Unbounded Hyperexpansive Weighted Composition Operators on $$L^{2} (\varSigma )$$ L 2 ( Σ )

The notion of a completely hyperexpansive operator has been introduced in [2] by Athavale. In this paper bounded and unbounded hyperexpansive composition operators are investigated. A unique semispectral measure is associated with a bounded completely hyperexpansive composition operator. Examples of bounded and unbounded densely defined $2$-isometric (completely hyperexpansive, $2$-hyperexpansive) composition operators with invariant domains are given.

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.

We present a unified approach to the problem of characterizing the Fredholm multiplication and composition operators on the Hardy spaceH 2.

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 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.

We study the notion of recurrence and some of its variations for linear operators acting on Banach spaces. We characterize recurrence for several classes of linear operators such as weighted shifts, composition operators and multiplication operators on classical Banach spaces. We show that on separable complex Hilbert spaces the study of recurrent operators reduces, in many cases, to the study of unitary operators. Finally, we study the notion of product recurrence and state some relevant open questions.

The study of systems of singular integral equations of CAUCHYtype, of TOEPLITZand WIENER‐HOPFoperators leads to the question of existence and representation of generalized factorizations of matrix functions Φ in [LP(Γ, σ)]m. This yields a corresponding factorization of the basic multiplication or translation invariant operatorA=A_CA+, respectively, which can be seen as a splitting of a bounded into unbounded operators. The present paper is devoted to the study of the nature of the induced intermediate space Z = imA+=imA−1, in particular, for Γ = ℝ and Φ ε ζ[Cβ(ℝ)]m×mwhich is of special interest in certain applications. As we know, this implies detailed results about the structure and the explicit asymptotic behaviour of solutions of boundary and transmission problems near singular points with a relation also to eigenvalue problems which result from the classical series expansion approach or from the Mellin symbol calculus (see [13]).

Information processing in the Block Design Task

Power bounded weighted composition operators on function spaces defined by local properties

We provide several characterizations of the bounded and the compact weighted composition operators from the Bloch space and the analytic Besov spaces (with ) into the Zygmund space . As a special case, we show that the bounded (resp., compact) composition operators from , , and to coincide. In addition, the boundedness and the compactness of the composition operator can be characterized in terms of the boundedness (resp., convergence to 0, under the boundedness assumption of the operator) of the Zygmund norm of the powers of the symbol.

Suppose $\\varphi$ is an analytic self-map of open unit disk $\\mathbf{D}$ and $\\psi$ is an analytic function on $\\mathbf{D}$. Then a weighted composition operator induced by $\\varphi$ with weight $\\psi$ is given by $(W_{\\psi, \\varphi}f)(z)= \\psi(z)f(\\varphi(z))$, for $z \\in \\mathbf{D}$ and $f$ analytic on $\\mathbf{D}$. Necessary and sufficient conditions are given for the boundedness and compactness of the weighted composition operators $W_{\\psi, \\varphi}$. In terms of fixed points in the closed unit disk $\\overline{\\mathbf{D}}$, conditions under which $W_{\\psi, \\varphi}$ is compact are given. Necessary conditions for the compactness of $C_{\\varphi}$ are given in terms of the angular derivative $\\varphi^{\\prime}(\\zeta)$ where $\\zeta$ is on the boundary of the unit disk. Moreover, we present sufficient conditions for the membership of composition operators in the Schatten $p$-class $S_{p}(H^s(\\beta_1), H^q(\\beta_2))$, where the inducing map has supremum norm strictly smaller than~$1$.

We state several equivalent noncommutative versions of the Cauchy-Riemann equations and characterize the unbounded operators on L 2 ( R ) L^{2}(\mathbf {R}) which satisfy them. These operators arise from the creation operator via a functional calculus involving a class of entire functions, identified by Newman and Shapiro, which act as unbounded multiplication operators on Bargmann-Segal space.

The purpose of this note is to present certain aspects of the theory of spectral operators in Grothendieck spaces with the Dunford-Pettis property, briefly, GDP-spaces, thereby elaborating on the recent note [10].For example, the sum and product of commuting spectral operators in such spaces are again spectral operators (cf. Proposition 2.1) and a continuous linear operator is spectral if and only if it has finite spectrum (cf. Proposition 2.2). Accordingly, if a spectral operator is of finite type, then its spectrum consists entirely of eigenvalues. Furthermore, it turns out that there are no unbounded spectral operators in such spaces (cf. Proposition 2.4). As a simple application of these results we are able to determine which multiplication operators in certain function spaces are spectral operators.

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).