by the norm of the weight function in the de Branges-Rovnyak space as- sociated to the symbol of the composition operator. As a corollary we obtain a new proof of the boundedness of composition operators on H 2 , and recover the standard upper bound for the norm. Sim- ilar arguments apply to weighted Bergman spaces. We also show that the positivity of a generalized de Branges-Rovnyak kernel is sufficient for the boundedness of a given composition operator on the standard functions spaces on the unit ball.

(Communicated by G. Sinnamon) Abstract. Suppose ϕ is an analytic self-map of open unit disk D and w is an analytic func- tion on D. Then a weighted composition operator induced by ϕ with weight w is given by (Ww,ϕ f)(z )= w(z)f(ϕ(z)) ,f orz ∈ D and f analytic on D.W efind a sufficient condition un- der which two composition operators lie in the same path component of C(H 2 ) ,a nd wefind as ufficient condition for the difference of such operators to be compact on H 2 (D) .T hen we provide another example that answers a question raised by Shapiro and Sundberg (18 )n ega- tively. Moreover, we characterize the Hilbert-Schmidt difference of two composition operators on H2(D).

In this paper, we investigate the conditions under which the Toeplitz composition operator on the Hardy space H2 becomes complex symmetric with respect to a certain conjugation. We also study various normality conditions for the Toeplitz composition operator on H2.

Toeplitz and asymptotic Toeplitz operators on [formula omitted

we revisit the Lorentz spaces <svg style="vertical-align:-2.29482pt;width:42.912498px;" id="M1" height="13.55" version="1.1" viewBox="0 0 42.912498 13.55" width="42.912498" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.55)"> <g transform="translate(72,-61.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝐿</tspan> <tspan style="font-size: 12.50px; " x="8.8521242" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="13.015623" y="0">𝑝</tspan> <tspan style="font-size: 12.50px; " x="19.129589" y="0">,</tspan> <tspan style="font-size: 12.50px; " x="24.330837" y="0">𝑞</tspan> <tspan style="font-size: 12.50px; " x="30.057213" y="0">)</tspan> </text> </g> </g> </svg> for <svg style="vertical-align:-2.29482pt;width:75.837502px;" id="M2" height="13.55" version="1.1" viewBox="0 0 75.837502 13.55" width="75.837502" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.55)"> <g transform="translate(72,-61.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝑝</tspan> <tspan style="font-size: 12.50px; " x="9.5772982" y="0">&gt;</tspan> <tspan style="font-size: 12.50px; " x="21.617687" y="0">1</tspan> <tspan style="font-size: 12.50px; " x="27.869186" y="0">,</tspan> <tspan style="font-size: 12.50px; " x="33.082939" y="0">𝑞</tspan> <tspan style="font-size: 12.50px; " x="42.272644" y="0">&gt;</tspan> <tspan style="font-size: 12.50px; " x="54.31303" y="0">0</tspan> </text> </g> </g> </svg> defined by G. G. Lorentz in the nineteen fifties and we show how the atomic decomposition of the spaces <svg style="vertical-align:-2.29482pt;width:43.5625px;" id="M3" height="13.55" version="1.1" viewBox="0 0 43.5625 13.55" width="43.5625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.55)"> <g transform="translate(72,-61.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝐿</tspan> <tspan style="font-size: 12.50px; " x="8.8521242" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="13.015623" y="0">𝑝</tspan> <tspan style="font-size: 12.50px; " x="19.129589" y="0">,</tspan> <tspan style="font-size: 12.50px; " x="24.330837" y="0">1</tspan> <tspan style="font-size: 12.50px; " x="30.582338" y="0">)</tspan> </text> </g> </g> </svg> obtained by De Souza in 2010 can be used to characterize the multiplication and composition operators on these spaces. These characterizations, though obtained from a completely different perspective, confirm the various results obtained by S. C. Arora, G. Datt and S. Verma in different variants of the Lorentz Spaces.

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).

The Invariant Subspace Problem (“ISP”) for Hilbert space operators is known to be equivalent to a question that, on its surface, seems surprisingly concrete: For composition operators induced on the Hardy space H 2 by hyperbolic automorphisms of the unit disc, is every nontrivial minimal invariant subspace one dimensional (i.e., spanned by an eigenvector)? In the hope of reviving interest in the contribution this remarkable result might offer to the studies of both composition operators and the ISP, I revisit some known results, weaken their hypotheses and simplify their proofs. Sample results: If φ is a hyperbolic disc automorphism with fixed points at a and β (both necessarily on the unit circle), and C φ the composition operator it induces on H 2, then for every \( f \in \sqrt {(z - \alpha )(z - \beta )} \) H 2, the doubly C φ-cyclic subspace generated by f contains many independent eigenvectors; more precisely, the point spectrum of C φ’s restriction to that subspace intersects the unit circle in a set of positive measure. Moreover, this restriction of C φ is hypercyclic (some forward orbit is dense). Under the stronger restriction \( f \in \sqrt {(z - \alpha )(z - \beta )} \) H p for some p > 2, the point spectrum of the restricted operator contains an open annulus centered at the origin.KeywordsComposition operatorhyperbolic automorphismInvariant Subspace ProblemMathematics Subject Classification (2000)Primary 47B33Secondary 47A15

Function spaces and its various applications are subjects of investigation in many research centers and we are glad that this special issue has received the remarkable attention by researchers in these areas. We have received papers from wide range of topics of function spaces such as interpolation methods for stochastic processes spaces, vector optimization problems in Banach spaces, Marcinkiewicz integral operator in Morrey spaces, Hardy’s operator in the variable exponent Lebesgue spaces, and differential equations of weakly parabolic type in Banach spaces. A number of papers are devoted to investigating different kinds of problems in Hardy spaces or Hardy subspaces, for example, Calderon-Zygumnd singular integral operators in Hardy spaces, weighted Hardy spaces related to the Schrodinger operator, and composition operator in hyperbolic Bloch-type spaces. Geometric problems such as modulus of convexity in some Banach sequence spaces, p-uniform convexity, and quniform smoothness in C spaces are also covered by this issue.